pseudo-differential operators and symmetries: background … · 2017-06-14 · pseudo-differential...

Pseudo-Differential OperatorsTheory and ApplicationsVol. 2

Managing Editor

M.W. Wong (York University, Canada)

Editorial Board

Luigi Rodino (Università di Torino, Italy)Bert-Wolfgang Schulze (Universität Potsdam, Germany)Johannes Sjöstrand (École Polytechnique, Palaiseau, France)Sundaram Thangavelu (Indian Institute of Science at Bangalore, India)Marciej Zworski (University of California at Berkeley, USA)

Pseudo-Differential Operators: Theory and Applications is a series ofmoderately priced graduate-level textbooks and monographs appea-ling to students and experts alike. Pseudo-differential operators are understood in a very broad sense and include such topics as harmonic analysis, PDE, geometry, mathematical physics, microlocal analysis, time-frequency analysis, imaging and computations. Modern trendsand novel applications in mathematics, natural sciences, medicine,scientific computing, and engineering are highlighted.

Michael Ruzhansky | Ville Turunen

Pseudo-Differential Operators and SymmetriesBackground Analysis and Advanced Topics

BirkhäuserBasel · Boston · Berlin

2000 Mathematics Subject Classification: 35Sxx, 58J40; 43A77, 43A80, 43A85

Library of Congress Control Number: 2009929498

Bibliographic information published by Die Deutsche BibliothekDie Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.

ISBN 978-3-7643-8513-2 Birkhäuser Verlag AG, Basel · Boston · Berlin

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Authors:

Michael RuzhanskyDepartment of MathematicsImperial College London180 Queen’s GateLondon SW7 2AZUnited Kingdome-mail: [email protected]

Ville TurunenInstitute of MathematicsHelsinki University of TechnologyP.O. Box 1100FI-02015 TKKFinlande-mail: [email protected]

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I Foundations of Analysis

A Sets, Topology and Metrics

A.1 Sets, collections, families . . . . . . . . . . . . . . . . . . . . . . . 9A.2 Relations, functions, equivalences and orders . . . . . . . . . . . . 12A.3 Dominoes tumbling and transfinite induction . . . . . . . . . . . 16A.4 Axiom of Choice: equivalent formulations . . . . . . . . . . . . . 17A.5 Well-Ordering Principle revisited . . . . . . . . . . . . . . . . . . 25A.6 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26A.7 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.8 Kuratowski’s closure . . . . . . . . . . . . . . . . . . . . . . . . . 35A.9 Complete metric spaces . . . . . . . . . . . . . . . . . . . . . . . 40A.10 Continuity and homeomorphisms . . . . . . . . . . . . . . . . . . 46A.11 Compact topological spaces . . . . . . . . . . . . . . . . . . . . . 49A.12 Compact Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . . 52A.13 Sequential compactness . . . . . . . . . . . . . . . . . . . . . . . 57A.14 Stone–Weierstrass theorem . . . . . . . . . . . . . . . . . . . . . . 62A.15 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.16 Connectedness and path-connectedness . . . . . . . . . . . . . . . 66A.17 Co-induction and quotient spaces . . . . . . . . . . . . . . . . . . 69A.18 Induction and product spaces . . . . . . . . . . . . . . . . . . . . 70A.19 Metrisable topologies . . . . . . . . . . . . . . . . . . . . . . . . . 74A.20 Topology via generalised sequences . . . . . . . . . . . . . . . . . 77

vi Contents

B Elementary Functional Analysis

B.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79B.1.1 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . 83

B.2 Topological vector spaces . . . . . . . . . . . . . . . . . . . . . . 85B.3 Locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . 87

B.3.1 Topological tensor products . . . . . . . . . . . . . . . . . 90B.4 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B.4.1 Banach space adjoint . . . . . . . . . . . . . . . . . . . . . 101B.5 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.5.1 Trace class, Hilbert–Schmidt, and Schatten classes . . . . 111

C Measure Theory and Integration

C.1 Measures and outer measures . . . . . . . . . . . . . . . . . . . . 116C.1.1 Measuring sets . . . . . . . . . . . . . . . . . . . . . . . . 116C.1.2 Borel regularity . . . . . . . . . . . . . . . . . . . . . . . . 124C.1.3 On Lebesgue measure . . . . . . . . . . . . . . . . . . . . 128C.1.4 Lebesgue non-measurable sets . . . . . . . . . . . . . . . . 133

C.2 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . 134C.2.1 Well-behaving functions . . . . . . . . . . . . . . . . . . . 134C.2.2 Sequences of measurable functions . . . . . . . . . . . . . 137C.2.3 Approximating measurable functions . . . . . . . . . . . . 141

C.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143C.3.1 Integrating simple non-negative functions . . . . . . . . . 144C.3.2 Integrating non-negative functions . . . . . . . . . . . . . 144C.3.3 Integration in general . . . . . . . . . . . . . . . . . . . . 147

C.4 Integral as a functional . . . . . . . . . . . . . . . . . . . . . . . . 152C.4.1 Lebesgue spaces Lp(μ) . . . . . . . . . . . . . . . . . . . . 152C.4.2 Signed measures . . . . . . . . . . . . . . . . . . . . . . . 158C.4.3 Derivatives of signed measures . . . . . . . . . . . . . . . 162C.4.4 Integration as functional on function spaces . . . . . . . . 169C.4.5 Integration as functional on Lp(μ) . . . . . . . . . . . . . 170C.4.6 Integration as functional on C(X) . . . . . . . . . . . . . 174

C.5 Product measure and integral . . . . . . . . . . . . . . . . . . . . 181

D Algebras

D.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191D.2 Topological algebras . . . . . . . . . . . . . . . . . . . . . . . . . 196D.3 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200D.4 Commutative Banach algebras . . . . . . . . . . . . . . . . . . . . 207D.5 C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213D.6 Appendix: Liouville’s Theorem . . . . . . . . . . . . . . . . . . . 217

Contents vii

Part II Commutative Symmetries

1 Fourier Analysis on Rn

1.1 Basic properties of the Fourier transform . . . . . . . . . . . . . . 2211.2 Useful inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 2291.3 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . 233

1.3.1 Fourier transform of tempered distributions . . . . . . . . 2331.3.2 Operations with distributions . . . . . . . . . . . . . . . . 2361.3.3 Approximating by smooth functions . . . . . . . . . . . . 239

1.4 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.4.1 Localisation of Lp-spaces and distributions . . . . . . . . . 2411.4.2 Convolution of distributions . . . . . . . . . . . . . . . . . 244

1.5 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2461.5.1 Weak derivatives and Sobolev spaces . . . . . . . . . . . . 2461.5.2 Some properties of Sobolev spaces . . . . . . . . . . . . . 2491.5.3 Mollifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . 2501.5.4 Approximation of Sobolev space functions . . . . . . . . . 253

1.6 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

2 Pseudo-differential Operators on Rn

2.1 Motivation and definition . . . . . . . . . . . . . . . . . . . . . . 2592.2 Amplitude representation of pseudo-differential operators . . . . 2632.3 Kernel representation of pseudo-differential operators . . . . . . . 2642.4 Boundedness on L2(Rn) . . . . . . . . . . . . . . . . . . . . . . . 2672.5 Calculus of pseudo-differential operators . . . . . . . . . . . . . . 271

2.5.1 Composition formulae . . . . . . . . . . . . . . . . . . . . 2712.5.2 Changes of variables . . . . . . . . . . . . . . . . . . . . . 2812.5.3 Principal symbol and classical symbols . . . . . . . . . . . 2822.5.4 Calculus proof of L2-boundedness . . . . . . . . . . . . . . 2842.5.5 Asymptotic sums . . . . . . . . . . . . . . . . . . . . . . . 285

2.6 Applications to partial differential equations . . . . . . . . . . . . 2872.6.1 Freezing principle for PDEs . . . . . . . . . . . . . . . . . 2882.6.2 Elliptic operators . . . . . . . . . . . . . . . . . . . . . . . 2892.6.3 Sobolev spaces revisited . . . . . . . . . . . . . . . . . . . 293

3 Periodic and Discrete Analysis

3.1 Distributions and Fourier transforms on Tn and Zn . . . . . . . . 2983.2 Sobolev spaces Hs(Tn) . . . . . . . . . . . . . . . . . . . . . . . . 3063.3 Discrete analysis toolkit . . . . . . . . . . . . . . . . . . . . . . . 309

3.3.1 Calculus of finite differences . . . . . . . . . . . . . . . . . 3103.3.2 Discrete Taylor expansion and polynomials on Zn . . . . . 3133.3.3 Several discrete inequalities . . . . . . . . . . . . . . . . . 319

viii Contents

3.3.4 Linking differences to derivatives . . . . . . . . . . . . . . 3213.4 Periodic Taylor expansion . . . . . . . . . . . . . . . . . . . . . . 3273.5 Appendix: on operators in Banach spaces . . . . . . . . . . . . . 329

4 Pseudo-differential Operators on Tn

4.1 Toroidal symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 3354.1.1 Quantization of operators on Tn . . . . . . . . . . . . . . 3354.1.2 Toroidal symbols . . . . . . . . . . . . . . . . . . . . . . . 3374.1.3 Toroidal amplitudes . . . . . . . . . . . . . . . . . . . . . 340

4.2 Pseudo-differential operators on Sobolev spaces . . . . . . . . . . 3424.3 Kernels of periodic pseudo-differential operators . . . . . . . . . . 3474.4 Asymptotic sums and amplitude operators . . . . . . . . . . . . . 3514.5 Extension of toroidal symbols . . . . . . . . . . . . . . . . . . . . 3564.6 Periodisation of pseudo-differential operators . . . . . . . . . . . 3604.7 Symbolic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 3674.8 Operators on L2(Tn) and Sobolev spaces . . . . . . . . . . . . . . 3744.9 Elliptic pseudo-differential operators on Tn . . . . . . . . . . . . 3764.10 Smoothness properties . . . . . . . . . . . . . . . . . . . . . . . . 3824.11 An application to periodic integral operators . . . . . . . . . . . . 3874.12 Toroidal wave front sets . . . . . . . . . . . . . . . . . . . . . . . 3894.13 Fourier series operators . . . . . . . . . . . . . . . . . . . . . . . . 3934.14 Boundedness of Fourier series operators on L2(Tn) . . . . . . . . 4054.15 An application to hyperbolic equations . . . . . . . . . . . . . . . 410

5 Commutator Characterisation of Pseudo-differential Operators

5.1 Euclidean commutator characterisation . . . . . . . . . . . . . . . 4135.2 Pseudo-differential operators on manifolds . . . . . . . . . . . . . 4165.3 Commutator characterisation on closed manifolds . . . . . . . . . 4215.4 Toroidal commutator characterisation . . . . . . . . . . . . . . . 423

Part III Representation Theory of Compact Groups

6 Groups

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4296.2 Groups without topology . . . . . . . . . . . . . . . . . . . . . . . 4306.3 Group actions and representations . . . . . . . . . . . . . . . . . 436

7 Topological Groups

7.1 Topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . 4457.2 Representations of topological groups . . . . . . . . . . . . . . . . 4497.3 Compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

Contents ix

7.4 Haar measure and integral . . . . . . . . . . . . . . . . . . . . . . 4537.4.1 Integration on quotient spaces . . . . . . . . . . . . . . . . 462

7.5 Peter–Weyl decomposition of representations . . . . . . . . . . . 4657.6 Fourier series and trigonometric polynomials . . . . . . . . . . . . 4747.7 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4787.8 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4797.9 Induced representations . . . . . . . . . . . . . . . . . . . . . . . 482

8 Linear Lie Groups

8.1 Exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . 4928.2 No small subgroups for Lie, please . . . . . . . . . . . . . . . . . 4968.3 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . 498

8.3.1 Universal enveloping algebra . . . . . . . . . . . . . . . . . 5068.3.2 Casimir element and Laplace operator . . . . . . . . . . . 510

9 Hopf Algebras

9.1 Commutative C∗-algebras . . . . . . . . . . . . . . . . . . . . . . 5159.2 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Part IV Non-commutative Symmetries

10 Pseudo-differential Operators on Compact Lie Groups

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52910.2 Fourier series on compact Lie groups . . . . . . . . . . . . . . . . 53010.3 Function spaces on the unitary dual . . . . . . . . . . . . . . . . 534

10.3.1 Spaces on the group G . . . . . . . . . . . . . . . . . . . . 53410.3.2 Spaces on the dual G . . . . . . . . . . . . . . . . . . . . . 53610.3.3 Spaces Lp(G) . . . . . . . . . . . . . . . . . . . . . . . . . 546

10.4 Symbols of operators . . . . . . . . . . . . . . . . . . . . . . . . . 55010.4.1 Full symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 55210.4.2 Conjugation properties of symbols . . . . . . . . . . . . . 556

10.5 Boundedness of operators on L2(G) . . . . . . . . . . . . . . . . . 55910.6 Taylor expansion on Lie groups . . . . . . . . . . . . . . . . . . . 56110.7 Symbolic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

10.7.1 Difference operators . . . . . . . . . . . . . . . . . . . . . 56310.7.2 Commutator characterisation . . . . . . . . . . . . . . . . 56610.7.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56710.7.4 Leibniz formula . . . . . . . . . . . . . . . . . . . . . . . . 570

10.8 Boundedness on Sobolev spaces Hs(G) . . . . . . . . . . . . . . . 57110.9 Symbol classes on compact Lie groups . . . . . . . . . . . . . . . 572

10.9.1 Some properties of symbols of Ψm(G) . . . . . . . . . . . 573

x Contents

10.9.2 Symbol classes Σm(G) . . . . . . . . . . . . . . . . . . . . 57510.10 Full symbols on compact manifolds . . . . . . . . . . . . . . . . . 57810.11 Operator-valued symbols . . . . . . . . . . . . . . . . . . . . . . . 579

10.11.1Example on the torus Tn . . . . . . . . . . . . . . . . . . 58910.12 Appendix: integral kernels . . . . . . . . . . . . . . . . . . . . . . 591

11 Fourier Analysis on SU(2)11.1 Preliminaries: groups U(1), SO(2), and SO(3) . . . . . . . . . . . 595

11.1.1 Euler angles on SO(3) . . . . . . . . . . . . . . . . . . . . 59711.1.2 Partial derivatives on SO(3) . . . . . . . . . . . . . . . . . 59811.1.3 Invariant integration on SO(3) . . . . . . . . . . . . . . . 598

11.2 General properties of SU(2) . . . . . . . . . . . . . . . . . . . . . 59911.3 Euler angle parametrisation of SU(2) . . . . . . . . . . . . . . . . 60011.4 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

11.4.1 Quaternions and SU(2) . . . . . . . . . . . . . . . . . . . 60311.4.2 Quaternions and SO(3) . . . . . . . . . . . . . . . . . . . 60411.4.3 Invariant integration on SU(2) . . . . . . . . . . . . . . . 60511.4.4 Symplectic groups . . . . . . . . . . . . . . . . . . . . . . 605

11.5 Lie algebra and differential operators on SU(2) . . . . . . . . . . 60711.6 Irreducible unitary representations of SU(2) . . . . . . . . . . . . 612

11.6.1 Representations of SO(3) . . . . . . . . . . . . . . . . . . 61511.7 Matrix elements of representations of SU(2) . . . . . . . . . . . . 61611.8 Multiplication formulae for representations of SU(2) . . . . . . . 62011.9 Laplacian and derivatives of representations on SU(2) . . . . . . 62411.10 Fourier series on SU(2) and on SO(3) . . . . . . . . . . . . . . . . 629

12 Pseudo-differential Operators on SU(2)12.1 Symbols of operators on SU(2) . . . . . . . . . . . . . . . . . . . 63112.2 Symbols of ∂+, ∂−, ∂0 and Laplacian L . . . . . . . . . . . . . . . 63412.3 Difference operators for symbols . . . . . . . . . . . . . . . . . . . 636

12.3.1 Difference operators on SU(2) . . . . . . . . . . . . . . . . 63612.3.2 Differences for symbols of ∂+, ∂−, ∂0 and Laplacian L . . . 64012.3.3 Differences for aσ∂0 . . . . . . . . . . . . . . . . . . . . . . 649

12.4 Symbol classes on SU(2) . . . . . . . . . . . . . . . . . . . . . . . 65612.5 Pseudo-differential operators on S3 . . . . . . . . . . . . . . . . . 66012.6 Appendix: infinite matrices . . . . . . . . . . . . . . . . . . . . . 662

Contents xi

13 Pseudo-differential Operators on Homogeneous Spaces

13.1 Analysis on closed manifolds . . . . . . . . . . . . . . . . . . . . . 66713.2 Analysis on compact homogeneous spaces . . . . . . . . . . . . . 66913.3 Analysis on K\G, K a torus . . . . . . . . . . . . . . . . . . . . . 67313.4 Lifting of operators . . . . . . . . . . . . . . . . . . . . . . . . . . 679

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

Preface

This monograph is devoted to the development of the theory of pseudo-differentialoperators on spaces with symmetries. Such spaces are the Euclidean space Rn, thetorus Tn, compact Lie groups and compact homogeneous spaces.

The book consists of several parts. One of our aims has been not only topresent new results on pseudo-differential operators but also to show parallelsbetween different approaches to pseudo-differential operators on different spaces.Moreover, we tried to present the material in a self-contained way to make itaccessible for readers approaching the material for the first time.

However, different spaces on which we develop the theory of pseudo-differen-tial operators require different backgrounds. Thus, while operators on the Eu-clidean space in Chapter 2 rely on the well-known Euclidean Fourier analysis,pseudo-differential operators on the torus and more general Lie groups in Chapters4 and 10 require certain backgrounds in discrete analysis and in the representationtheory of compact Lie groups, which we therefore present in Chapter 3 and in PartIII, respectively. Moreover, anyone who wishes to work with pseudo-differential op-erators on Lie groups will certainly benefit from a good grasp of certain aspects ofrepresentation theory. That is why we present the main elements of this theory inPart III, thus eliminating the necessity for the reader to consult other sources formost of the time. Similarly, the backgrounds for the theory of pseudo-differentialoperators on S3 and SU(2) developed in Chapter 12 can be found in Chapter 11presented in a self-contained way suitable for immediate use.

However, it was still not a simple matter to make a self-contained presentationof these theories without referring to basics of the more general analysis. Thus, inhoping that this monograph may serve as a guide to different aspects of pseudo-differential operators, we decided to include the basics of analysis that are certainlyuseful for anyone working with pseudo-differential operators.

Overall, we tried to supplement all the material with exercises for learningthe ideas and practicing the techniques. They range from elementary problems tomore challenging ones. In fact, on many occasions where other authors could say“it is easy to see” or “one can check”, we prefer to present it as an exercise. Atthe same time, more challenging exercises also serve as an excellent way to presentmore aspects of the discussed material.

xiv Preface

We would like to thank Professor G. Vainikko, who introduced V. Turunento pseudo-differential equations on circles [137], leading naturally to the non-commutative setting of the doctoral thesis. The thesis work was crucially in-fluenced by a visit to M.E. Taylor in spring 2000. We are grateful to ProfessorM.W. Wong for suggesting that we write this monograph, to our students for giv-ing us useful feedback on the background material of the book, and to Dr. J. Wirthfor reading the manuscript and for his useful feedback and numerous comments,which led to clarifications of the presentation, especially of the material from Sec-tion 10.3. Most of the work was carried out at the pleasant atmospheres providedby Helsinki University of Technology and Imperial College London. Moreover, overthe years, we have outlined substantial parts of the monograph elsewhere: partic-ularly, we appreciate the hospitality of University of North Carolina at ChapelHill, University of Torino and Osaka University. The work of M. Ruzhansky wassupported in part by EPSRC grants EP/E062873/01 and EP/G007233/1. Thetravels of V. Turunen were financed by the Magnus Ehrnrooth Foundation, by theVilho, Yrjo and Kalle Vaisala Foundation of the Finnish Academy of Science andLetters, and by the Finnish Cultural Foundation. Finally, our loving thanks go toour families for all the encouragement and understanding that we received whileworking on this monograph.

March 2009 Michael Ruzhansky, LondonVille Turunen, Helsinki

Introduction

Historical notes

Pseudo-differential operators (ΨDO) can be considered as natural extensions oflinear partial differential operators, with which they share many essential proper-ties. The study of pseudo-differential operators grew out of research in the 1960son singular integral operators; being a relatively young subject, the theory is onlynow reaching a stable form.

Pseudo-differential operators are generalisations of linear partial differentialoperators, with roots entwined deep down in solving differential equations.

Among the most influential predecessors of the theory of pseudo-differentialoperators one must mention the works of Solomon Grigorievich Mikhlin, AlbertoCalderon and Antoni Szczepan Zygmund. Around 1957, anticipating novel meth-ods, Alberto Calderon proved the local uniqueness theorem of the Cauchy problemof a partial differential equation. This proof involved the idea of studying the al-gebraic theory of characteristic polynomials of differential equations.

Another landmark was set in ca. 1963, when Michael Atiyah and IsadoreSinger presented their celebrated index theorem. Applying operators, which nowa-days are recognised as pseudo-differential operators, it was shown that the geo-metric and analytical indices of Erik Ivar Fredholm’s “Fredholm operator” ona compact manifold are equal. In particular, these successes by Calderon andAtiyah–Singer motivated developing a comprehensive theory for these newly foundtools. The Atiyah–Singer index theorem is also tied to the advent of K-theory, asignificant field of study in itself.

The evolution of the pseudo-differential theory was then rapid. In 1963, PeterLax proposed some singular integral representations using Jean Baptiste JosephFourier’s “Fourier series”. A little later, Joseph Kohn and Louis Nirenberg pre-sented a more useful approach with the aid of Fourier integral operators and namedtheir representations pseudo-differential operators. Showing that these operatorsform an algebra, they derived a broad theory, and their results were applied byPeter Lax and Kurt Otto Friedrichs in boundary problems of linear partial differ-ential equations. Other related studies were conducted by Agranovich, Bokobza,Kumano-go, Schwartz, Seeley, Unterberger, and foremost, by Lars Hormander,

2 Introduction

who coined the modern pseudo-differential theory in 1965, leading into a vast rangeof methods and results. The efforts of Kohn, Nirenberg, and Hormander gave birthto symbol analysis, which is the basis of the theory of pseudo-differential operators.

It is interesting how the ideas of symbol analysis have matured over about 200years. Already Joseph Lagrange and Augustin Cauchy studied the assignment of acharacteristic polynomial to the corresponding differential operator. In the 1880s,Oliver Heaviside developed an operational calculus for the solution of ordinarydifferential equations met in the theory of electrical circuits. A more sophisticatedproblem of this kind, related to quantum mechanics, was solved by Hermann Weylin 1927, and eventually the concept of the symbol of an operator was introducedby Solomon Grigorievich Mikhlin in 1936. After all, there is nothing new underthe sun.

Since the mid-1960s, pseudo-differential operators have been widely appliedin research on partial differential equations: along with new theorems, they haveprovided a better understanding of parts of classical analysis including, for in-stance, Sergei Lvovich Sobolev’s “Sobolev spaces”, potentials, George Green’s“Green functions”, fundamental solutions, and the index theory of elliptic oper-ators. Furthermore, they appear naturally when reducing elliptic boundary valueproblems to the boundary. Briefly, modern mathematical analysis has gained valu-able clarity with the unifying aid of pseudo-differential operators. Fourier integraloperators are more general than pseudo-differential operators, having the samestatus in the study of hyperbolic equations as pseudo-differential operators havewith respect to elliptic equations.

A natural approach to treat pseudo-differential operators on n-dimensionalC∞-manifolds is to use the theory of Rn locally: this can be done, since the classesof pseudo-differential operators are invariant under smooth changes of coordinates.However, on periodic spaces (tori) Tn, this could be a clumsy way of thinking, asthe local theory is plagued with rather technical convergence and local coordi-nate questions. The compact group structure of the torus is important from theharmonic analysis point of view.

In 1979 (and 1985) Mikhail Semenovich Agranovich (see [3]) presented anappealing formulation of pseudo-differential operators on the unit circle S1 us-ing Fourier series. Hence, the independent study of periodic pseudo-differentialoperators was initiated. The equivalence of local and global definitions of peri-odic pseudo-differential operators was completely proven by William McLean in1989. By then, the global definition was widely adopted and used by Agranovich,Amosov, D.N. Arnold, Elschner, McLean, Saranen, Schmidt, Sloan, and Wendlandamong others. Its effectiveness has been recognised particularly in the numericalanalysis of boundary integral equations.

The literature on pseudo-differential operators is extensive. At the time ofwriting of this paragraph (28 January 2009), a search on MathSciNet showed1107 entries with words “pseudodifferential operator” in the title (among which33 are books), 436 entries with words “pseudo-differential operator” in the title

Introduction 3

(among which 37 are books), 3971 entries with words “pseudodifferential opera-tor” anywhere (among which 417 are books), and 1509 entries with words “pseudo-differential operator” anywhere (among which 151 are books). Most of these worksare devoted to the analysis on Rn and thus we have no means to give a comprehen-sive overview there. Thus, the emphasis of this monograph is on pseudo-differentialoperators on the torus, on Lie groups, and on spaces with symmetries, in whichcases the literature is much more limited.

Periodic pseudo-differential operators

It turns out that the pseudo-differential and periodic pseudo-differential theoriesare analogous, the periodic case actually being more discernible.

Despite the intense research on periodic integral equations, the theory ofperiodic pseudo-differential operators has been difficult to find in the literature. Onthe other hand, the wealth of publications on general pseudo-differential operatorsis cumbersome for the periodic case, and it is too easy to get lost in the midst ofirrelevant technical details.

In the sequel the elementary properties of periodic pseudo-differential opera-tors are studied. The prerequisites for understanding the theory are more modestthan one might expect. Of course, a basic knowledge of functional analysis is nec-essary, but the simple central tools are Gottfried Wilhelm von Leibniz’ “Leibnizformula”, Brook Taylor’s “Taylor expansion”, and Jean Baptiste Joseph Fourier’s“Fourier transform”. In the periodic case, these familiar concepts of the classicalcalculus are to be expressed in discrete forms using differences and summationinstead of derivatives and integration.

Our working spaces will be the Sobolev spaces Hs(Tn) on the compact torusgroup Tn. These spaces ideally reflect smoothness properties, which are of fun-damental significance for pseudo-differential operators, as the traditional operatortheoretic methods fail to be satisfactory – pseudo-differential operators and peri-odic pseudo-differential operators do not form any reasonable normed algebra.

The structure of the treatment of periodic pseudo-differential operators is thefollowing: first, introduction of necessary functional analytic prerequisites, thendevelopment of useful tools for analysis of series and periodic functions, and afterthat the presentation of the theory of periodic pseudo-differential operators. Thefocus of the study is on symbolic analysis.

The techniques of the extension of symbols and the periodisation of operatorsallow one effectively to relate the Euclidean and the periodic theories, and to useone to derive results in the other. However, we tried to reduce a reliance on suchideas, keeping in mind the development of the subject on Lie groups where sucha relation is not readily available. From this point of view, analysis on the toruscan be viewed rather as a special case of analysis on a Lie group than the periodicEuclidean case.

4 Introduction

The main justification of this work on the torus, from the authors’ point ofview, is the unification and development of the global theory of periodic pseudo-differential operators. It becomes evident how elegant this theory is, especiallywhen compared to the theory on Rn; and as such, periodic pseudo-differentialoperators may actually serve as a nice first introduction to the general theory ofpseudo-differential operators. For those who have already acquainted themselveswith pseudo-differential operators this work may still offer another aspect of theanalysis. Thus, there is a hope that these tools will find various uses.

Although we decided not to discuss Fourier integral operators on Rn, wedevote some efforts to analysing operators that we call Fourier series operators.These are analogues of Fourier integral operators on the torus and we study them interms of toroidal quantization. The main new difficulty here is that while pseudo-differential operators do not move the wave front sets of distributions, this is nolonger the case for Fourier series operators. Thus, we are quickly forced to makeextensions of functions from an integer lattice to Euclidean space on the frequencyside. The analysis presented here shows certain limitation of the use of Fourierseries operators; however, we succeed in establishing elements of calculus for themand discuss an application to hyperbolic partial differential equations.

Pseudo-differential operators on Lie groups

Non-commutative Lie groups and homogeneous spaces play important roles indifferent areas of mathematics. Some fundamental examples include spheres Sn,which are homogeneous spaces under the action of the orthogonal groups. Theimportant special case is the three-dimensional sphere S3 which happens to bealso a group. However, while the general theory of pseudo-differential operatorsis available on such spaces, it presents certain limitations. First, working in localcoordinates often makes it very complicated to keep track of the global geometricfeatures. For example, a fundamental property that spheres are fixed by rotationsbecomes almost untraceable when looking at it in local coordinates. Another lim-itation is that while the local approach yields an invariant notion of the principalsymbol, the full symbol is not readily available. This presents profound compli-cations in applying the theory of pseudo-differential operators to problems onmanifolds that depend on knowledge of the full symbol of an operator.

In general, it is a natural idea to build pseudo-differential operators out ofsmooth families of convolution operators on Lie groups. There have been manyworks aiming at the understanding of pseudo-differential operators on Lie groupsfrom this point of view, e.g., the works on left-invariant operators [121, 78, 40],convolution calculus on nilpotent Lie groups [77], L2-boundedness of convolutionoperators related to Howe’s conjecture [57, 41], and many others.

However, in this work, we strive to develop the convolution approach into asymbolic quantization, which always provides a much more convenient frameworkfor the analysis of operators. For this, our analysis of operators and their symbols

Introduction 5

is based on the representation theory of Lie groups. This leads to a description ofthe full symbol of a pseudo-differential operator on a Lie group as a sequence ofmatrices of growing sizes equal to the dimensions of the corresponding representa-tions of the group. We also characterise, in terms of the introduced quantizations,standard Hormander classes Ψm on Lie groups. One of the advantages of the pre-sented approach is that we obtain a notion of full (global) symbols which matchesthe underlying Fourier analysis on the group in a perfect way. For a group G, sucha symbol can be interpreted as a mapping defined on the space G × G, where Gis the unitary dual of a compact Lie group G. In a nutshell, this analysis can beregarded as a non-commutative analogue of the Kohn–Nirenberg quantization ofpseudo-differential operators that was proposed by Joseph Kohn and Louis Niren-berg in [68] in the Euclidean setting. As such, the present research is perhaps mostclosely related to the work of Michael Taylor [128], who, however, in his analysisused an exponential mapping to rely on pseudo-differential operators on a Lie al-gebra which can be viewed as a Euclidean space with the corresponding standardtheory of pseudo-differential operators. However, the approach developed in thiswork is different from that of [128, 129] in the sense that we rely on the groupstructure directly and thus are not restricted to neighbourhoods of the neutralelement, thus being able to approach global symbol classes directly. Some aspectsof the analysis presented in this part appeared in [99].

As an important example, the approach developed here gives us quite de-tailed information on the global quantization of operators on the three-dimensionalsphere S3. More generally, we note that if we have a closed simply-connected three-dimensional manifold M , then by the recently resolved Poincare conjecture thereexists a global diffeomorphism M � S3 � SU(2) that turns M into a Lie groupwith a group structure induced by S3 (or by SU(2)). Thus, we can use the approachdeveloped for SU(2) to immediately obtain the corresponding global quantizationof operators on M with respect to this induced group product. In fact, all theformulae remain completely the same since the unitary dual of SU(2) (or S3 in thequaternionic R4) is mapped by this diffeomorphism as well. An interesting featureof the pseudo-differential operators from Hormander’s classes Ψm on these spacesis that they have matrix-valued full symbols with a remarkable rapid off-diagonaldecay property.

We also introduce a general machinery with which we obtain global quantiza-tion on homogeneous spaces using the one on the Lie group that acts on the space.Although we do not yet have general analogues of the diffeomorphic Poincare con-jecture in higher dimensions, this already covers cases when M is a convex surfaceor a surface with positive curvature tensor, as well as more general manifolds interms of their Pontryagin class, etc.

6 Introduction

Conventions

Each part or a chapter of the book is preceded by a short introduction explainingthe layout and conventions. However, let us mention now several conventions thathold throughout the book.

Constants will be usually denoted by C (sometimes with subscripts), andtheir values may differ on different occasions, even when appearing in subsequentestimates. Throughout the book, the notation for the Laplace operator is L inorder not to confuse it with difference operators which are denoted by �.

In Chapters 3 and 4 we encounter a notational difficulty that both frequen-cies and multi-indices are integers with different conventions for norms than arenormally used in the literature. To address this issue, there we let |α| = |α|�1 bethe �1-norm (of the multi-index α) and ‖ξ‖ = ‖ξ‖�2 be the Euclidean �2 norm (ofthe frequency ξ ∈ Zn). However, in other chapters we write a more traditional |ξ|for the length of the vector ξ in Rn, and reserve the notation ‖ · ‖X for a normin a normed space X. However, there should be no confusion with this notationsince we usually make it clear which norm we use. In Part IV, ξ = ξ(x) stands fora representation, so that we can still use the usual notation σ(x, ξ) for symbols.

Part I

Foundations of Analysis

Part I of the monograph contains preliminary material that could be useful foranyone working in the theory of pseudo-differential operators.

The material of the book is on the intersection of classical analysis withthe representation theory of Lie groups. Aiming at making the presentation self-sufficient we include preliminary material that may be used as a reference forconcepts developed later. In any case, the material presented in this part may beused either as a reference or as an independent textbook on the foundations ofanalysis.

Throughout the book, we assume that the reader has survived undergraduatecalculus courses, so that concepts like partial derivatives and the Riemann integralare familiar. Otherwise, the prerequisites for understanding the material in thisbook are quite modest. We shall start with a naive version of a set theory, metricspaces, topology, functional analysis, measure theory and integration in Lebesgue’ssense.

Chapter A

Sets, Topology and Metrics

First, we present the basic notations and properties of sets, used elsewhere in thebook. The set theory involved is “naive”, sufficient for our purposes; for a thoroughtreatment, see, e.g., [46]. The sets of integer, rational, real or complex numberswill be taken for granted, we shall not construct them.

Let us first list some abbreviations that we are going to use:

• “P and Q” means that both properties P and Q are true.• “P or Q” means that at least one of the properties P and Q is true.• “P ⇒ Q” reads “If P then Q”, meaning that “P is false or Q is true”.

Equivalently “Q⇐ P”, i.e., “Q only if P”.• “P ⇐⇒ Q” is “P ⇒ Q and P ⇐ Q”, reading “P if and only if Q”.• “∃x” reads “There exists x”.• “∃!x” reads “There exists a unique x”.• “∀x” reads “For every x”.• “P := Q” or “Q =: P” reads “P is defined to be Q”.

A.1 Sets, collections, families

Naively, a set (or a collection or a family) A consists of points (or elements ormembers) x.Example. Sets of points, like a collection of coins, a family of two parents and threechildren, a flock of sheep, a pack of wolves, or a crowd of protesters.Example. Points in a set, like the members of a parliament, the flowers in a bundle,or the stars in a constellation.

We denote x ∈ A if the element x belongs to the set A, and x ∈ A if x doesnot belong to A. A set A is a subset of a set B, denoted by A ⊂ B or B ⊃ A, if

∀x : x ∈ A⇒ x ∈ B.

10 Chapter A. Sets, Topology and Metrics

Sets A,B are equal, denoted by A = B, if A ⊂ B and B ⊂ A, i.e.,

∀x : x ∈ A ⇐⇒ x ∈ B.

If A ⊂ B and A = B then A is called a proper subset of B.Remark A.1.1 (Notation for numbers). The sets of integer, rational, real andcomplex numbers are respectively Z, Q, R and C; let N = Z+ and R+ stand forthe corresponding subsets of (strictly) positive numbers. Then

Z+ ⊂ Z ⊂ Q ⊂ R ⊂ C.

We also write N0 = N ∪ {0}.There are various ways for expressing sets. Sometimes all the elements can

be listed:

• The empty set ∅ = {} is the unique set without elements: ∀x : x ∈ ∅.• Set {x} consists of a single element x ∈ {x}.• Set {x, y} = {y, x} consists of elements x and y. And so on. Yet {x} ={x, x} = {x, x, x} etc.

A set consisting of those elements for which property P holds can be denoted by

{x : P (x)} = {x | P (x)} .

A set consisting of finitely many elements x1, . . . , xn could be denoted by

{x1, . . . , xn} = {xk : k ∈ {1, . . . , n}}=

{xk | k ∈ Z+ : k ≤ n

}= {xk}n

k=1 ,

and the infinite set of positive integers by

Z+ = {1, 2, 3, 4, 5, · · · } .

The power set P(X) consists of all the subsets of X,

P(X) = {A : A ⊂ X}

Example. For the set X = {1}, we have

P(X) = {∅, {1}} ,

P(P(X)) = {∅, {∅}, {1}, {∅, {1}}} ,

and we leave it as an exercise to find P(P(P(X))), which contains 24 = 16 elementsin this case.

A.1. Sets, collections, families 11

Example. Always at least ∅, X ∈ P(X). If x ∈ X, then {x} ∈ P(X) and {{x}} ∈P(P(X)),

x = {x} = {{x}} = · · · ,x ∈ {x} ∈ {{x}} ∈ · · · .

However, we shall allow neither x ∈ x nor x ∈ x; consider Russell’s paradox: givenx = {a : a ∈ a}, is x ∈ x?

For A,B ⊂ X, let us define the union A∪B, the intersection A∩B and thedifference A \B by

A ∪B := {x : x ∈ A or x ∈ B} ,

A ∩B := {x : x ∈ A and x ∈ B} ,

A \B := {x : x ∈ A and x ∈ B} .

The complement Ac of A in X is defined by Ac := X\A.

Example. If A = {1, 2} and B = {2, 3} then A ∪ B = {1, 2, 3}, A ∩ B = {2} andA \B = {1}.Example. R \Q is the set of irrational numbers.

Exercise A.1.2. Show that

(A ∪B) ∪ C = A ∪ (B ∪ C),(A ∩B) ∩ C = A ∩ (B ∩ C),(A ∪B) ∩ C = (A ∩ C) ∪ (B ∩ C),(A ∩B) ∪ C = (A ∪ C) ∩ (B ∪ C).

Notice that in the latter two cases above, the order of the parentheses is essen-tial. On the other hand, the associativity in the first two equalities allows us toabbreviate A ∪B ∪ C := (A ∪B) ∪ C and A ∩B ∩ C := (A ∩B) ∩ C and so on.

Definition A.1.3 (Index sets). Let I be any set and assume that for every i ∈ Iwe are given a set Ai. Then I is an index set for the collection of sets Ai.

Definition A.1.4 (Unions and intersections of families). For a family A ⊂ P(X),the union

⋃A and the intersection⋂A are defined by⋃

A =⋃

B∈AB := {x | ∃B ∈ A : x ∈ B} ,⋂

A =⋂

B∈AB := {x | ∀B ∈ A : x ∈ B} .

Example. If A = {B,C} then⋃A = B ∪ C and

⋂A = B ∩ C.


Notice that if A ⊂ B ⊂ P(X) then

∅ ⊂⋃A ⊂

⋃B ⊂ X and ∅ ⊂

⋂B ⊂

⋂A ⊂ X.

Especially, for ∅ ⊂ P(X) we have⋃∅ = ∅ and

⋂∅ = X. (A.1)

Notice that A ∪ B =⋃{A,B} and A ∩ B =

⋂{A,B}. For unions (and similarlyfor intersections), the following notations are also commonplace:⋃

j∈K

Aj :=⋃{Aj | j ∈ K},

n⋃k=1

Ak :=⋃{Ak | k ∈ Z+ : 1 ≤ k ≤ n},

∞⋃k=1

Ak :=⋃{Ak | k ∈ Z+}.

Example.3⋂

k=1

Ak = A1 ∩A2 ∩A3.

Exercise A.1.5 (de Morgan’s rules). Prove de Morgan’s rules:

X \⋃

j∈K

Aj =⋂

j∈K

(X \Aj),

X \⋂

j∈K

Aj =⋃

j∈K

(X \Aj).

A.2 Relations, functions, equivalences and orders

The Cartesian product of sets A and B is

A×B = {(x, y) : x ∈ A, y ∈ B} ,

where the elements (x, y) := {x, {x, y}} are ordered pairs: if x = y then (x, y) =(y, x), whereas {x, y} = {y, x}. A relation from A to B is a subset R ⊂ A × B.We write xRy if (x, y) ∈ R, saying “x is in relation R to y”; analogously, x R ymeans (x, y) ∈ R (“x is not in relation R to y”).

Functions. A relation f ⊂ X × Y is called a function (or a mapping) from X toY , denoted by

f : X → Y or Xf

→ Y,

A.2. Relations, functions, equivalences and orders 13

if for each x ∈ X there exists a unique y ∈ Y such that (x, y) ∈ f :

∀x ∈ X ∃!y ∈ Y : (x, y) ∈ f ;

in this case, we write

y := f(x) or x �→ f(x) = y.

Intuitively, a function f : X → Y is a rule taking x ∈ X to f(x) ∈ Y . Functions

Xf→ Y and Y

g→ Z yield a composition Xg◦f→ Z by g ◦ f(x) := g(f(x)). The

restriction of f : X → Y to A ⊂ X is f |A : A→ Y defined by f |A(x) := f(x).Example. The characteristic function of a set E ∈ P(X) is χE : X → R definedby

χE(x) :=

{1, if x ∈ E,

0, if x ∈ E.

Definition A.2.1 (Injections, surjections, bijections). A function f : X → Y is

• an injection if f(x1) = f(x2) implies x1 = x2,• a surjection if for every y ∈ Y there exists x ∈ X such that f(x) = y,• and a bijection if it is both injective and surjective, and in this case we may

define the inverse function f−1 : Y → X such that f(x) = y if and only ifx = f−1(y).

Definition A.2.2. (Image and preimage) A function f : X → Y begets functions

f+ : P(X)→ P(Y ), f+(A) = f(A) := {f(x) ∈ Y : x ∈ A} ,

f− : P(Y )→ P(X), f−(B) = f−1(B) := {x ∈ Y : f(x) ∈ B}.

Sets f(A) and f−1(B) are called the image of A ⊂ X and the preimage of B ⊂ Y ,respectively.

Exercise A.2.3. Let f : X → Y , A ⊂ X and B ⊂ Y . Show that

A ⊂ f−1(f(A)) and f(f−1(B)) ⊂ B.

Give examples showing that these subsets can be proper.

Exercise A.2.4. Let f : X → Y , A0 ⊂ X, B0 ⊂ Y , A ⊂ P(X) and B ⊂ P(Y ).Show that ⎧⎪⎨⎪⎩

f(⋃A) =

⋃A∈A f(A),

f(⋂A) ⊂ ⋂

A∈A f(A),f(X \A0) ⊃ Y \ f(A0),

where the subsets can be proper, while⎧⎪⎨⎪⎩f−1(

⋃B) =⋃

B∈B f−1(B),f−1(

⋂B) =⋂

B∈B f−1(B),f−1(Y \B0) = X \ f−1(B0).


These set-operation-friendly properties of f−1 : P(Y )→ P(X) will be encounteredlater in topology and measure theory.

Definition A.2.5 (Induced and co-induced families). Let f : X → Y , A ⊂ P(X)and B ⊂ P(Y ). Then f is said to induce the family f−1(B) ⊂ P(X) and toco-induce the family D ⊂ P(Y ), where

f−1(B) :={f−1(B) | B ∈ B

},

D :={B ⊂ Y | f−1(B) ∈ A

}.

Equivalences

Definition A.2.6 (Equivalence relation). A subset ∼ of X × X is an equivalencerelation on X if it is

1. reflexive: x ∼ x (for all x ∈ X);2. symmetric: if x ∼ y then y ∼ x (for all x, y ∈ X);3. transitive: if x ∼ y and y ∼ z then x ∼ z (for all x, y, z ∈ X).

The equivalence class of x ∈ X is

[x] := {y ∈ X | x ∼ y} ,

and the equivalence classes form the quotient space

X/ ∼ := {[x] | x ∈ X} .

Notice that x ∈ [x] ⊂ X, that [x]∩[y] = ∅ if [x] = [y], and that X =⋃

x∈X [x].Example. Clearly, the identity relation = is an equivalence relation on X, andf(x) := {x} defines a natural bijection f : X → X/ =.Example. Let X and Y denote the sets of all women and men, respectively. Forsimplicity, we may assume the disjointness X ∩Y = ∅. Let Isolde, Juliet ∈ X andRomeo, Tristan ∈ Y . For a, b ∈ X ∪ Y , let x ∼ y if and only if a and b are of thesame gender. Then

Y = [Tristan] = [Romeo] = [Juliet] = [Isolde] = X,

X ∪ Y = [Romeo] ∪ [Juliet],(X ∪ Y )/ ∼ = {[Romeo], [Juliet]} .

Exercise A.2.7. Let us define a relation ∼ in the Euclidean plane R2 by setting(x1, x2) ∼ (y1, y2) if and only if x1−y1, x2−y2 ∈ Z. Show that ∼ is an equivalencerelation. What is the equivalence class of the origin (0, 0) ∈ R2? What is commonbetween a doughnut and the quotient space here?

Exercise A.2.8. Let us define a relation ∼ in the punctured Euclidean spaceR3 \ {(0, 0, 0)} by setting (x1, x2, x3) ∼ (y1, y2, y3) if and only if (x1, x2, x3) =(ty1, ty2, ty3) for some t ∈ R+. Prove that ∼ is an equivalence relation. What iscommon between a sphere and the quotient space here?

A.2. Relations, functions, equivalences and orders 15

Orders

Definition A.2.9 (Partial order). A non-empty set X is partially ordered if thereis a partial order ≤ on X. That is, ≤ is a relation from X to X, such that it is

1. reflexive: x ≤ x (for all x ∈ X);2. anti-symmetric: if x ≤ y and y ≤ x then x = y (for all x, y ∈ X);3. transitive: if x ≤ y and y ≤ z then x ≤ z (for all x, y, z ∈ X).

We say that y is greater than x (or x is less than y), denoted by x < y, if x ≤ yand x = y.

Example. The set R of real numbers has the usual order ≤. Naturally, any ofits non-empty subsets, e.g., Z+ ⊂ R, inherits the order. The set [−∞,+∞] =R ∪ {−∞,+∞} has the order ≤ extended from R, with conventions −∞ ≤ x andx ≤ +∞ for every x ∈ [−∞,+∞].Example. Let us order X = P(S) by inclusion. That is, for A,B ⊂ S, let A ≤ Bif and only if A ⊂ B.Example. Let X, Y be sets, where Y has a partial order ≤. We may introduce anew partial order for all functions f, g : X → Y by setting

f ≤ gdefinition

⇐⇒ ∀x ∈ X : f(x) ≤ g(x).

This partial order is commonplace especially when Y = R or Y = [−∞,∞].

Definition A.2.10 (Chains and total order). A non-empty subset K ⊂ X is a chainif x ≤ y or y ≤ x for all x, y ∈ K. The partial order is total (or linear) if the wholeset X is a chain.

Example. [−∞,+∞] is a chain with the usual partial order. Thereby also itssubsets are chains, e.g., R and Z+. If {Aj : j ∈ J} ⊂ P(S) is a chain thenAj ⊂ Ak or Ak ⊂ Aj for each j, k ∈ J . Moreover, P(S) is not a chain if S hasmore than one element.

Definition A.2.11 (Bounds). Let ≤ be a partial order on X. The sets of upper andlower bounds of A ⊂ X are defined, respectively, by

↑ A := {x ∈ X | ∀a ∈ A : a ≤ x} ,

↓ A := {x ∈ X | ∀a ∈ A : x ≤ a} .

If x ∈ A∩ ↑ A then it is the maximum of A, denoted by x = max(A). If x ∈ A∩ ↓ Athen it is the minimum of A, denoted by x = min(A). If A∩ ↑ {z} = {z} thenthe element z ∈ A is called maximal in A. Similarly, if A∩ ↓ {z} = {z} then theelement z ∈ A is called minimal in A. If sup(A) := min(↑ A) ∈ X exists, it iscalled the supremum of A, and if inf(A) := max(↓ A) ∈ X exists, it is the infimumof A.


Remark A.2.12. Notations like

supk≥1

xk = supk∈Z+

xk = sup{xk : k ∈ Z+}

are quite common.Example. The minimum in Z+ is 1, but there is no maximal element. For eachA ⊂ [−∞,∞], the infimum and the supremum exist.Example. Let X = P(S). Then max(X) = S and min(X) = ∅. If A ⊂ X thensup(A) =

⋃A and inf(A) =⋂A. For each x ∈ S, element S \{x} ∈ X is maximal

in the subset X \ {S}.Definition A.2.13 (lim sup and lim inf). Let xk ∈ X for each k ∈ Z+. If thefollowing supremums and infimums exist, let

lim supk→∞

xk := inf{sup{xk : j ≤ k} | j ∈ Z+

},

lim infk→∞

xk := sup{inf{xk : j ≤ k} | j ∈ Z+

}.

Example. Let Ek ∈ P(X) for each k ∈ Z+. Then

lim supk→∞

Ek =∞⋂

j=1

∞⋃k=j

Ek,

lim infk→∞

Ek =∞⋃

j=1

∞⋂k=j

Ek.

Exercise A.2.14. Let A = lim supk→∞

Ek and B = lim infk→∞

Ek as in the example above.

Show thatχA = lim sup

k→∞χEk

and χB = lim infk→∞

χEk,

where χE : X → R is the characteristic function of E ⊂ X.

A.3 Dominoes tumbling and transfinite induction

The principle of mathematical induction can be compared to a sequence of domi-noes, falling over one after another when the first tumbles down. More precisely,

if 1 ∈ S ⊂ Z+ and n ∈ S ⇒ n + 1 ∈ S for every n ∈ Z+, then S = Z+.

The Transfinite Induction Principle generalises this, working on any well-orderedset.

Definition A.3.1 (Well-ordered sets). A partially ordered set X is said to be wellordered, if min(A) exists whenever ∅ = A ⊂ X.

A.4. Axiom of Choice: equivalent formulations 17

Example. With its usual order, Z+ is well ordered. With their usual orders, Z, Rand [−∞,+∞] are not well ordered. With the inclusion order, P(S) is not wellordered, if there is more than one element in S.

Theorem A.3.2 (Transfinite Induction Principle). Let X be well ordered and S ⊂X. Assume that for each x ∈ X it holds that x ∈ S if

{y ∈ X : y < x} ⊂ S.

Then S = X.

Exercise A.3.3. Prove the Transfinite Induction Principle.

Exercise A.3.4 (Transfinite =⇒ mathematical induction). Check that in the caseX = Z+, the Transfinite Induction Principle is the usual mathematical induction.

The value of the Transfinite Induction Principle might be limited, as we haveto assume the well-ordering of the underlying set. Actually, many (but not all)working mathematicians assume that

every non-empty set can be well ordered,

which is the so-called Well-Ordering Principle. Is such a principle likely to be true?After all, for example on sets R or P(Z+), can we imagine what well-orderingsmight look like? All the elementary tools which we use in our mathematical rea-soning should be at least believable, so maybe the Well-Ordering Principle doesnot appear as a satisfying set theoretic axiom. Could we perhaps prove or disproveit from other, intuitively more reliable principles? We shall return to this questionlater.

A.4 Axiom of Choice: equivalent formulations

In this section we shall consider how to calculate the number of points in a set,and what infinity might mean in general.

Choosing. We may always choose one point out of a non-empty set, no matterhow many points there are around. But sometimes we need infinitely many tasksdone at once. For instance, we might want to choose a point from each of the non-empty subsets A ⊂ X in no time at all: as a tool, we need the Axiom of Choicefor X.

Definition A.4.1 (Choice function). Let X = ∅. A mapping f : P(X) → X iscalled a choice function on X if f(A) ∈ A whenever ∅ = A ⊂ X.

Example. Let X = {p, q} where p = q. Let f : P(X) → X such that f(X) = p =f({p}) and f({q}) = q. Then f is a choice function on X.

The following Axiom of Choice should be considered as an axiom or a fun-damental principle. In this section we discuss its implications.


Axiom A.4.2 (Axiom of Choice). For every non-empty set there exists a choicefunction.

Exercise A.4.3. Prove that a choice function exists on a well-ordered set. Thus theWell-Ordering Principle implies the Axiom of Choice.

The Axiom of Choice might look more convincing than the Well-OrderingPrinciple. Yet, we should be careful, as we are dealing with all kinds of sets, aboutwhich our intuition might be deficient. The Axiom of Choice might be plausiblefor X = Z+, or maybe even for X = R, but can we be sure whether it is true ingeneral? Nevertheless, let us add Axiom A.4.2 to our set-theoretic tool box.

There are plenty of equivalent formulations for the Axiom of Choice. In thesequel, we present some variants, starting with the “Axiom of Choice for CartesianProducts”, to be presented soon.

Definition A.4.4 (Cartesian product). Let Xj be a set for each j ∈ J . The Carte-sian product is defined to be∏

j∈J

Xj :={

f | f : J →⋃j∈J

Xj and ∀j ∈ J : f(j) ∈ Xj

}.

If Xj = X for each j ∈ J , we write XJ :=∏

j∈J Xj . The elements f ∈ XJ arethen functions f : J → X. Moreover, let Xn := XZn , where

Zn := {k ∈ Z+ | k ≤ n}.Exercise A.4.5. Give an example of a bijection

g : X1 ×X2 →∏

j∈{1,2}Xj ,

especially in the case X ×X → X2. Thereby X1 ×X2 can be identified with theCartesian product

∏j∈{1,2}Xj .

Exercise A.4.6. Give a bijection g : P(X)→ {0, 1}X .

The following Theorem A.4.7 is a consequence of the Axiom of Choice A.4.2.However, by Exercise A.4.8, Theorem A.4.7 also implies the Axiom of Choice, andthus it could have been taken as an axiom itself.

Theorem A.4.7 (Axiom of Choice for Cartesian Products). The Cartesian productof non-empty sets is non-empty.

Exercise A.4.8. Show that the Axiom of Choice is equivalent to the Axiom ofChoice for Cartesian Products.

Theorem A.4.9 (Hausdorff Maximal Principle). Any chain is contained in a max-imal chain.

Proof. Let (X,≤) be a partially ordered set with a chain C0 ⊂ X. Let

T := {C | C ⊂ X is a chain such that C0 ⊂ C} .


Now C0 ∈ T , so T = ∅. Let f : P(X) → X be a choice function for X. Let usdefine s : T → T such that s(C) = C if C ∈ T is maximal, and if C ∈ T is notmaximal then

s(C) := C ∪ {f({x ∈ X \ C : C ∪ {x} ∈ T })} ;

in this latter case, the chain s(C) is obtained by adding one element to the chainC. The claim follows if we can show that C = s(C) for some C ∈ T . Let U ⊂ Tbe a tower if

• C0 ∈ U ,• ⋃K ∈ U for any chain K ⊂ U ,• s(U) ⊂ U . In other words: if A ∈ U then s(A) ∈ U .

For instance, T is a tower. Let V be the intersection of all towers. Clearly, V isa tower, in fact the minimal tower. It will turn out that

⋃V ∈ T is a maximalchain. This follows if we can show that V ′ ⊂ V is a tower, where

V ′ := {C ∈ V | ∀B ∈ V : B ⊂ C or C ⊂ B} ,

since the minimality would imply V = V ′. Clearly, C0 ∈ V ′, and if K ⊂ V ′ is achain then

⋃K ∈ V ′. Let C ∈ V ′; we have to show that s(C) ∈ V ′. This follows,if we can show that 〈C〉 ⊂ V is a tower, where

〈C〉 := {A ∈ V | A ⊂ C or s(C) ⊂ A} .

Clearly, C0 ∈ 〈C〉, and if K ⊂ 〈C〉 is a chain then⋃K ∈ 〈C〉. Let A ∈ 〈C〉;

we have to show that s(A) ∈ 〈C〉, i.e., show that s(A) ⊂ C or s(C) ⊂ s(A).Since C ∈ V ′, we have s(A) ⊂ C or C ⊂ s(A). Suppose the non-trivial case“C ⊂ s(A) and A ⊂ C”. Since s(A) = A ∪ {x} for some x ∈ X, we must haves(A) = C or C = A. The proof is complete. �Theorem A.4.10 (Zorn’s Lemma). A partially ordered set where every chain hasan upper bound has a maximal element.

Exercise A.4.11 (Hausdorff Maximal Principle ⇐⇒ Zorn’s Lemma). Show thatthe Hausdorff Maximal Principle is equivalent to Zorn’s Lemma.

Theorem A.4.12 (Zorn’s Lemma =⇒ Axiom of Choice). Zorn’s Lemma impliesthe Axiom of Choice.

Proof. Let X be a non-empty set. Let

P := {f | f : P(A)→ A is a choice function for some A ⊂ X} .

Now P = ∅, because ({x} �→ x) : P({x}) → {x} belongs to P for any x ∈ X. Letus endow P with the partial order ≤ by inclusion:

f ≤ gdefinition

⇐⇒ f ⊂ g


(here recall that f ∈ P is a subset f ⊂ P(A) × A for some A ⊂ X). SupposeC = {fj : j ∈ J} ⊂ P is a chain. Then it is easy to verify that⋃

C =⋃j∈J

fj ∈ P

is an upper bound for C, so according to Zorn’s Lemma there exists a maximalelement f ∈ P , which is a choice function for some A ⊂ X. We have to show thatA = X. On the contrary, suppose B ⊂ X such that B ∈ P(A). Take x ∈ B. Thenf ⊂ f ∪ {(B, x)} ∈ P , which would contradict the maximality of f . Hence f mustbe a choice function for A = X. �

How many points? Intuitively, cardinality measures the number of the elementsin a set. Cardinality is a relative concept: sets A,B are compared by whetherthere is an injection, a surjection or a bijection from one to another. The mostinteresting results concern infinite sets.

Definition A.4.13 (Cardinality). Sets A,B have the same cardinality, denoted by

|A| = |B| (or A ∼ B),

if there exists a bijection f : A → B. If there exists C ⊂ B such that |A| = |C|,we write

|A| ≤ |B|.Moreover, |A| ≤ |B| = |A| is abbreviated by

|A| < |B|.

The cardinality of a set A is often also denoted by card(A).

Exercise A.4.14. Let |A| = |B|. Show that |P(A)| = |P(B)|.Exercise A.4.15. Show that |Z+| = |Z|.Remark A.4.16. Clearly for every set A,B,C we have

A ∼ A,

A ∼ B ⇐⇒ B ∼ A,

A ∼ B and B ∼ C =⇒ A ∼ C;

formally this is an equivalence relation, though we may have difficulties whendiscussing the “set of all sets”. Notice that |A| ≤ |B| means that there is aninjection f : A → B, and in this case we may identify set A with f(A) ⊂ B.Obviously,

|A| ≤ |B| ≤ |C| =⇒ |A| ≤ |C|.It is less obvious whether |A| = |B| when |A| ≤ |B| ≤ |A|:


Theorem A.4.17 (Schroder–Bernstein). Let |X| ≤ |Y | and |Y | ≤ |X|. Then|X| = |Y |.

Proof. Let f : X → Y and g : Y → X be injections. Let X0 := X and X1 := g(Y ).Define inductively {Xk : k ∈ Z+} ⊂ P(X) by

Xk+2 := g(f(Xk)).

Let X∞ :=⋂∞

k=0 Xk. Now X∞ ⊂ Xk+1 ⊂ Xk for each k ≥ 0. Moreover,

Xk \Xk+1 ∼{

X0 \X1, if k is odd,

X1 \X2, if k is even,

so that

X = X∞ ∪∞⋃

k=0

(Xk \Xk+1)

∼ X∞ ∪∞⋃

k=0

(Xk+1 \Xk+2)

= X1

∼ Y.

Thus X ∼ Y . �

The following Law of Trichotomy is equivalent to the Axiom of Choice,though we derive it only as a corollary to Zorn’s Lemma:

Theorem A.4.18 (The Law of Trichotomy). Let X, Y be sets. Then exactly one ofthe following holds:

|X| < |Y |, |X| = |Y |, |Y | < |X|.

Proof. Assume the non-trivial case X, Y = ∅. Let us define

J := {f | A ⊂ X, f : A→ Y injective} .

Clearly, J = ∅. Thus we may define a partial order ≤ on J by

g ≤ h ⇐⇒ g ⊂ h;

notice here that g, h ⊂ X×Y . Let K ⊂ J be a chain. Then it has an upper bound⋃K ∈ J . Hence by Zorn’s Lemma, there exists a maximal element f ∈ J . Nowf : A→ Y is injective, where A ⊂ X. If A = X then

|X| ≤ |Y |.


If f(A) = Y then|Y | = |A| ≤ |X|.

So let us suppose that A = X and f(A) = Y . Then take x0 ∈ X \ A and y0 ∈Y \ f(A). Define

g : A ∪ {x0} → Y, g(x) :=

{f(x), if x ∈ A,

y0, if x = x0.

Then g ∈ J and f ≤ g = f , which contradicts the maximality of f . TherebyA = X or f(A) = Y , meaning

|X| ≤ |Y | or |Y | ≤ |X|.

Finally, if |X| ≤ |Y | and |Y | ≤ |X| then |X| = |Y | by Theorem A.4.17. �

There is no greatest cardinality:

Theorem A.4.19 (No greatest cardinality). Let X be a set. Then |X| < |P(X)|.

Proof. If X = ∅ then P(X) = {∅}, and the only injection from X to P(X) is thenthe empty relation, which is not a bijection. Assume that X = ∅. Then function

f : X → P(X), f(x) := {x}

is an injection, establishing |X| ≤ |P(X)|. To get a contradiction, assume thatX ∼ P(X), so that there exists a bijection g : X → P(X). Let

A := {x ∈ X : x ∈ g(x)} .

Let x0 := g−1(A). Now x0 ∈ A if and only if

x0 ∈ g(x0) = A,

which is a contradiction. �

Definition A.4.20 (Counting). Let A,B,C, D be sets. For n ∈ Z+, let

Zn := {k ∈ Z+ | k ≤ n} = {1, . . . , n}.

We say that |∅| = 0, |Zn| = n,A is finite if |A| = n for some n ∈ Z+ ∪ {0}.B is infinite if it is not finite.C is countable if |C| ≤ |Z+|.D is uncountable if it is not countable.


Remark A.4.21. To strive for transparency in the proofs in this section, let usforget the Law of Trichotomy, which would provide short-cuts like

|X| < |Y | ⇐⇒ |Y | ≤ |X|.

The reader may easily simplify parts of the reasoning using this. The reader is alsoencouraged to find out where we use the Axiom of Choice or some other nontrivialtools.

Proposition A.4.22. Let A,B be sets. Then |A| < |Z+| ≤ |B| if and only if A isfinite and B is infinite.

Proof. Let A = ∅ be finite, so A ∼ Zn ⊂ Z+ for some n ∈ Z+. Hence |A| ≤ |Z+|.If f : Z+ → A then f(n+1) ∈ f(Zn), so f is not injective, especially not bijective.Thus |Z+| ≤ |A| and |A| < |Z+|. Consequently, if |Z+| ≤ |B| then B is infinite.

Let B be infinite. Take x1 ∈ B = ∅. Let An = {x1, . . . , xn} ⊂ B be a finiteset. Inductively, take xn+1 ∈ B \An = ∅. Define{

g : Z+ → B,

g(n) := xn.

Now g is injective. Hence |Z+| ≤ |B|.Let B ⊂ Z+ be infinite. Define h : Z+ → B inductively by{

h(1) := min(B),h(n + 1) := min (B \ {h(1), . . . , h(n)}) .

Now h is a bijection: |B| = |Z+|. So if |A| < |Z+| then A is finite. �Proposition A.4.23. Let C,D be sets. Then |C| ≤ |Z+| < |D| if and only if C iscountable and D is uncountable.

Proof. Property |C| ≤ |Z+| is just the definition of countability. Let D be uncount-able, i.e., |D| ≤ |Z+|. By Proposition A.4.22, D is not finite, i.e., it is infinite, i.e.,|Z+| ≤ |D|. Because of |Z+| = |D|, we have |Z+| < |D|.

Let |Z+| < |D|. By Proposition A.4.22, D is infinite, i.e., |D| < |Z+|. Becauseof |Z+| = |D|, we have even |D| ≤ |Z+|, i.e., D is uncountable. �Remark A.4.24. Let us collect the results from Propositions A.4.22 and A.4.23:For sets A,B,C, D, {

|A| < |Z+| ≤ |B|,|C| ≤ |Z+| < |D|

if and only if A is finite, B is infinite, C is countable, and D is uncountable. Inthe proofs, we used induction, i.e., well-ordering for Z+.

Proposition A.4.25 (Cantor). Let Ak ⊂ X be a countable subset for each k ∈ Z+.Then

⋃∞k=1 Ak is countable.


Proof. We may enumerate the elements of each countable Ak:

Ak :={akj : j ∈ Z+

},

A1 = {a11, a12, a13, a14, · · · } ,

A2 = {a21, a22, a23, a24, · · · } ,

A3 = {a31, a32, a33, a34, · · · } ,

A4 = {a41, a42, a43, a44, · · · } ,

...

Their union is enumerated by

∪∞k=1Ak = {a11,

a21, a12,

a31, a22, a13,

a41, a32, a23, a14, · · · }=

{ak−j+1,j : 1 ≤ j ≤ k, k ∈ Z+

}. �

Exercise A.4.26. Show that the set Q of rational numbers is countably infinite.

Exercise A.4.27 (Algebraic numbers). A number λ ∈ C is called algebraic if p(λ) =0 for some non-zero polynomial p with integer coefficients, i.e., if some polynomial

p(z) =n∑

k=0

akzk,

where n ∈ Z+, {ak}nk=0 ⊂ Z and an = 0. Let A ⊂ C be the set of algebraic

numbers. Show that Q ⊂ A, that A is countable, and give an example of a numberλ ∈ (R ∩ A) \Q.

Proposition A.4.28. |R| = |P(Z+)|.

Proof. Let us define

f : R→ P(Q), f(x) := {r ∈ Q : r < x} .

Obviously f is injective, hence |R| ≤ |P(Q)|. By Exercise A.4.26, |Q| = |Z+|,implying |P(Q)| = |P(Z+)|. On the other hand, let us define

g : P(Z+)→ R, g(A) :=∑k∈A

10−k.

For instance, 0 = g(∅) ≤ g(A) ≤ g(Z+) = 1/9. Nevertheless, g is injective, imply-ing |P(Z+)| ≤ |R|. This completes the proof. �Exercise A.4.29. Let X be an uncountable set. Show that there exists an uncount-able subset S ⊂ X such that X \ S is also uncountable.

A.5. Well-Ordering Principle revisited 25

A.5 Well-Ordering Principle revisited

Trivially, the Well-Ordering Principle implies the Axiom of Choice. Actually, thereis the reverse implication, too:

Theorem A.5.1 (Well-Ordering Principle). Every non-empty set can be wellordered.

Proof. Let X = ∅. Let

P := {(Aj ,≤j) | j ∈ J, Aj ⊂ X, (Aj ,≤j) well-ordered} .

Clearly, P = ∅. Define a partial order ≤ on P by inclusion:

(Aj ,≤j) ≤ (Ak,≤k) definition⇐⇒ ≤j⊂≤k .

Take a chain C ⊂ P . Let

B :=⋃

(Aj ,≤j)∈C

Aj , ≤B :=⋃

(Aj ,≤j)∈C

≤j .

Then (B,≤B) ∈ P is an upper bound for the chain C ⊂ P , so there existsa maximal element (A,≤A) ∈ P by Zorn’s Lemma A.4.10. Now, if there wasx ∈ X \A, then we easily see that A∪ {x} could be well ordered by ≤x for which≤A⊂≤x, which would contradict the maximality of (A,≤A). Therefore A = X hasbeen well ordered. �

Although we already know that the Well-Ordering Principle and the Haus-dorff Maximal Principle are equivalent, let us demonstrate how to use transfiniteinduction in a related proof:

Proposition A.5.2 (Well-Ordering Principle =⇒ Hausdorff Maximal Principle).The Well-Ordering Principle implies the Hausdorff Maximal Principle.

Proof. Let (X,≤) be well ordered, i.e., there exists min(A) ∈ A whenever ∅ = A ⊂X. Let ≤0 be a partial order on X. Let us define f : X → P(X) by transfiniteinduction in the following way:

f(x) :=

{{x}, if {x} ∪ f({y : y < x}) is a chain with respect to ≤0,

∅ otherwise.

Then f(X) ⊂ P(X) is a maximal chain. �Exercise A.5.3. Fill in the details in the proof of Proposition A.5.2.

Remark A.5.4 (Formulations of the Axiom of Choice). Collecting earlier resultsand exercises, we see that the following claims are equivalent: the Axiom of Choice,the Axiom of Choice for Cartesian Products, the Hausdorff Maximal Principle,Zorn’s Lemma, and the Well-Ordering Principle. The Law of Trichotomy wasderived as a corollary to these, but it is actually another equivalent formulationfor the Axiom of Choice (see, e.g., [124]).


Remark A.5.5 (Continuum Hypothesis). When working in analysis, one does notoften pay much attention to the underlying set theoretic foundations. Yet, thereare many deep problems involved. For instance, it can be shown that there is thesmallest uncountable cardinality |Ω|, i.e., whenever S is uncountable then

|Z+| < |Ω| ≤ |S|.So |Ω| ≤ |R|. A natural question is whether |Ω| = |R|? Actually, in year 1900,David Hilbert proposed the so-called Continuum Hypothesis

|Ω| = |R|.The Generalised Continuum Hypothesis is that if X, Y are infinite sets and |X| ≤|Y | ≤ |P(X)| then |X| = |Y | or |Y | = |P(X)|. Without going into details, let (ZF)denote the Zermelo–Fraenkel axioms for set theory, (AC) the Axiom of Choice, and(CH) the Generalised Continuum Hypothesis. From 1930s to 1960s, Kurt Godeland Paul Cohen discovered that:

1. Within (ZF) one cannot prove whether (ZF) is consistent.2. (ZF+AC+CH) is consistent if (ZF) is consistent.3. (AC) is independent of (ZF).4. (CH) is independent of (ZF+AC).

The reader will be notified, whenever we apply (AC) or its equivalents (which isnot that often); in this book, we shall not need (CH) at all.

A.6 Metric spaces

Definition A.6.1 (Metric space). A function d : X ×X → [0,∞) is called a metricon the set X if for every x, y, z ∈ X we have

d(x, y) = 0 ⇐⇒ x = y (non-degeneracy);d(x, y) = d(y, x) (symmetry);d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).

Then (X, d) (or simply X when d is evident) is called a metric space. Sometimesa metric is called a distance function. When x ∈ X and r > 0,

Br(x) := {y ∈ X | d(x, y) < r}is called the x-centered open ball of radius r. If we want to emphasise that the ballis taken with respect to metric d, we will write Bd(x, r).

Remark A.6.2. In a metric space (X, d),∞⋃

k=1

Bk(x) = X and∞⋂

k=1

B1/k(x) = {x}.

A.6. Metric spaces 27

Example (Discrete metric). The mapping d : X ×X → [0,∞) defined by

d(x, y) :=

{1, if x = y,

0, if x = y

is called the discrete metric on X. Here

Br(x) =

{X, if 1 < r,

{x}, if 0 < r ≤ 1.

Example. Normed vector spaces form a very important class of metric spaces, seeDefinition B.4.1.

Exercise A.6.3. For 1 ≤ p <∞,

dp(x, y) = ‖x− y‖p :=

⎛⎝ n∑j=1

|xj − yj |p⎞⎠1/p

defines a metric dp : Rn × Rn → [0,∞). Function

d∞(x, y) = max1≤j≤n

|xj − yj |

also turns Rn into a metric space. Unless otherwise mentioned, the space Rn isendowed with the Euclidean metric d2 (distance “as the crow flies”).

Exercise A.6.4 (Sup-metric). Let a < b and let B([a, b]) be the space of all boundedfunctions f : [a, b]→ R. Show that function

d∞(f, g) = supy∈[a,b]

|f(y)− g(y)|

turns B([a, b]) into a metric space. It is called the sup-metric.

Remark A.6.5 (Metric subspaces). If A ⊂ X and d : X ×X → [0,∞) is a metricthen the restriction

d|A×A : A×A→ [0,∞)

is a metric on A, with Bd|A×A(x, r) = A ∩Bd(x, r).

Exercise A.6.6. Let a < b and let C([a, b]) be the space of all continuous func-tions f : [a, b] → R. Show the following statements: The function d∞(f, g) =supy∈[a,b] |f(y)−g(y)| turns (C([a, b]), d∞) into a metric subspace of (B([a, b]), d∞).The space C([a, b]) also becomes a metric space with metric

dp(f, g) =

(∫ b

a

|f(y)− g(y)|p dy

)1/p

,

for any 1 ≤ p <∞. However, B([a, b]) with these dp is not a metric space.


Definition A.6.7 (Diameter and bounded sets). The diameter of a set A ⊂ X in ametric space (X, d) is

diam(A) := sup {d(x, y) | x, y ∈ A} ,

with convention diam(∅) = 0. A set A ⊂ X is said to be bounded, if diam(A) <∞.

Example. diam({x}) = 0, diam({x, y}) = d(x, y), and

diam({x, y, z}) = max {d(x, y), d(y, z), d(x, z)} .

Exercise A.6.8. Show that diam(Br(x)) ≤ 2r, so that balls are bounded.

Definition A.6.9 (Distance between sets). The distance between sets A,B ⊂ X is

dist(A,B) := inf {d(x, y) | x ∈ A, y ∈ B} ,

with the convention that dist(A, ∅) =∞.

Exercise A.6.10. Show that A ∩Br(x) = ∅ if and only if dist({x}, A) < r.

We note that the function dist(A,B) does not define a metric on subsets ofX. For example:

Exercise A.6.11. Give an example of sets A,B ⊂ R2 for which dist(A,B) = 0 eventhough A ∩B = ∅. Here we consider naturally the Euclidean metric.

Exercise A.6.12. Show that set S in a metric space (X, d) is bounded if and onlyif there exist some a ∈ X and r > 0 such that S ⊂ Br(a).

Lemma A.6.13. Let S be a bounded set in a metric space (X, d) and let c ∈ X.Then S ⊂ BR(c) for some R > 0.

Proof. Since S is a bounded set, there exist some a ∈ X and r > 0 such thatS ⊂ Br(a). Consequently, for all x ∈ S we have

d(x, c) ≤ d(x, a) + d(a, c) < r + d(a, c),

so the statement follows with R = r + d(a, c). �Proposition A.6.14. The union of finitely many bounded sets in a non-empty met-ric space is bounded.

Proof. Let S1, . . . , Sn be bounded sets in a non-empty metric space (X, d). Let ustake some c ∈ X. Then by Lemma A.6.13 there exists some Ri, i = 1, . . . , n, suchthat Si ⊂ BRi

(c). If we take R = max{R1, . . . , Rn}, then we have Si ⊂ BRi(c) ⊂

BR(c), which implies that ∪ni=1Si ⊂ BR(c) is bounded. �

Remark A.6.15. We note that the union of infinitely many bounded sets does nothave to be bounded. For example, the union of sets Si = (0, i) ⊂ R, i ∈ N, is notbounded in (R, d∞).

A.7. Topological spaces 29

Usually, the topological properties can be characterised with generalised se-quences (or nets). Now, we briefly study this phenomenon in metric topology,where ordinary sequences suffice.

Definition A.6.16 (Sequences). A sequence in a set A is a mapping x : Z+ → A.We write xk := x(k) and

x = (xk)k∈Z+ = (xk)∞k=1 = (x1, x2, x3, . . .).

Notice that x = {x1, x2, x3, · · · } = {xk : k ∈ Z+}.Definition A.6.17 (Convergence). Let (X, d) be a metric space. A sequence x :Z+ → X converges to a point p ∈ X, if lim

k→∞d(xk, p) = 0, i.e.,

∀ε > 0 ∃kε ∈ Z+ : k ≥ kε ⇒ d(xk, p) < ε.

In such a case, we write limk→∞

xk = p or xk → p or xkd−−−−→

k→∞p etc.

Clearly, xk → p as k →∞ if and only if

∀ε > 0 ∃N : k ≥ N ⇒ xk ∈ Bε(p).

We now collect some properties of limits.

Proposition A.6.18 (Uniqueness of limits in metric spaces). Let (X, d) be a metricspace. If xk → p and xk → q as k →∞, then p = q.

Proof. Let ε > 0. Since xk → p and xk → q as k → ∞, it follows that there aresome numbers N1, N2 such that d(xk, p) < ε for all k > N1 and such that d(xk, q) <ε for all k > N2. Hence by the triangle inequality for all k > max{N1, N2} wehave d(p, q) ≤ d(p, xk) + d(xk, q) < 2ε. Since this conclusion is true for any ε > 0,it follows that d(p, q) = 0 and hence p = q. �

A.7 Topological spaces

Previously, a metric provided a way of measuring distances between sets. Thebranch of mathematics called topology can be thought as a way to describe “qual-itative geography of a set” without referring to specific numerical distance values.We begin by considering properties of metric spaces that motivate the definitionof topology which follows after them.

Definition A.7.1 (Open sets and neighbourhoods). A set U ⊂ X in a metric spaceX is said to be open if for every x ∈ U there is some ε > 0 such that Bε(x) ⊂ U .For a point x ∈ X, any open set containing x is called an open neighbourhood of x.

Proposition A.7.2. Every ball Br(a) in a metric space (X, d) is open.


Proof. Let x ∈ Br(a). Then the number ε = r − d(x, a) > 0 is positive, andBε(x) ⊂ Br(a). Indeed, for any y ∈ Bε(x) we have d(y, a) ≤ d(y, x) + d(x, a) <ε + d(x, a) = r. �Proposition A.7.3. Let (X, d) be a metric space. Then xk → p as k → ∞ ifand only if every open neighbourhood of x contains all but finitely many of thepoints xk.

Proof. “If” implication is immediate because balls are open. On the other hand,let p ∈ U where U is an open set. Then there is some ε > 0 such that Bε(p) ⊂ U .Now, if xk → p as k → ∞, there is some N such that for all k > N we havexk ∈ Bε(p) ⊂ U , implying the statement. �Definition A.7.4 (Continuous mappings in metric spaces). Let (X1, d1) and(X2, d2) be two metric spaces, let f : X1 → X2, and let a ∈ X1. Then f is said tobe continuous at a if for every ε > 0 there is some δ > 0 such that d1(x, a) < δimplies d2(f(x), f(a)) < ε. The mapping f is said to be continuous (on X1) if itis continuous at all points of X1.

Example. Let X1 = C([a, b]) and X2 = R be equipped with the sup-metrics d1

and d2, respectively. Then mapping Φ : X1 → X2 defined by Φ(h) =∫ b

ah(y)dy is

continuous.

Definition A.7.5 (Preimage). Let f : X1 → X2 be a mapping and let S ⊂ X2 beany subset of X2. Then the preimage of S under f is defined by

f−1(S) = {x ∈ X1 : f(x) ∈ S}.Theorem A.7.6. Let (X1, d1), (X2, d2) be metric spaces and let f : X1 → X2. Thenthe following statements are equivalent:

(i) f is continuous on X1;(ii) for every a ∈ X1 and every ball Bε(f(a)) ⊂ X2 there is a ball Bδ(a) ⊂ X1

such that Bδ(a) ⊂ f−1(Bε(f(a)));(iii) for every open set U ⊂ X2 its preimage f−1(U) is open in X1.

Proof. First, let us show the equivalence of (i) and (ii). Condition (i) is equivalentto saying that for every ε > 0 there is δ > 0 such that d1(x, a) < δ impliesd2(f(a), f(x)) < ε. In turn this is equivalent to saying that for every ε > 0there is δ > 0 such that x ∈ Bδ(a) implies f(x) ∈ Bε(f(a)), which means thatBδ(a) ⊂ f−1(Bε(f(a))).

To show that (ii) implies (iii), let us assume that f is continuous and thatU ⊂ X2 is open. Take x ∈ f−1(U). Then f(x) ∈ U and since U is open there issome ε > 0 such that Bε(f(a)) ⊂ U . Consequently, by (ii), there exists δ > 0 suchthat Bδ(a) ⊂ f−1(Bε(f(a))) ⊂ f−1(U), implying that f−1(U) is open.

Finally, let us show that (iii) implies (ii). We observe that by (iii) for everya ∈ X1 and every ε > 0 the set f−1(Bε(f(a))) is an open set containing a. Hencethere is some δ > 0 such that Bδ(a) ⊂ f−1(Bε(f(a))), completing the proof. �


Theorem A.7.7. Let X be a metric space. We have the following properties of opensets in X:

(T1) ∅ and X are open sets in X.(T2) The union of any collection of open subsets of X is open.(T3) The intersection of a finite collection of open subsets of X is open.

Proof. It is obvious that the empty set ∅ is open. Moreover, for any x ∈ X andany ε > 0 we have Bε(x) ⊂ X, implying that X is also open.

To show (T2), suppose that we have a collection {Ai}i∈I of open sets in X,for an index set I. Let a ∈ ∪i∈IAi. Then there is some j ∈ I such that a ∈ Aj andsince Aj is open there is some ε > 0 such that Bε(a) ⊂ Aj ⊂ ∪i∈IAi, implying(T2).

To show (T3), assume that A1, . . . , An is a finite collection of open sets and leta ∈ ∩n

i=1Ai. It follows that for every i = 1, . . . , n we have a ∈ Ai and hence there isεi > 0 such that Bεi

(a) ⊂ Ai. Let now ε = min{ε1, . . . , εn}. Then Bε(a) ⊂ Ai forall i and hence Bε(a) ⊂ ∩n

i=1Ai implying that the intersection of Ai’s is open. �

Definition A.7.8 (Topology). A family of sets τ ⊂ P(X) is called a topology on theset X if

1.⋃U ∈ τ for every collection U ⊂ τ , and

2.⋂U ∈ τ for every finite collection U ⊂ τ .

Then (X, τ) (or simply X when τ is evident) is called a topological space; a setA ⊂ X is called open (or τ -open) if A ∈ τ , and closed (or τ -closed) if X \ A ∈ τ .Let the collection of τ -closed sets be denoted by

τ∗ = {X \ U : U ∈ τ}.

Then the axioms of the topology become naturally complemented:

1.⋂A ∈ τ∗ for every collection A ⊂ τ∗, and

2.⋃A ∈ τ∗ for every finite collection A ⊂ τ∗.

Remark A.7.9. Recall our natural conventions (A.1) for the union and the inter-section of the empty family. Thereby τ ⊂ P(X) is a topology if and only if thefollowing conditions hold:(T1) ∅, X ∈ τ ,(T2)

⋃U ∈ τ for every non-empty collection U ⊂ τ , and(T3) U ∩ V ∈ τ for every U, V ∈ τ .

Consequently, for any topology of X, the subsets ∅ ⊂ X and X ⊂ X are alwaysboth open and closed.

Proposition A.7.3 motivates the following notion of convergence in topologicalspaces.


Definition A.7.10 (Convergence in topological spaces). Let (X, τ) be a topologicalspace. We say that a sequence xk converges to p as k → ∞, and write xk → pas k → ∞, if every open neighbourhood of p contains all but finitely many ofpoints xk.

Proposition A.7.11. Let X and Y be topological spaces and let f : X → Y becontinuous. If xk → p in X as k →∞ then f(xk)→ f(p) in Y as k →∞.

Proof. Let U be an open set in Y containing f(p). Then p ∈ f−1(U) and f−1(U)is open in X, implying that there is N such that xk ∈ f−1(U) for all k > N .Consequently, f(xk) ∈ U for all k > N implying that f(xk) → f(p) in Y ask →∞. �Corollary A.7.12. Any metric space is a topological space by Theorem A.7.7. Thecanonical topology of a metric space (X, d) is the family τ consisting of all sets in(X, d) which are open according to Definition A.7.1. This canonical metric topologywill be denoted by τd or by τ(d). Metric convergence in (X, d) is equivalent to thetopological convergence in the canonical metric topology (X, τd).

Remark A.7.13. Notice that the intersection of any finite collection of τ -open setsis τ -open. On the other hand, it may well be that a countably infinite intersectionof open sets is not open. In a metric space (X, d),

∞⋂k=1

B1/k(x) = {x}.

Now {x} ∈ τd if and only if {x} = Br(x) for some r > 0.

Corollary A.7.14 (Properties of closed sets). Let X be a topological space. Wehave the following properties of closed sets in X:

(C1) ∅ and X are closed in X.(C2) The intersection of any collection of closed subsets of X is closed.(C3) The union of a finite collection of closed subsets of X is closed.

Proof. Let Ai, i ∈ I, be any collection of subsets of X. The corollary followsimmediately from Remark A.7.9 and de Morgan’s rules

X\⋃i∈I

Ai =⋂i∈I

(X\Ai), X\⋂i∈I

Ai =⋃i∈I

(X\Ai),

see Exercise A.1.5. �Definition A.7.15 (Comparing metric topologies). Let d1, d2 be two metrics on aset X. The topology τ(d1) defined by d1 is said to be stronger than topology τ(d2)defined by d2 if τ(d1) ⊃ τ(d2). In this case the topology τ(d2) is also said to beweaker than τ(d1). Metrics d1, d2 on a set X are said to be equivalent if they definethe same topology τ(d1) = τ(d2).


Proposition A.7.16 (Criterion for comparing metric topologies). Let d1, d2 be twometrics on a set X such that there is a constant C > 0 such that d2(x, y) ≤Cd1(x, y) for all x, y ∈ X. Then τ(d2) ⊂ τ(d1), i.e., every d2-open set is alsod1-open.

Consequently, if there is a constant C > 0 such that

C−1d1(x, y) ≤ d2(x, y) ≤ Cd1(x, y), (A.2)

for all x, y ∈ X, then metrics d1 and d2 are equivalent. Such metrics are calledLipschitz equivalent.

Sometimes such metrics are called just equivalent, however we use the term“Lipschitz” to distinguish this equivalence from the one in Definition A.7.15.

Proof. Fixing the constant C > 0 from (A.2), we observe that d1(x, y) < r impliesd2(x, y) < Cr, which means that Bd1(x, r) ⊂ Bd2(x, Cr). Let now U ∈ τ(d2)and let x ∈ U . Then there is some ε > 0 such that Bd2(x, ε) ⊂ U implying thatBd1(x, ε/C) ⊂ U . Hence U ∈ τ(d1). �

Exercise A.7.17. Prove that the metrics dp, 1 ≤ p ≤ ∞, from Exercise A.6.3, areall Lipschitz equivalent. The corresponding topology is called the Euclidean metrictopology on Rn.

Definition A.7.18 (Relative topology). Let (X, τ) be a topological space and letA ⊂ X. Then we define the relative topology on A by

τA = {U ∩A : U ∈ τ}.

Proposition A.7.19 (Relative topology is a topology). Any subset A of a topologicalspace (X, τ) when equipped with the relative topology τA is a topological space.

Proof. We have to check the properties (T1)–(T3) of Remark A.7.9. It is easy tosee that ∅ = ∅∩A ∈ τA and that A = X∩A ∈ τA. To show (T2), let Vi ∈ τA, i ∈ I,be a family of sets from τA. Then there exist sets Ui ∈ τ such that Vi = Ui ∩ A.Consequently, we have

⋃i∈I

Vi =⋃i∈I

(Ui ∩A) =

(⋃i∈I

Ui

)∩A ∈ τA.

To show (T3), let V1, . . . , Vn be a family of sets from τA. It follows that there existsets Ui ∈ τ such that Vi = Ui ∩A. Consequently, we have

n⋂i=1

Vi =n⋂

i=1

(Ui ∩A) =

(n⋂

i=1

Ui

)∩A ∈ τA,

completing the proof. �


Remark A.7.20 (Metric subspaces). Let (X, d) be a metric space with canonicaltopology τ(d). Let Y ⊂ X be a subset of X and let us define dY = d|Y×Y . Thenτ(dY ) = τ(d)Y , i.e., the canonical topology of the metric subspace coincides withthe relative topology of the metric space.

Definition A.7.21 (Product topology). Let (X1, τ1) and (X2, τ2) be topologicalspaces. A subset of X1 × X2 is said to be open in the product topology if it is aunion of sets of the form U1 × U2, where U1 ∈ τ1, U2 ∈ τ2. The collection of allsuch open sets is denoted by τ1 ⊗ τ2.

Proposition A.7.22 (Product topology is a topology). The set X1 × X2 with thecollection τ1 ⊗ τ2 is a topological space.

Proof. We have to check properties (T1)–(T3) of Remark A.7.9. It is easy to seethat ∅ = ∅ × ∅ ∈ τ1 ⊗ τ2 and that X1 ×X2 ∈ τ1 ⊗ τ2.

To show (T2), assume that Aα ∈ τ1 ⊗ τ2 for all α ∈ I. Then each Aα is aunion of sets of the form U1 × U2 with U1 ∈ τ1, U2 ∈ τ2. Consequently, the union∪α∈IAα is a union of sets of the same form and does, therefore, also belong toτ1 ⊗ τ2.

To show (T3), even for n sets, assume that Ai ∈ τ1 ⊗ τ2, for all i = 1, . . . , n.By definition there exist collections U i

αi∈ τ1, V i

αi∈ τ2, αi ∈ Ii, i = 1, . . . , n, such

thatAi =

⋃αi∈Ii

(U iαi× V i

αi), i = 1, . . . , n.

Consequently,

n⋂i=1

Ai =⋃

αi∈Ii, 1≤i≤n

((∩n

j=1Ujαi

)× (∩nj=1V

jαi

))∈ τ1 ⊗ τ2,

completing the proof. �Theorem A.7.23 (Topologies on R2). The product topology on R × R is the Eu-clidean metric topology of R2.

Proof. We start by proving that every set open in the product topology of R2 isalso open in the Euclidean topology of R2. First we note that any open set in R inthe Euclidean topology is a union of open intervals, i.e., every open set U can bewritten as U = ∪x∈UBεx(x), where Bεx(x) is an open ball centred at x with someεx > 0. Then we note that every open rectangle in R2 is open in the Euclideantopology. Indeed, any rectangle R = (a, b)× (c, d) in R2 can be written as a unionof balls, i.e.,

R = ∪x∈RBεx(x),

with balls Bεx(x) taken with respect to d2, with some εx > 0, implying that R is

open in the Euclidean topology of R2. Finally, we note that any open set A in theproduct topology is a union of sets of the form U1 ×U2, where U1, U2 are open in

A.8. Kuratowski’s closure 35

R. Consequently, writing both U1 and U2 as unions of open intervals, we obtain Ais a union of open rectangles in R2, which we showed to be open in the Euclideantopology, implying in turn that A is also open in the Euclidean topology of R2.

Conversely, let us prove that every set open in the Euclidean topology of R2

is also open in the product topology of R2. First we note that clearly every discBε(x) in R2 can be written as a union of open rectangles and is, therefore, openin the product topology of R2. Consequently, every open set U in the Euclideantopology can be written as U = ∪x∈UBεx

(x) for some εx > 0, so that it is alsoopen in the product topology as a union of open sets. �

A.8 Kuratowski’s closure

In this section we describe another approach to topology based on Kuratowski’sclosure operator. This provides another (and perhaps more intuitive) approach tosome notions of the previous section.

Definition A.8.1 (Metric interior, closure, boundary, etc.). In a metric space(X, d), the metric closure of A ⊂ X is

A = cld(A) := {x ∈ X | ∀r > 0 : A ∩Br(x) = ∅} .

In other words, x ∈ cld(A) ⇐⇒ dist({x}, A) = 0 (i.e., “x is close to A”). This isalso equivalent to saying that every ball around x contains point(s) of A.

The metric interior intd(A), the metric exterior extd(A) and the metricboundary ∂d(A) are defined by

intd(A) := X \ cld(X \A),extd(A) := X \ cld(A),

∂d(A) := cld(A) ∩ cld(X \A).

Notice that in this way, we have defined mappings

cld, intd, extd, ∂d : P(X)→ P(X).

Exercise A.8.2. Let (X, d) be a metric space and A ⊂ X. Prove the followingclaims:

intd(A) = {x ∈ X | ∃r > 0 : Br(x) ⊂ A} ,

∂d(A) = cld(A) \ intd(A),X = intd(A) ∪ ∂d(A) ∪ extd(A).

Consequently, prove that cld(A) is closed for any set A ⊂ X.


Definition A.8.3 (Metric topology). Let (X, d) be a metric space. Then

τd := intd(P(X)) = {intd(A) | A ⊂ X}

is called the metric topology or the family of metrically open sets. The correspond-ing family of metrically closed sets is

τ∗d := cld(P(X)) = {cld(A) | A ⊂ X} .

By the following Lemma A.8.4, we have

• a set C ⊂ X is metrically closed if and only if C = cld(C),• a set U ⊂ X is metrically open if and only if U = intd(U).

Lemma A.8.4. Let (X, d) be a metric space and A ⊂ X. Then

cld (cld(A)) = cld(A), (A.3)intd (intd(A)) = intd(A). (A.4)

Proof. Let C = cld(A). Trivially, C ⊂ cld(C). Let x ∈ cld(C). Let r > 0. Takey ∈ C ∩Br(x), and then z ∈ A ∩Br(y). Hence

d(x, z) ≤ d(x, y) + d(y, z) < 2r,

so x ∈ C. Thus (A.3) is obtained. By the definition of the metric interior, (A.3)implies (A.4). �Definition A.8.5 (Topological interior, closure, boundary, etc.). Let τ be a topologyon X. For A ⊂ X, the interior intτ (A) is the largest open subset of A, and theclosure A = clτ (A) is the smallest closed set containing A. That is,

A = intτ (A) :=⋃{U ∈ τ | U ⊂ A} ,

clτ (A) :=⋂{S ∈ τ∗ | A ⊂ S} .

These define mappings intτ , clτ : P(X) → P(X). The boundary ∂τ (A) of a setA ⊂ X is defined by

∂τ (A) := clτ (A) ∩ clτ (X \A).

A set A ⊂ X is dense if clτ (A) = X. The topological space (X, τ) is separable if ithas a countable dense subset. A point x ∈ X is an isolated point of a set A ⊂ Xif A ∩ U = {x} for some U ∈ τ . A point y ∈ X is an accumulation point of a setB ⊂ X if (B∩V )\{y} = ∅ for every V ∈ τ . A neighbourhood of x ∈ X is any openset U ⊂ X containing x. The family of neighbourhoods of x ∈ X is denoted by

Vτ (x) := {U ∈ τ | x ∈ U}

(or simply V(x), when τ is evident).


Remark A.8.6. Intuitively, the closure clτ (A) ⊂ X contains those points that areclose to A. Clearly,

τ = {intτ (A) | A ⊂ X} ,

τ∗ = {clτ (A) | A ⊂ X} .

Moreover, U ∈ τ if and only if U = intτ (U), and C ∈ τ∗ if and only if C = clτ (C).

Exercise A.8.7. Prove that

∂τ (A) = clτ (A) \ intτ (A).

Exercise A.8.8. Let τd be the metric topology of a metric space (X, d). Show thatintd = intτd

and that cld = clτd.

Proposition A.8.9 (A characterisation of open sets). Let A be a subset of a topo-logical space X. Then A is open if and only if for every x ∈ A there is an open setUx containing x such that Ux ⊂ S.

Proof. If A is open we can take Ux = A for every x ∈ A. Conversely, writingA = ∪x∈AUx by property (T2) of open sets we get that A is open if all Ux areopen. �

Proposition A.8.10 (A characterisation of closures). Let A be a subset of a topo-logical space X. Then x ∈ A if and only if every open set containing x contains apoint of A.

Proof. We will prove that x ∈ A if and only if there is an open set U such thatx ∈ U but A ∩ U = ∅. Since A is defined as the intersection of all closed setscontaining A, it follows that x ∈ A means that there is a closed set C such thatA ⊂ C and x ∈ C. Set U = X\U is then the required set. �

Definition A.8.11 (Closure operator). Let X be a set. A closure operator on X isa mapping c : P(X)→ P(X) satisfying Kuratowski’s closure axioms:

1. c(∅) = ∅,2. A ⊂ c(A),

3. c(c(A)) = c(A),

4. c(A ∪B) = c(A) ∪ c(B).

Instead of a closure operator c : P(X) → P(X), we could study an interioroperator i : P(X)→ P(X), related to each other by

i(S) = X \ c(X \ S),c(A) = X \ i(X \A).


Kuratowski’s closure axioms become interior axioms:

1. i(X) = X,

2. i(S) ⊂ S,

3. i(i(S)) = i(S),

4. i(S ∩ T ) = i(S) ∩ i(T ).

Theorem A.8.12. Let (X, τ) be a topological space. Then the mappings intτ , clτ :P(X)→ P(X) are interior and closure operators, respectively.

Proof. Obviously, intτ (X) = X and intτ (A) ⊂ A. Moreover, intτ (U) = U forU ∈ τ , and intτ (A) ∈ τ , because τ is a topology. Hence intτ (intτ (A)) = intτ (A).Finally, {

intτ (A ∩B) ⊂ intτ (A) ⊂ A,

intτ (A ∩B) ⊂ intτ (B) ⊂ B,

yielding

intτ (intτ (A ∩B)) ⊂ intτ (intτ (A) ∩ intτ (B)) ⊂ intτ (A ∩B),

where intτ (A) ∩ intτ (B) ∈ τ , so that intτ (A ∩B) = intτ (A) ∩ intτ (B). �

Theorem A.8.13. Let i : P(X) → P(X) be an interior operator. Then the familyτi = i(P(X)) = {i(A) : A ⊂ X} is a topology. Moreover, i = intτi .

Proof. First,∅ = i(∅) ∈ τi, X = i(X) ∈ τi.

Second, if A,B ∈ τi then A ∩ B = i(A) ∩ i(B) = i(A ∩ B) ∈ τi. Third, letA = {Aj : j ∈ J} ⊂ τi. Now

⋃A Aj=i(Aj)=

⋃j∈J

i(Aj)i(Aj)⊂i(

⋃ A)⊂ i

(⋃A

)⊂

⋃A.

Thus⋃A = i(

⋃A) ∈ τi. Next,

intτi(A) =

⋃{U ∈ τi | U ⊂ A}

=⋃{i(B) | i(B) ⊂ A, B ⊂ X} .

Here we see that i(A) ⊂ intτi(A). Moreover, if i(A) ⊂ i(B) ⊂ A then i(A) =

i(i(A)) ⊂ i(i(B)) = i(B) ⊂ i(A). Hence i(A) = intτi(A). �

Remark A.8.14. Above we have seen how topologies and closure operators (orinterior operators) on a set are in bijective correspondence.


Exercise A.8.15. For each j ∈ J , let τj be a topology on X. Prove that τ =⋂j∈J

τj

is a topology. Give an example, where⋃j∈J

τj is not a topology.

Definition A.8.16 (Base of topology). Let (X, τ) be a topological space. A familyB ⊂ P(X) is called a base (or basis) for the topology τ if any open set is a unionof some members of B, i.e.,

τ ={⋃

B′ : B′ ⊂ B}

.

A family A ⊂ P(X) is called a subbase (or subbasis) for the topology τ if{⋂A′ : A′ ⊂ A is finite

}is a base for the topology. A topology is called second countable if it has a countablebase.

Example. Trivially a topology τ is a base for itself, as U =⋃{U} for every U ∈ τ .

If (X, d) is a metric space then

B := {Br(x) | x ∈ X, r > 0}

constitutes a base for τd.

Exercise A.8.17. Let A ⊂ P(X). Show that there is the minimal topology τA onX such that A ⊂ τA: more precisely, if σ is a topology on X for which A ⊂ σ,then τA ⊂ σ.

Exercise A.8.18. Let τA be as in the previous exercise. Prove that a base for thistopology is provided by

B ={⋂

A′ : A′ ⊂ A ∪ {X} is finite}

.

Finally, we give another proof of Corollary A.7.12 that metric spaces aretopological spaces using the introduced notions of interior and closure.

Theorem A.8.19 (Metric topology is a topology). Any metric topology is a topol-ogy.

Proof. Let τd be the metric topology of (X, d). By Lemma A.8.4, U ∈ τd if andonly if U = intd(U). Now ∅, X ∈ τd, because{

intd(∅) = {x ∈ X | ∃r > 0 : Br(x) ⊂ ∅} = ∅,intd(X) = {x ∈ X | ∃r > 0 : Br(x) ⊂ X} = X.

Next, if Br(x) ⊂ U and Bs(x) ⊂ V then Bmin{r,s}(x) ⊂ U ∩ V . Thus if U, V ∈ τd

then U ∩ V ∈ τd. Finally, if Br(x) ⊂ Uk for some k ∈ J then Br(x) ⊂ ⋃j∈J Uj .

Thus if {Uj : j ∈ J} ⊂ τd then⋃

j∈J Uj ∈ τd. �


Exercise A.8.20 (Product topology). Let X, Y be topological spaces with basesBX ,BY , respectively. Show that sets

{U × V | U ∈ BX , V ∈ BY }

form a base for the product topology of X × Y = {(x, y) | x ∈ X, y ∈ Y } fromDefinition A.7.21.

The metric topology (but not only, cf. topological spaces with countabletopology bases) can be characterised by the limits of sequences:

Theorem A.8.21. Let (X, d) be a metric space, p ∈ X and A ⊂ X. Then p ∈ cd(A)if and only if some sequence x : Z+ → A converges to p.

Proof. Let xk → p, where xk ∈ A for each k ∈ Z+. That is,

∀ε > 0 ∃kε ∈ Z+ : k ≥ kε ⇒ xk ∈ Bd(p, ε).

Thus A ∩Bd(p, ε) = ∅ for every ε > 0. Thereby p ∈ cd(A).Let p ∈ cd(A), that is A ∩ Br(x) = ∅ for all r > 0. For each k ∈ Z+, take

xk ∈ A ∩ Bd(p, 1/k). Now (xk)∞k=1 is a sequence in A, converging to p, becaused(xk, p) < 1/k. �

A.9 Complete metric spaces

In this section we discuss complete metric spaces, give a sample application toFredholm integral equations using Banach’s Fixed Point Theorem, and show thatevery metric space can be “completed” and such a completion is essentially unique.Later, we will revisit this topic again to show completeness of R and Rn in Theo-rem A.13.10 and Corollary A.13.11. Completeness in topological vector spaces willbe discussed in Section B.2.

Definition A.9.1 (Cauchy sequences and completeness). Let (X, d) be a metricspace. A sequence x : Z+ → X is a Cauchy sequence if

∀ε > 0 ∃kε ∈ Z+ : i, j ≥ kε ⇒ d(xi, xj) < ε.

A metric space is called complete if all Cauchy sequences converge.

Example. The Euclidean metric space (Rn, d) is complete (see Corollary A.13.11),but its dense subset Qn is not (metric of course inherited from d). For instance,Napier’s constant e ∈ R \Q is obtained as the limit of numbers

∑kj=0 1/j! ∈ Q.

Lemma A.9.2 (Properties of Cauchy sequences). We have the following properties:

(1) Every convergent sequence is a Cauchy sequence.(2) Every Cauchy sequence is bounded.(3) If a Cauchy sequence has a convergent subsequence, it converges to the same

limit.

A.9. Complete metric spaces 41

Proof. We assume that a metric space (X, d) is non-empty. To prove (1), let xk →p. We want to show that (xk)∞k=1 is a Cauchy sequence. Let ε > 0. Take kε ∈ Z+

such that d(xk, p) < ε if k ≥ kε. Let i, j ≥ kε. Then

d(xi, xj) ≤ d(xi, p) + d(p, xj) < 2ε.

To prove (2), let (xk)∞k=1 be a Cauchy sequence. Take ε = 1. Then there is somek such that for i, j ≥ k we have d(xi, xj) < 1. Let us now fix some a ∈ X. Thenfor i > k we have

d(a, xi) ≤ d(a, xk+1) + d(xk+1, xi) < ρ + 1,

with ρ = d(a, xk+1). Setting R := max{d(a, x1), . . . , d(a, xk), ρ}, we get that xi ∈BR+1(a) for all i.

To prove (3), let (xn)∞n=1 be a Cauchy sequence, with a convergent subse-quence xni

→ p ∈ X. Fix some ε > 0. Then there is some k such that for alln, m ≥ k we have d(xn, xm) < ε. At the same time, there is some N such that forni > N , we have d(xni , p) < ε. Consequently, for n ≥ max{k,N}, we get

d(xn, p) ≤ d(xn, xni) + d(xni , p) < 2ε,

which means that xn → p as n→∞. �Theorem A.9.3. Let (X, d) be a complete metric space, and A ⊂ X. Then(A, d|A×A) is complete if and only if A ⊂ X is closed.

Proof. Let A ⊂ X be closed. Take a Cauchy sequence x : Z+ → A. Due to thecompleteness of (X, d), x converges to a point p ∈ X. Now p ∈ A, because A isclosed. Thus (A, d|A×A) is complete.

Suppose (A, d|A×A) is complete. We have to show that cd(A) = A. Takep ∈ cd(A). For each k ∈ Z+, take xk ∈ A∩Bd(p, 1/k). Clearly, xk → p, so (xk)∞k=1

is a Cauchy sequence in A. Due to the completeness of (A, d|A×A), xk → a forsome a ∈ A. Because the limits in X are unique, p = a ∈ A. Thus A = cd(A) isclosed. �

We now show one application of the notion of completeness to solving integralequations.

Definition A.9.4 (Pointwise convergence of functions). Let fn : [a, b] → R be asequence of functions and let f : [a, b] → R. Then we say that fn converges to fpointwise on [a, b] if fn(x)→ f(x) as n→∞ for all x ∈ [a, b]. In other words, thismeans that

∀x ∈ [a, b] ∀ε > 0 ∃N = N(ε, x) : n > N =⇒ |fn(x)− f(x)| < ε.

As before, by C([a, b]) we denote the space of all continuous functions f :[a, b]→ R. By default we always equip it with the sup-metric d∞.


Exercise A.9.5. Find a sequence of continuous functions fn ∈ C([0, 1]) such thatfn → f pointwise on [0, 1], but f : [0, 1]→ R is not continuous on [0, 1].

To remedy this situation, we introduce another notion of convergence offunctions:

Definition A.9.6 (Uniform convergence of functions). Let fn : [a, b] → R be asequence of functions and let f : [a, b] → R. Then we say that fn converges to funiformly on [a, b] if

∀ε > 0 ∃N = N(ε) : ∀n > N x ∈ [a, b] =⇒ |fn(x)− f(x)| < ε.

The difference with the pointwise convergence here is that the same index Nworks for all x ∈ [a, b].

Theorem A.9.7. Let fn ∈ C([a, b]) be a sequence of continuous functions, let f :[a, b] → R, and suppose that fn converges to f uniformly on [a, b]. Then f iscontinuous on [a, b].

Proof. Fix ε > 0. Since fn → f uniformly, there is some N = N(ε) such that forall n > N and all x ∈ [a, b] we have |fn(x) − f(x)| < ε. Let c ∈ [a, b]. We willshow that f is continuous at c. Since every function fn is continuous at c, there issome δ = δ(n) > 0 such that |x− c| < δ implies |fn(x)− fn(c)| < ε. Taking somen > N , we get

|f(x)− f(c)| ≤ |f(x)− fn(x)|+ |fn(x)− fn(c)|+ |fn(c)− f(c)| < 3ε

for all |x− c| < δ, implying that f is continuous at c. �

This result extends to uniform limits of continuous functions on general topo-logical spaces, see Exercise C.2.18.

Proposition A.9.8 (Metric uniform convergence). We have fn → f in metric space(C([a, b]), d∞) if and only if fn → f uniformly on [a, b].

Proof. Convergence fn → f in metric space (C([a, b]), d∞) means that for everyε > 0 there is N such that for all n > N we have supx∈[a,b] |fn(x) − f(x)| < ε.But this means that |fn(x) − f(x)| < ε for all x ∈ [a, b], which is the uniformconvergence. �

Theorem A.9.9 (Completeness of continuous functions). Space C([a, b]) with sup-metric d∞ is complete.

Proof. Let fn ∈ C([a, b]) be a Cauchy sequence. Fix ε > 0. Then there is some Nsuch that for all m,n > N we have

supx∈[a,b]

|fn(x)− fm(x)| < ε. (A.5)


Therefore, for each x ∈ [a, b] the sequence (fn(x))∞n=1 is a Cauchy sequence in R.If we use that R is complete (see Theorem A.13.10), it converges to some point inR, which we call f(x). Thus, for every x ∈ [a, b] we have fn(x)→ f(x) as n→∞.Passing to the limit as n → ∞ in (A.5), we obtain supx∈[a,b] |f(x) − fm(x)| ≤ ε,which means that d∞(f, fm) ≤ ε, completing the proof. �Theorem A.9.10 (Banach’s Fixed Point Theorem). Let (X, d) be a non-emptycomplete metric space, let k < 1 be a constant, and let f : X → X be such that

d(f(x), f(y)) ≤ k d(x, y) (A.6)

for all x, y ∈ X. Then there exists a unique point a ∈ X such that a = f(a).

A mapping f satisfying (A.6) with some constant k < 1 is called a contrac-tion. A point a such that a = f(a) is called a fixed point of f .

Exercise A.9.11. Show that the conditions of Theorem A.9.10 are indispensable.For example, the conclusion of Theorem A.9.10 fails if X is not complete. Showthat it also fails if k ≥ 1. Finally, give an example of a function f : X → Xsatisfying

d(f(x), f(y)) < d(x, y)

instead of (A.6) on a complete metric space X = ∅ such that f does not have fixedpoints.

Proof of Theorem A.9.10. First we observe that f is continuous. Indeed, ifd(x, y) < ε, it follows that d(f(x), f(y)) ≤ kd(x, y) < kε < ε. We now construct acertain Cauchy sequence, whose limit will be the required fixed point of f . Takeany x0 ∈ X. For all n ≥ 0, define xn+1 = f(xn). Then for all n ≥ 1 we have

d(xn+1, xn) = d(f(xn+1), f(xn)) ≤ kd(xn, xn−1),

implying that d(xn+1, xn) ≤ knd(x1, x0). Consequently, for n > m ≥ 1, we have

d(xn, xm) ≤ d(xn, xn−1) + · · ·+ d(xm+1, xm)≤ (kn−1 + · · ·+ km)d(x1, x0)

≤ km∞∑

i=0

ki d(x1, x0)

=km

1− kd(x1, x0).

Since k < 1 it follows that d(xn, xm) → 0 as n, m → ∞ which means that xn isa Cauchy sequence. Since X is complete, xn → a for some a ∈ X. We claim thata is a fixed point of f . Indeed, since xn → a and since f is continuous, we havef(xn) → f(a) by Proposition A.7.11. Therefore, xn+1 → f(a) as n → ∞, and bythe uniqueness of limits in metric spaces Proposition A.6.18 we have f(a) = a.

Finally, let us show that there is only one fixed point. Suppose that f(a) = aand f(b) = b. It follows that d(a, b) = d(f(a), f(b)) ≤ kd(a, b) and since k < 1, wemust have d(a, b) = 0 and hence a = b. �


Corollary A.9.12 (Fredholm integral equations). Let p : [0, 1]→ R be continuous,p ≥ 0, and such that

∫ 1

0p(t) dt < 1. Let g ∈ C([0, 1]). Then there exists a unique

function f ∈ C([0, 1]) such that

f(x) = g(x)−∫ x

0

f(t) p(t) dt.

Proof. As usual, let us equip C([0, 1]) with the sup-metric d∞, and let us defineT : C([0, 1])→ C([0, 1]) by

(Tf)(x) = g(x)−∫ x

0

f(t) p(t) dt.

We claim that Y is a contraction, which together with the completeness of C([0, 1])in Theorem A.9.9 and Banach’s Fixed Point Theorem A.9.10 would imply thestatement. We have

d∞(Tf, Tg) = supx∈[0,1]

∣∣∣∣∫ x

0

(f(t)− g(t)) p(t) dt

∣∣∣∣ ≤ supx∈[0,1]

∫ x

0

|f(t)− g(t)| p(t) dt

=∫ 1

0

|f(t)− g(t)| p(t) dt ≤ supx∈[0,1]

|f(x)− g(x)|∫ 1

0

p(t) dt

≤ kd(f, g),

where k =∫ 1

0p(t) dt < 1. �

Finally we will show that every metric space can be “completed” to becomea complete metric space and such “completion” is essentially unique.

Definition A.9.13 (Completion). Let (X, d) be a metric space. A complete metricspace X∗ is said to be a completion of X if X is a topological subspace of X∗ andif X = X∗ (i.e., if X is dense in X∗).

Remark A.9.14. Completion of a metric space can be defined in another way: acomplete metric space (X∗, d∗) is a completion of (X, d) if there exists an isom-etry ι : X → X∗ such that the image ι(X) is dense in X∗. In the proof ofTheorem A.9.15, we are actually using this idea: there X∗ is the family of Cauchysequences in X, and the points of X are naturally identified with the constantsequences.

Theorem A.9.15 (Completions of metric spaces). Every metric space (X, d) has acompletion. This completion is unique up to an isometry leaving X fixed.

Proof. Existence. We will construct a completion as a space of equivalence classesof Cauchy sequences in X. Thus, we will call Cauchy sequences (xn)∞n=1 and(x′n)∞n=1 equivalent if d(xn, x′n) → 0 as n → ∞. One can readily see that thisis an equivalence relation as in Definition A.2.6, and we define X∗ to be the spaceof equivalence classes of such Cauchy sequences. Space X∗ has a metric d∗ de-fined as follows. For x∗, y∗ ∈ X∗, pick some representatives (xn)∞n=1 ∈ x∗ and


(yn)∞n=1 ∈ y∗, and setd∗(x∗, y∗) := lim

n→∞ d(xn, yn). (A.7)

We first check that d∗ is a well-defined function on X∗, namely that the limit in(A.7) exists and that it is independent of the choice of representatives on equiv-alence classes x∗ and y∗. To check that the limit exists, we use the fact that xn

and yn are Cauchy sequences, so for n and m sufficiently large we can estimate

|d(xn, yn)− d(xm, ym)|= |d(xn, yn)− d(xn, ym) + d(xn, ym)− d(xm, ym)|≤ |d(xn, yn)− d(xn, ym)|+ |d(xn, ym)− d(xm, ym)|≤ d(yn, ym) + d(xn, xm),

and the latter goes to zero as n, m → ∞. It follows that the sequence of realnumbers (d(xn, yn))∞n=1 is a Cauchy sequence in R, and hence converges becauseR is complete by Theorem A.13.10 (which will be proved later).

Let us now show that d∗(x∗, y∗) is independent of the choice of representativesfrom x∗ and y∗. Let us take (xn)∞n=1, (x

′n)∞n=1 ∈ x∗ and (yn)∞n=1, (y

′n)∞n=1 ∈ y∗.

Then by a calculation similar to the one before we can show that

|d(xn, yn)− d(x′n, y′n)| ≤ d(xn, x′n) + d(yn, y′n),

which implies that limn→∞ d(xn, yn) = limn→∞ d(x′n, y′n).We now claim that (X∗, d∗) is a metric space. Non-degeneracy and symmetry

in Definition A.6.1 are straightforward. The triangle inequality for d∗ follows fromthat for d. Indeed, passing to the limit as n → ∞ in the inequality d(xn, zn) ≤d(xn, yn) + d(yn, zn), we get d∗(x∗, z∗) ≤ d∗(x∗, y∗) + d∗(y∗, z∗).

Next we will verify that X∗ is a completion of X. We first have to checkthat (X, d) is a topological subspace of (X∗, d∗). We observe that for every x ∈X its equivalence class contains the convergent constant sequence (xn = x)∞n=1,and hence any equivalent Cauchy sequence must be also convergent. Thus, theclass x∗ consists of all sequences (xn)∞n=1 convergent to x. Now, if x, y ∈ X and(xn)∞n=1 ∈ x∗, (yn)∞n=1 ∈ y∗, we have xn → x and yn → y as n → ∞, andhence d(x, y) = limn→∞ d(xn, yn) = d∗(x∗, y∗). Therefore, the mapping x �→ x∗

is an isometry from X to X∗ and hence X is a topological subspace of X∗ if weidentify it with its image under this isometry. Thus, in the sequel we will no longerdistinguish between X and its image in X∗.

We next show that X is dense in X∗. Let x∗ ∈ X∗, let ε > 0, and let(xn)∞n=1 ∈ x∗. Since xn is a Cauchy sequence, there is some N such that forall n, m > N we have d(xn, xm) < ε. Letting m → ∞, we get d∗(xn, x∗) =limm→∞ d(xn, xm) ≤ ε. Therefore, any neighbourhood of x∗ contains a point ofX, which means that X = X∗ by Proposition A.8.10.

Finally, we show that (X∗, d∗) is complete. First we observe that by theconstruction of X∗ any Cauchy sequence (xn)∞n=1 of points of X converges tox∗ ∈ X∗, for x∗ � (xn)∞n=1. Second, for any Cauchy sequence x∗n of points in


X∗ there is an equivalent sequence xn of points of X because X = X∗. Indeed,for every n there is a point xn ∈ X such that d∗(xn, x∗n) < 1

n . Sequence xn isthen a Cauchy sequence and by the first part of this argument it converges to itsequivalence class x∗ in X∗. Therefore, x∗n also converges to x∗ in (X∗, d∗).

Uniqueness. We want to show that if (X∗, d∗) and (X∗∗, d∗∗) are two comple-tions of X then there is a bijection f : X∗ → X∗∗ such that f(x) = x for all x ∈ X,and such that f(x∗) = x∗∗, f(y∗) = y∗∗ implies that d∗(x∗, y∗) = d∗∗(x∗∗, y∗∗).We define f in the following way. For x∗ ∈ X∗, in view of the density of X in X∗,there exists a sequence xn ∈ X such that xn → x∗ in (X∗, d∗). Therefore, xn is aCauchy sequence in X, and since X∗∗ is also a completion of X and is complete,it has some limit in X∗∗, so that xn → x∗∗ in (X∗∗, d∗∗). One can readily see thatthis x∗∗ is independent of the choice of sequence xn convergent to x∗. We definef by setting x∗∗ = f(x∗).

By construction it is clear that f(x) = x for all x ∈ X. Moreover, let xn →x∗ and yn → y∗ in (X∗, d∗) and let xn → x∗∗ and yn → y∗∗ in (X∗∗, d∗∗).Consequently,

d∗(x∗, y∗) = limn→∞ d∗(xn, yn) = lim

n→∞ d(xn, yn)

= limn→∞ d∗∗(xn, yn) = d∗∗(x∗∗, y∗∗),


A.10 Continuity and homeomorphisms

Recall that an expression like “(X, τ) is a topological space” is often abbreviatedby “X is a topological space”. In the sequel, to simplify notation, we may usethe same letter c for the closure operators of different topological spaces: thatis, if A ⊂ X and B ⊂ Y , c(A) is the closure in the topology of X, and c(B) isthe closure in the topology of Y . If needed, we shall express which topologies aremeant. In reading the following definition, recall how we have interpreted x ∈ c(A)as “x ∈ X is close to A ⊂ X”:

Definition A.10.1 (Continuous mappings). A mapping f : X → Y is continuousat point x ∈ X if

x ∈ c(A) =⇒ f(x) ∈ c (f(A))

for every A ⊂ X. A mapping f : X → Y is continuous if it is continuous at everypoint x ∈ X, i.e.,

f (c(A)) ⊂ c (f(A))

for every A ⊂ X. If precision is needed, we may emphasize the topologies involvedand, instead of mere continuity, speak specifically about (τX , τY )-continuity. Theset of continuous functions from X to Y is often denoted by C(X, Y ), with con-vention C(X) = C(X, R) (or C(X) = C(X, C)).

A.10. Continuity and homeomorphisms 47

Exercise A.10.2. Let c ∈ R. Let f, g : X → R be continuous, where we use theEuclidean metric topology on R. Show that the following functions X → R are thencontinuous: cf , f + g, fg, |f |, max{f, g}, min{f, g} (here, e.g., max{f, g}(x) :=max{f(x), g(x)} etc.). Moreover show that if g(x) = 0 then f/g is continuous atx ∈ X.

Exercise A.10.3. Let (X1, τ1) and (X2, τ2) be topological spaces. Show that amapping f : X1 → X2 is continuous at x ∈ X1 if and only if

∀V ∈ Vτ2(f(x)) ∃U ∈ Vτ1(x) : f(U) ⊂ V.

Exercise A.10.4. Let (X, dX) and (Y, dY ) be metric spaces, p ∈ X and f : X → Y .Show that the following conditions are equivalent:

1. f is continuous at p ∈ X (with respect to the metric topologies).2. ∀ε > 0 ∃δ > 0 ∀w ∈ X : dX(p, w) < δ ⇒ dY (f(p), f(w)) < ε.3. f(xk)→ f(p) whenever xk → p.

Theorem A.10.5. Let f : X → Y . Then f is continuous if and only if f−1(V ) ∈ τX

for every V ∈ τY .

Remark A.10.6. The continuity criterion here might be read as: “preimages of opensets are open”. Sometimes this condition is taken as the definition of continuity off . Equivalently, by taking complements, this means “preimages of closed sets areclosed”.

Proof. Let us assume that “preimages of closed sets are closed”. Then A′ =f−1(c(f(A))) is closed, and A ⊂ A′, so c(A) ⊂ c(A′) = A′. Hence

f(c(A)) ⊂ f(A′) ⊂ c(f(A)).

Property f(c(A)) ⊂ c(f(A)) means the continuity of f : X → Y .Conversely, let f : X → Y be continuous. Let A = f−1(c(B)), where B ⊂ Y .

Then f(c(A)) ⊂ c(f(A)) ⊂ c(c(B)) = c(B), so

c(A) ⊂ f−1(f(c(A))) ⊂ f−1(c(B)) = A.

Therefore c(A) = A, i.e., A is closed. �

Corollary A.10.7. Let f : X → Y , and let τY be a topology on Y . Then f−1(τY ) ={f−1(V ) | V ∈ τY } is a topology on X. Moreover, f is (τX , τY )-continuous if andonly if f−1(τY ) ⊂ τX .

Exercise A.10.8. Prove Corollary A.10.7. The topology f−1(τY ) is called the topol-ogy induced from τY by f . Show that the relative topology on a subset A ⊂ Xof a topological space X in Definition A.7.18 is induced by the identity mappingA→ X.


Definition A.10.9 (Induced topology). Let F be a family of mappings f : X → Y ,where (Y, τY ) is a topological space. Then⋂

f∈Ff−1(τY ) ⊂ P(X)

is the topology induced from τY by F .

Proposition A.10.10. Let X, Y, Z be topological spaces and let f : X → Y andg : Y → Z be continuous. Then g ◦ f : X → Z is continuous.

Proof. We will use Theorem A.10.5. Let U be open in Z. Then g−1(U) is openin Y and hence (g ◦ f)−1(U) = f−1(g−1(U)) is open in X, implying that g ◦ f iscontinuous. �Exercise A.10.11. Prove Proposition A.10.10 directly from Definition A.10.1.

Definition A.10.12 (Homeomorphisms and topological equivalence). A bijectivemapping f : X → Y is a homeomorphism if both f and f−1 are continuous. Inthis case we say that the corresponding topological spaces (X, τX) and (Y, τY ) arehomeomorphic. Homeomorphic spaces are also called topologically equivalent. Aproperty which holds in all topologically equivalent spaces is called a topologicalproperty.

Example. Any two open intervals in R are topologically equivalent. For a set X,properties “X has five elements” or “all subsets of X are open” are topologicalproperties.Remark A.10.13. A homeomorphism is a topological isomorphism: homeomorphicspaces are topologically the same. As the saying goes, a topologist is a person whodoes not know the difference between a doughnut and a coffee cup. Let us denotebriefly X ≈ Y when (X, τX) and (Y, τY ) are homeomorphic. It is easy to see thatwe have an equivalence

X ≈ X,

X ≈ Y =⇒ Y ≈ X,

X ≈ Y and Y ≈ Z =⇒ X ≈ Z.

Analogously, there is a concept of metric space isomorphisms: a bijective map-ping f : X → Y between metric spaces (X, dX), (Y, dY ) is called an isometricisomorphism if dY (f(a), f(b)) = dX(a, b) for every a ∈ X and b ∈ Y .Example. The reader may check that (x �→ x/(1 + |x|)) : R ≈ (−1, 1). Usingalgebraic topology, one can prove that Rm ≈ Rn if and only if m = n (this is nottrivial!).Example. Any isometric isomorphism is a homeomorphism. Clearly the unboundedR and the bounded (−1, 1) are not isometrically isomorphic. An orthogonal linearoperator A : Rn → Rn is an isometric isomorphism, when Rn is endowed with the

A.11. Compact topological spaces 49

Euclidean norm. The forward shift operator on �p(Z) is an isometric isomorphism,but the forward shift operator on �p(N) is only a non-surjective isometry.

Exercise A.10.14. Let (X, dX) and (Y, dY ) be metric spaces. Recall that f : X → Yis continuous if and only if

∀a ∈ X ∀ε > 0 ∃δ > 0 ∀b ∈ X : dX(a, b) < δ =⇒ dY (f(a), f(b)) < ε.

A function f : X → Y is uniformly continuous if

∀ε > 0 ∃δ > 0 ∀a, b ∈ X : dX(a, b) < δ =⇒ dY (f(a), f(b)) < ε,

and Lipschitz-continuous if

∃C <∞ ∀a, b ∈ X : dY (f(a), f(b)) ≤ C dX(a, b).

Prove that Lipschitz-continuity implies uniform continuity, and that uniform con-tinuity implies continuity; give examples showing that these implications cannotbe reversed.

Theorem A.10.15. Two metrics d1, d2 on a set X are equivalent if and only if theidentity mapping from (X, d1) to (X, d2) is a homeomorphism.

Proof. Let id(x) = x be the identity mapping from (X, d1) to (X, d2). Sinceid−1(U) = U for any set U , the forward implication follows from the definition ofa continuous mapping and that of equivalent metrics. On the other hand, supposethe identity map is a homeomorphism. Again, since id−1(U) = U we get thatevery set open in (X, d2) is open in (X, d1) since id is continuous. The converse istrue since id−1 is also continuous. �

A.11 Compact topological spaces

Eventually, we will mainly concentrate on compact Hausdorff spaces, but in thissection we deal with more general classes of topological spaces.

Definition A.11.1 (Coverings). Let X be a set and K ⊂ X. A family U ⊂ P(X)is called a cover of K if

K ⊂⋃U ;

if the cover U is a finite set, it is called a finite cover. A cover U of K ⊂ X has asubcover U ′ ⊂ U if U ′ itself is a cover of K. In a topological space, an open coverrefers to a cover consisting of open sets.

Definition A.11.2 (Compact sets). Let (X, τ) be a topological space. A subsetK ⊂ X is compact (more precisely τ -compact) if every open cover of K has afinite subcover. We say that (X, τ) is a compact space if X itself is τ -compact.A topological space is locally compact if each of its points has a neighbourhoodwhose closure is compact.


Remark A.11.3. Briefly, in a topological space (X, τ), K ⊂ X is compact if andonly if the following holds: given any family U ⊂ τ such that K ⊂ ⋃U , thereexists a finite subfamily U ′ ⊂ U such that K ⊂ ⋃U ′.Remark A.11.4. Let us consider a tongue-in-cheek geographical-zoological ana-logue for compactness: In a space or universe (X, τ), let non-empty open sets cor-respond to territories of angry animals; recall the metaphor that a point x ∈ U ∈ τis “far away from (i.e., not close to) the set X \ U”. Compactness of an islandK ⊂ X means that any given territorial cover U has a finite subcover U ′: alreadya finite number of beasts governs the whole island.

Example.

1. If τ1 and τ2 are topologies of X, τ1 ⊂ τ2, and (X, τ2) is a compact space then(X, τ1) is a compact space.

2. (X, {∅, X}) is a compact space.

3. If |X| =∞ then (X,P(X)) is not a compact space, but it is locally compact.Clearly any space with a finite topology is compact. Even though a compacttopology can be of any cardinality, it is in a sense “not far away from beingfinite”.

4. A metric space is compact if and only if it is sequentially compact (i.e., everysequence contains a converging subsequence, see Theorem A.13.4).

5. A subset X ⊂ Rn is compact if and only if it is closed and bounded (Heine–Borel Theorem A.13.7).

6. Theorem B.4.21 due to Frigyes Riesz asserts that a closed ball in a normedvector space over C (or R) is compact (i.e., the space is locally compact) ifand only if the vector space is finite-dimensional.

Of course, we may work with a complemented version of the compactnesscriterion in terms of closed sets:

Proposition A.11.5 (Finite intersection property). A topological space X is com-pact if and only if the closed sets in X have the finite intersection property, whichmeans that any collection {Fα}α of closed sets in X with ∩αFα = ∅ has a finitesubcollection {Fi}n

i=1 ⊂ {Fα}α such that ∩ni=1Fi = ∅.

Proof. Defining Uα = X\Fα, we observe that condition ∩αFα = ∅ means that{Uα}α is an open covering of X. The condition that X is compact means that anysuch covering has some finite subcollection {Ui}n

i=1 with ∪ni=1Ui = X, which in

turn means that ∩ni=1Fi = ∅. �

Proposition A.11.6 (Characterisation of compact subspaces). Let (X, τ) be a topo-logical space and let Y ⊂ X. Topological subspace (Y, τY ) is compact if and only ifevery collection {Uα}α∈I of sets Uα ∈ τ with

⋃α∈I Uα ⊃ Y has a finite subcollec-

tion that covers Y .

A.11. Compact topological spaces 51

Proof. Assume that (Y, τY ) is compact and let {Uα}α∈I be a collection of setsUα ∈ τ with

⋃α∈I Uα ⊃ Y . Then the collection {Uα ∩ Y }α∈I is an open cover of

(Y, τY ) and hence has a finite subcover {Ui ∩ Y }ni=1. The corresponding collection

{Ui}ni=1 is a finite subcollection of {Uα}α∈I that covers Y .Conversely, let {Vα}α∈I ⊂ τY be an open cover of Y . Then there exist sets

Uα ∈ τ such that Vα = Uα∩Y . Consequently, {Uα}α∈I ⊂ τ is a cover of Y , and byassumption it has a finite subcollection {Ui}n

i=1 that covers Y . The correspondingcollection {Vi}n

i=1 is then a finite open cover of Y . �Exercise A.11.7. Show that a finite set in a topological space is compact.

Exercise A.11.8. Let x ∈ Rn and r > 0. Show that the open ball Br(x) ⊂ Rn isnot compact in the Euclidean metric topology.

Exercise A.11.9. Prove that a union of two compact sets is compact.

Proposition A.11.10. Let (X, τ) be a topological space, K ⊂ X compact and S ⊂ Xclosed. Then K ∩ S is compact.

Proof. Let U be an open cover of K ∩S. Then U ∪{X \S} is an open cover of K,thus having a finite subcover U ′. Then U ′∩U ⊂ U is a finite subcover of K∩S. �Proposition A.11.11 (Some properties of compact sets). We have the followingproperties:

(1) A closed subset of a compact topological space is compact.(2) A compact subset of a metric space is bounded (and closed).

Proof. To prove (1), let Y be a closed subset of a compact topological space (X, τ).Let {Uα}α∈I ⊂ τ be an open cover of Y . Since Y is closed, its complement X\Yis open, and collection {X\Y, Uα}α∈I is an open cover of X. Since X is compact,it has a finite subcover and since X\Y is disjoint from Y , removing X\Y (ifnecessary) from this subcover we obtain a finite subcover of Y .

To prove (2), let Y be a compact subspace of a metric space (X, d). A col-lection of unit balls {B1(y)}y∈Y is an open cover of Y , and hence it has a finitesubcover, say {B1(yi)}n

i=1. Applying Proposition A.6.14 we obtain that Y mustbe bounded. �Proposition A.11.12. Let X be a compact space and f : X → Y continuous. Thenf(X) ⊂ Y is compact.

Proof. Let V be an open cover of f(X). Then U := {f−1(V ) | V ∈ V} is an opencover of X, thus having a finite subcover U ′ = {f−1(V ) | V ∈ V ′}, where V ′ ⊂ V isa finite collection. Then f(X) is covered by V ′ ⊂ V: if y ∈ f(X) then y = f(x) forsome x ∈ X, so x ∈ f−1(V0) for some V0 ∈ V ′, so y=f(x)∈f(f−1(V0))⊂V0. �Corollary A.11.13. Let f : X → R be a continuous mapping from a compacttopological space X to R equipped with the Euclidean topology. Then f(X) is abounded subset of R.


Theorem A.11.14 (Product of compact spaces is compact). Let X, Y be compacttopological spaces. Then X × Y in the product topology is compact.

Proof. Let C = {Wα}α∈I be an open cover of X × Y in the product topology. Inparticular, it means that each Wα is a union of “rectangles” of the form U × Ywhere U and V are open in X and Y , respectively. For every (x, y) there is a“rectangle” Uy

x × V yx and the corresponding set W y

x such that

(x, y) ∈ Uyx × V y

x ⊂W yx ∈ C.

For every x ∈ y, collection {V yx }y∈Y is an open covering of Y which then must

have some finite subcover, which we denote by {V yi(x)x }n(x)

i=1 . Set Ux = ∩n(x)i=1 U

yi(x)x

is open in X and collection {W yi(x)x }n(x)

i=1 is a cover of Ux × Y .In turn, the collection {Ux}x∈X is an open cover of X which then must

have some finite subcover, which we denote by {Uxj}m

j=1. We now claim that the

collection {W yi(xj)xj }ij ⊂ C is a finite cover of X×Y . Indeed, for every (x, y) ∈ X×Y

there is some Uxj that contains x, and then there is some Vyj(xj)xj that contains y,

implying that (x, y) ∈Wyj(xj)xj . �

Lemma A.11.15. Let (X, τ) be a compact space and S ⊂ X infinite. Then S hasan accumulation point.

Proof. Recall that x ∈ X is an accumulation point of S ⊂ X if

∀U ∈ τ : x ∈ U =⇒ (S ∩ U) \ {x} = ∅.Suppose S ⊂ X has no accumulation points, i.e.,

∀x ∈ X ∃Ux ∈ τ : x ∈ Ux and S ∩ Ux ⊂ {x}.Now U = {Ux : x ∈ X} is an open cover of X, having a finite subcover U ′ ⊂ Uby compactness. Then

S = S ∩(⋃

U ′)

=⋃

Ux∈U ′(S ∩ Ux) .

Here the union is finite, and S ∩ Ux ⊂ {x} in each case. Thus S is finite. �

A.12 Compact Hausdorff spaces

Next we are going to witness how beautiful compact Hausdorff topologies are.Among topological spaces, Hausdorff spaces are those where points are distinc-tively separated by open neighbourhoods; this happens especially in metric topol-ogy. Roughly, Hausdorff spaces have enough open sets to distinguish between anytwo points, while compact spaces “do not have too many open sets”. Combiningthese two properties, compact Hausdorff spaces form a useful class of topologicalspaces.

A.12. Compact Hausdorff spaces 53

Definition A.12.1 (Hausdorff spaces). A topological space (X, τ) is called a Haus-dorff space if for each a, b ∈ X, where a = b, there exists U, V ∈ τ such that a ∈ U ,b ∈ V and U ∩ V = ∅.Example.

1. If τ1 and τ2 are topologies of X, τ1 ⊂ τ2, and (X, τ1) is a Hausdorff spacethen (X, τ2) is a Hausdorff space.

2. (X,P(X)) is a Hausdorff space.3. If X has more than one point and τ = {∅, X} then (X, τ) is not Hausdorff.4. Clearly any metric space (X, d) is a Hausdorff space; if x, y ∈ X, x = y, then

Br(x) ∩Br(y) = ∅, when r ≤ d(x, y)/2.5. The distribution spaces D′(Rn), S ′(Rn) and E ′(Rn) are non-metrisable Haus-

dorff spaces.

Theorem A.12.2. In Hausdorff spaces, we have the following properties:

(1) Every convergent sequence has a unique limit.(2) All finite sets are closed.(3) Every topological subspace is also Hausdorff.(4) A compact subspace of a Hausdorff space is closed.(5) A subset of a compact Hausdorff space is compact if and only if it is closed.

Proof. To prove (1), let xn be a sequence such that xn → p and xn → q asn → ∞. Assume p = q. Then there exist open sets U, V such that p ∈ U, q ∈ Vand U ∩V = ∅. Consequently, there are numbers N and M such that for all n > Nwe have xn ∈ U and for all n > M we have xn ∈ V , which yields a contradiction.

To prove (2), in view of property (C3) of Corollary A.7.14 it is enough toshow that one-point sets {x} in a Hausdorff topological space X are closed. Forevery y ∈ X\{x} there exist open disjoint sets Uy � x and Vy � y. Since x ∈ Vy itfollows that Vy ⊂ X\{x} and hence X\{x} =

⋃y∈X\{x} Vy, implying that X\{x}

is open.To prove (3), let Y be a subset of a Hausdorff topological space (X, τ) and

let τY be the relative topology on Y . Let a, b ∈ Y be such that a = b. Since (X, τ)is Hausdorff there exist open disjoint sets U, V ∈ τ such that a ∈ U and b ∈ V .Consequently, a ∈ U ∩ Y ∈ τY and b ∈ V ∩ Y ∈ τY , and U ∩ Y and V ∩ Y aredisjoint, implying that (Y, τY ) is Hausdorff.

To prove (4), let Y be a compact subspace of a topological space X. If Y = Xthe statement is trivial. Assuming that Y = X, let us take some x ∈ X\Y . Thenfor every y ∈ Y there are open disjoint sets Uy � x and Vy � y. The collection{Vy}y∈Y is a covering of Y , and hence by Proposition A.11.6 there is a finitecollection Vy1 , . . . , Vyn

still covering Y . Then set Ux = ∩ni=1Uyi

is open, x ∈ Ux,and Ux ∩ Y = ∅. Therefore, X\Y = ∪x∈X\Y Ux is open and hence Y is closed.

Statement (5) follows immediately from (4) and property (1) of Proposi-tion A.11.11. �


Theorem A.12.3 (Hausdorff property is a topological property). Let f : X1 → X2

be an injective and continuous mapping between topological spaces (X1, τ1) and(X2, τ2). If (X2, τ2) is Hausdorff then (X1, τ1) is also Hausdorff. Consequently,the Hausdorff property is a topological property.

Proof. Let x, y ∈ X1 be such that x = y. Since f is injective, we have f(x) = f(y)and since (X2, τ2) is Hausdorff there exist open disjoint sets U, V ∈ τ2 such thatf(x) ∈ U and f(y) ∈ V . Since f is continuous, sets f−1(U) and f−1(V ) are opendisjoint neighbourhoods of x and y in X1, respectively, implying that (X1, τ1)is also Hausdorff. That the Hausdorff property is a topological property followsimmediately from this. �Exercise A.12.4 (Product of Hausdorff spaces). If (X1, τ1) and (X1, τ1) are Haus-dorff topological spaces, show that (X1 × X2, τ1 ⊗ τ2) is a Hausdorff topologicalspace.

Theorem A.12.5. Let X be a Hausdorff space, A,B ⊂ X compact subsets, andA ∩ B = ∅. Then there exist open sets U, V ⊂ X such that A ⊂ U , B ⊂ V , andU ∩ V = ∅.

Proof. The proof is trivial if A = ∅ or B = ∅. So assume x ∈ A and y ∈ B. SinceX is a Hausdorff space and x = y, we can choose neighbourhoods Uxy ∈ V(x) andVxy ∈ V(y) such that Uxy ∩ Vxy = ∅. The collection P = {Vxy | y ∈ B} is an opencover of the compact set B, so that it has a finite subcover

Px = {Vxyj | 1 ≤ j ≤ nx} ⊂ P

for some nx ∈ N. Let

Ux :=nx⋂j=1

Uxyj.

Now O = {Ux | x ∈ A} is an open cover of the compact set A, so that it has afinite subcover

O′ = {Uxi | 1 ≤ i ≤ m} ⊂ O.

Then define

U :=⋃O′, V :=

m⋂i=1

⋃Pxi

.

It is an easy task to check that U and V have the desired properties. �Corollary A.12.6. Let X be a compact Hausdorff space, x ∈ X, and W ∈ V(x).Then there exists U ∈ V(x) such that U ⊂W .

Proof. Now {x} and X \ W are closed sets in a compact space, thus they arecompact. Since these sets are disjoint, there exist open disjoint sets U, V ⊂ Xsuch that x ∈ U and X \W ⊂ V ; i.e., x ∈ U ⊂ X \ V ⊂ W. Hence x ∈ U ⊂ U ⊂X \ V ⊂W . �

A.12. Compact Hausdorff spaces 55

Proposition A.12.7. Let (X, τX) be a compact space and (Y, τY ) a Hausdorff space.Any bijective continuous mapping f : X → Y is a homeomorphism.

Proof. Let U ∈ τX . Then X \U is closed, hence compact. Consequently, f(X \U)is compact, and due to the Hausdorff property f(X \ U) is closed. Therefore(f−1)−1(U) = f(U) is open. �

Corollary A.12.8. Let X be a set with a compact topology τ2 and a Hausdorfftopology τ1. If τ1 ⊂ τ2 then τ1 = τ2.

Proof. The identity mapping (x �→ x) : X → X is a continuous bijection from(X, τ2) to (X, τ1). �

A more direct proof of the corollary. Let U ∈ τ2. Since (X, τ2) is compact andX \ U is τ2-closed, X \ U must be τ2-compact. Now τ1 ⊂ τ2, so that X \ U isτ1-compact. (X, τ1) is Hausdorff, implying that X \ U is τ1-closed, thus U ∈ τ1;this yields τ2 ⊂ τ1. �

Definition A.12.9 (Separating points). A family F of mappings X → C is saidto separate the points of the set X if there exists f ∈ F such that f(x) = f(y)whenever x = y.

Definition A.12.10 (Support). The support of a function f ∈ C(X) is the set

supp(f) := {x ∈ X | f(x) = 0}.

Let f ∈ C(X) such that 0 ≤ f ≤ 1. Notations

K ≺ f, f ≺ U

mean, respectively, that K ⊂ X is compact and χK ≤ f , and that U ⊂ X is openand supp(f) ⊂ U .

Theorem A.12.11 (Urysohn’s Lemma). Let X be a compact Hausdorff space, A,B ⊂ X closed non-empty sets, A ∩ B = ∅. Then there exists f ∈ C(X) andU ⊂ X \A such that B ≺ f ≺ U . Especially, we find f such that

0 ≤ f ≤ 1, f(A) = {0}, f(B) = {1}.

Proof. The set Q ∩ [0, 1] is countably infinite; let φ : N→ Q ∩ [0, 1] be a bijectionsatisfying φ(0) = 0 and φ(1) = 1. Choose open sets U0, U1 ⊂ X such that

A ⊂ U0 ⊂ U0 ⊂ U1 ⊂ U1 ⊂ X \B.

Then we proceed inductively as follows: Suppose we have chosen open sets Uφ(0),Uφ(1), . . ., Uφ(n) such that

φ(i) < φ(j)⇒ Uφ(i) ⊂ Uφ(j).


Let us choose an open set Uφ(n+1) ⊂ X such that

φ(i) < φ(n + 1) < φ(j)⇒ Uφ(i) ⊂ Uφ(n+1) ⊂ Uφ(n+1) ⊂ Uφ(j)

whenever 0 ≤ i, j ≤ n. Let us define

r < 0⇒ Ur := ∅, s > 1⇒ Us := X.

Hence for each q ∈ Q we get an open set Uq ⊂ X such that

∀r, s ∈ Q : r < s⇒ Ur ⊂ Us.

Let us define a function f : X → [0, 1] by

f(x) := inf{r : x ∈ Ur}.

Clearly 0 ≤ f ≤ 1, f(A) = {0} and f(B) = {1}.Let us prove that f is continuous. Take x ∈ X and ε > 0. Take r, s ∈ Q such

thatf(x)− ε < r < f(x) < s < f(x) + ε;

then f is continuous at x, since x ∈ Us \ Ur and for every y ∈ Us \ Ur we have|f(y)− f(x)| < ε. Thus f ∈ C(X). �Corollary A.12.12. Let X be a compact space. Then C(X) separates the points ofX if and only if X is Hausdorff.

Exercise A.12.13. Prove the previous corollary.

Definition A.12.14 (Partition of unity). A partition of unity on K ⊂ X in atopological space (X, τ) is a family F = {φj : X → [0, 1] | j ∈ J} of continuousfunctions such that

χK ≤∑j∈J

φj ≤ 1,

where the sum is required to be locally finite: for each x ∈ X there exists U ∈ V(x)such that supp(φj) ⊂ X \ U for all but finitely many φj ∈ F . Moreover, if nowφj ≺ Uj for all j ∈ J , where U = {Uj : j ∈ J} is an open cover of X, then F iscalled a partition of unity on K subordinate to U .

Corollary A.12.15 (Partition of unity). Let U be an open cover of a compact setK ⊂ X in a Hausdorff space (X, τ). Then there exists a partition of unity on Ksubordinate to U .

Proof. Assume the non-trivial case K = ∅. Take a finite subcover U ′ = {Uj | 1 ≤j ≤ n} ⊂ U . For x ∈ K, take j ∈ {1, . . . , n} such that x ∈ Uj ; then chooseVx ∈ V(x) such that Vx ⊂ Uj . Then O = {Vx | x ∈ K} is an open cover of K, thushaving a finite subcover O′ ⊂ O. Let

Kj :=⋃{V ∈ O′ : V ⊂ Uj}.

A.13. Sequential compactness 57

Urysohn’s Lemma provides functions fj ∈ C(X) satisfying Kj ≺ fj ≺ Uj . Againby Urysohn’s Lemma, there exists g ∈ C(X) such that

n⋃j=1

Kj ≺ g ≺{

x ∈ X :n∑

k=1

fk(x) > 0

}.

Notice that K ⊂ ⋃nj=1 Kj . Let

φj := fj/(1− g +n∑

k=1

fk).

Then {φj ∈ C(X)}nj=1 provides a desired partition of unity. �

Exercise A.12.16. In a compact metric space (X, d), Urysohn’s Lemma is mucheasier to obtain: When A,B ⊂ X are closed and non-empty such that A ∩B = ∅,define f : X → R by

f(x) := min{

1,dist(A, {x})dist(A,B)

}.

Show that f is continuous, 0 ≤ f ≤ 1, f(A) = {0} and f(B) = 1.

Definition A.12.17 (Equicontinuity). Let X be a topological space. A family F ofmappings f : X → C is called equicontinuous at p ∈ X if for every ε > 0 thereexists a neighbourhood U ⊂ X of p such that |f(x) − f(p)| < ε whenever f ∈ Fand x ∈ U .

Exercise A.12.18. Prove the following Theorem A.12.19. (Hint: a bounded se-quence of numbers has a convergent subsequence. . . )

Theorem A.12.19 (Arzela-Ascoli Theorem). Let K ⊂ Rn be compact. For eachj ∈ Z+, let fj : K → C be continuous, and assume that F = {fj | j ∈ Z+} isequicontinuous on K. If F is bounded, i.e.,

supx∈K,j∈Z+

|fj(x)| <∞,

then there is a subsequence {fjk| k ∈ Z+} that converges uniformly on K.

A.13 Sequential compactness

In this section, a metric space (X, d) is endowed with its canonical metric topol-ogy τd.

Proposition A.13.1 (Closed and bounded if compact in metric). Let (X, d) be ametric space, and let K ⊂ X be compact. Then K is closed and bounded.


Proof. Let us assume K = ∅, to avoid a triviality. Let x0 ∈ X. Then U ={Bk(x0) | k ∈ Z+} is an open cover of K. Due to compactness of K, there isa subcover U ′ = {Bk(x0) | k ∈ S}, where S ⊂ Z+ is finite. Now

K ⊂⋃U ′ =

⋃k∈S

Bk(x0) = Bmax(S)(x0).

Therefore diam(K) ≤ 2 max(S) <∞, so K is bounded.We have to prove that K is closed. Let x ∈ X \K. Then

V :={Bd(x,y)/2(y) | y ∈ K

}is an open cover of K. By compactness, there is a finite subcover

V ′ ={Bd(x,yj)/2(yj)

}n

j=1.

Let r := min {d(x, yj)/2}nj=1. Then

Br(x) ∩K ⊂n⋃

j=1

(Br(x) ∩Bd(x,yj)/2(yj)

)= ∅,

so x ∈ cd(K). Thereby K = cd(K) is closed. �

Exercise A.13.2. Give an example of a bounded non-compact metric space.

Definition A.13.3 (Sequential compactness). A metric space is sequentially com-pact if each of its sequences has a converging subsequence. That is, given a sequence(xk)∞k=1 in a sequentially compact metric space (X, d), there is a converging se-quence (xkj )

∞j=1, where kj+1 > kj ∈ Z+ for each j ∈ Z+.

Theorem A.13.4 (Compact ⇔ sequentially compact in metric spaces). A metricspace (X, d) is compact if and only if it is sequentially compact.

Proof. Let us assume that X = ∅ is compact. Take a sequence (xk)∞k=1 in X. Ifthe set {xk : k ∈ Z+} is finite, there exists y ∈ X such that y = xk for infinitelymany k ∈ Z+. Then a desired convergent subsequence is given by (y, y, y, . . .). Nowassume that the set S := {xk : k ∈ Z+} is infinite, so it has an accumulation pointp ∈ X by Lemma A.11.15. Take k1 ∈ Z+ such that xk1 ∈ S ∩ B1(p). Inductively,take kj+1 > kj ∈ Z+ such that xkj+1 ∈ S∩B1/j(p). Then d(p, xkj+1) < 1/j →j→∞0, so xkj

→j→∞ p. We have proven that a compact metric space is sequentiallycompact.

Now let (X, d) be sequentially compact. We want to show that its metrictopology is compact. Take an open cover U = {Uα : α ∈ A} of X. We claim that

∃ε0 > 0 ∀x ∈ X ∃α ∈ A : Bε0(x) ⊂ Uα. (A.8)


Let us prove this by deducing a contradiction from the logically negated assump-tion

∀ε > 0 ∃x ∈ X ∀α ∈ A : Bε(x) ⊂ Uα.

This would especially imply

∀k ∈ Z+ ∃xk ∈ X ∀α ∈ A : B1/k(xk) ⊂ Uα.

This gives us a sequence (xk)∞k=1, which by sequential compactness has a subse-quence (xkj )

∞j=1 converging to a point p ∈ X. Since U covers X, we have p ∈ Uαp

for some αp ∈ A. Since Uαp is open, Bε(p) ⊂ Uαp for some ε > 0. But for largeenough j,

B1/kj(xkj ) ⊂ Bε(p) ⊂ Uαp .

This is a contradiction, so (A.8) must be true. Now we claim that

X can be covered with finitely many open balls of radius ε0. (A.9)

What happens if (A.9) is not true? Then take x1 ∈ X, and inductively

xk+1 ∈ X \k⋃

j=1

Bε0(xj) = ∅,

where the non-emptiness of the set is due to the counter-assumption.Now d(xj , xk) ≥ ε0 > 0 if j = k, so the sequence (xk)∞k=1 does not have

a convergent subsequence. But this would contradict the sequential compactness.Hence (A.9) must be true. �

Exercise A.13.5. Think why the compactness of X follows from (A.8) and (A.9).

Exercise A.13.6. Show that a compact metric space is complete.

Corollary A.13.7 (Heine–Borel Theorem). Let Rn be endowed with its Euclideantopology. Then K ⊂ Rn is compact if and only if it is closed and bounded.

Proof. In any metric topology, compactness implies closedness and boundedness,see Proposition A.13.1. So let S ⊂ Rn be non-empty, closed and bounded. Weshall prove that it is sequentially compact. Take a sequence (xk)∞k=1 in S. Byboundedness, there exist a, b ∈ R such that S ⊂ [a, b]n =: Q1. That is, Q1 is aclosed cube of sidelength b− a.

Now we chop Q1 inductively into pieces. When the cube Qj has been cho-sen, we decompose Qj “dyadically” to a union of 2n cubes Qj+1,m (here m ∈{1, . . . , 2n}), whose interiors are disjoint and whose sidelengths are 2−j(b − a).Choose Qj ∈ {Qj+1,m : j ∈ {1, . . . , 2n}} such that xk ∈ Qj+1 for infinitely manyk ∈ Z+.


We construct the convergent subsequence (xkj)∞j=1 inductively. Let k1 := 1.

Take kj+1 > kj ∈ Z+ such that xkj+1 ∈ Qj+1. Now (xkj)∞j=1 is a Cauchy sequence,

because {Q1 ⊃ Q2 ⊃ Q3 ⊃ · · · ⊃ Qj ⊃ Qj+1 ⊃ · · · ,diam(Qj+1) =

√n 2−j(b− a)→j→∞ 0.

Due to the completeness of Rn, the Cauchy sequence (xkj)∞j=1 of S ⊂ Rn converges

to a point p ∈ Rn. But p ∈ S, because S is closed. Thus S is sequentially compact.�

Corollary A.13.8. Let (X, τ) be a compact topological space and f : X → R con-tinuous. Then there exist max(f(X)),min(f(X)) ∈ R.

Proof. Assume that X = ∅. By Proposition A.11.12, f(X) ⊂ R is compact. Bythe Heine–Borel Theorem A.13.7, equivalently f(X) ⊂ R is closed and bounded.Thereby sup(f(X)), inf(f(X)) ∈ f(X). �

We note that the Heine–Borel theorem can also be proved without referringto the sequential compactness. For simplicity, we show this in the one-dimensionalcase.

Theorem A.13.9 (Heine-Borel Theorem in 1D). Closed intervals [a, b] are compactin R in the Euclidean topology.

Proof. We will assume a < b since otherwise the statement is trivial. For an opencovering C = {Uα}α∈I of [a, b] let S ⊂ [a, b] be defined by

S = {x ∈ [a, b] : [a, x] can be covered by finitely many sets from C}.

The statement of the theorem will follow if we show that b ∈ S. Since S = 0 inview of a ∈ S and since S ⊂ [a, b] is bounded, we can define c = sup S so thatc ∈ [a, b]. The statement of the theorem will follow if we show that c ∈ S and thatc = b.

To show that c ∈ S, we observe that since c ∈ [a, b], there is some set Uc ∈ Csuch that c ∈ Uc. Since Uc is open, there is some ε > 0 such that (c−ε, c] ⊂ Uc. Atthe same time, since c− ε < c = sup S, the closed interval [a, c− ε] can be coveredby finitely many sets from C by the definition of S and c. Consequently, adding Uc

to this finite collection of sets from C we obtain a finite covering of [a, c], implyingthat c ∈ S.

To show that c = b, let us assume that c < b. As before, let Uc be such thatc ∈ Uc ∈ C. Since Uc is open and c < b, there is ε > 0 such that [c, c + 2ε) ⊂ Uc.Since c ∈ S, closed interval [a, c] can be covered by finitely many sets from C, andadding Uc to this finite collection we obtain a finite covering of [a, c + ε] which isa contradiction with c = sup S. �Theorem A.13.10 (R is complete). The real line R with the Euclidean metric iscomplete.


Proof. Let xn be a Cauchy sequence in R. By Lemma A.9.2, (2), the set {xn}∞n=1

is bounded, i.e., there are some a, b ∈ R such that {xn}∞n=1 ⊂ [a, b]. By the Heine–Borel Theorem in Corollary A.13.7 or in Theorem A.13.9 the interval [a, b] iscompact, and by Exercise A.13.6 it must be complete. Therefore, xn must have aconvergent subsequence, and since it is a Cauchy sequence, the whole sequence isconvergent by Lemma A.9.2, (3). �

Corollary A.13.11 (Rn is complete). The space Rn is complete with respect to anyof the Lipschitz equivalent metrics dp, 1 ≤ p ≤ ∞.

Proof. Since all metrics dp are Lipschitz equivalent, it is enough to take one, e.g.,d∞. Writing xk = (x(1)

k , . . . , x(n)k ) and d∞(xk, xl) = max1≤i≤n |x(i)

k − x(i)l |, we

have that d∞(xk, xl) < ε implies |x(i)k − x

(i)l | < ε for all i = 1, . . . , n. Thus, if

xk ∈ Rn is a Cauchy sequence in Rn, it follows that x(i)k is a Cauchy sequence

in R for all i, and hence it has a limit, say x(i), for all i, by Theorem A.13.10.Writing x = (x(1), . . . , x(n)), we claim that xk → x as k → ∞. Indeed, let ε > 0.Then for all i there is a number Ni such that k > Ni implies |x(i)

k − x(i)| < ε.Therefore, for k > max1≤i≤n Ni, we have |x(i)

k − x(i)| < ε for all i, which meansthat d∞(xk, x) < ε. �

Alternative proof of Theorem A.13.4. We now state several results of indepen-dent importance that will give another proof of Theorem A.13.4.

Lemma A.13.12 (Lebesgue’s covering lemma). Let C be an open covering of asequentially compact metric space (X, d). Then there is ε > 0 such that every ballwith radius ε is contained in some set from the covering C. Such ε is called aLebesgue number of the covering C.

Proof. Suppose that no such ε > 0 exists. It means that for every n ∈ N thereis a ball B1/n(xn) which is not contained in any set from C. Let (xnj )

∞j=1 be a

convergent subsequence of (xn)∞n=1 with some limit x ∈ X, so that xnj→ X as

j →∞. Let U ∈ C be a set in C containing x. Since U is open, there is some δ > 0such that B2δ(x) ⊂ U . Now let N be one of indices nj such that d(xN , x) < δand such that 1

N < δ. We claim that B1/N (xN ) ⊂ B2δ(x) which would be acontradiction with our choice of the sequence xn and the fact that B2δ(x) ⊂ U .Indeed, if y ∈ B1/N (xN ), we have

d(y, x) ≤ d(y, xN ) + d(xN , x) <1N

+ δ < 2δ,

which means that y ∈ B2δ(x), completing the proof. �

Lemma A.13.13 (Totally bounded metric spaces). Let X be a sequentially compactmetric space. Then X is totally bounded, which means that for every ε > 0 thereare finitely many balls in X with radius ε that cover X.


Proof. Suppose that there is ε > 0 such that no finitely many balls in X with radiusε cover X. We will now construct a sequence of points in X with no convergentsubsequence. Let x1 ∈ X be an arbitrary point. Let x2 be any point in X\Bε(x1).Inductively, suppose we have points x1, . . . , xn ∈ X such that xj ∈ X\∪j−1

i=1 Bε(xi).Since the collection {Bε(xi)}n

i=1 does not cover X, we can always choose some xn ∈X\ ∪n−1

i=1 Bε(xi). All points in this sequence have the property that d(xn, xk) ≥ εfor any n, k ∈ N which means that sequence (xn)∞n=1 can not have any convergentsubsequence. �

Exercise A.13.14. In general, a metric space is said to be totally bounded if

∀ε > 0 ∃ {xj | j ∈ {1, . . . , nε}} ⊂ X : X =nε⋃

j=1

Bε(xj).

Show that a metric space (X, d) is compact if and only if it is bounded and totallybounded.

Alternative proof of Theorem A.13.4. Let (X, d) be a metric space. First we willprove that if X is compact it is sequentially compact. Let (xn)∞n=1 be a sequenceof points in x. Define An = {xn, xn+1, xn+2, . . . }, so that A1 = {xn}∞n=1. LetFn = An. Clearly Fn is a closed set and the intersection of any finite number ofsets Fn is non-empty since it contains AN for some N . Since X is compact, by thefinite intersection property in Proposition A.11.5 we have ∩∞n=1Fn = ∅. Let nowx ∈ ∩∞n=1Fn, so that x ∈ Fn = An for all n. Using a characterisation of closuresin Proposition A.8.10, it follows that every open ball B1/j(x) contains a pointxnj

∈ Anjwith nj as large as we want. Therefore, we have a subsequence {xnj

} of{xn} such that d(xnj

, x) < 1/j, which means that it is a convergent subsequenceof {xn}.

Let us now prove that if X is sequentially compact it is compact. Let C be anopen cover of X and let ε > 0 be its Lebesgue number according to Lemma A.13.12.By Lemma A.13.13, X is totally bounded, so that it is covered by finitely manyballs {Bε(xi)}n

i=1. Since ε is a Lebesgue number, for every i = 1, . . . , n, thereis some Ui ∈ C such that Bε(xi) ⊂ Ui. Consequently, {Ui}n

i=1 must be a coverfor X. �

A.14 Stone–Weierstrass theorem

In the sequel we study densities of subalgebras in C(X). These results will beapplied in characterising function algebras among Banach algebras. For materialconcerning algebras we refer to Chapter D. First we study continuous functionson [a, b] ⊂ R:

Theorem A.14.1 (Weierstrass Theorem (1885)). Polynomials are dense in C([a,b]).

A.14. Stone–Weierstrass theorem 63

Proof. Evidently, it is enough to consider the case [a, b] = [0, 1]. Let f ∈ C([0, 1]),and let g(x) = f(x)− (f(0)+ (f(1)− f(0))x); then g ∈ C(R) if we define g(x) = 0for x ∈ R \ [0, 1]. For n ∈ N let us define kn : R→ [0,∞) by

kn(x) :=

⎧⎨⎩(1−x2)n∫ 1

−1(1−t2)n dt, when |x| < 1,

0, when |x| ≥ 1.

Then define Pn := g ∗ kn (convolution of g and kn), that is

Pn(x) =∫ ∞

−∞g(x− t) kn(t) dt =

∫ ∞

−∞g(t) kn(x− t) dt =

∫ 1

0

g(t) kn(x− t) dt,

and from this last expression we see that Pn is a polynomial on [0, 1]. Notice thatPn is real valued if f is real valued. Take any ε > 0. The function g is uniformlycontinuous, so that there exists δ > 0 such that

∀x, y ∈ R : |x− y| < δ ⇒ |g(x)− g(y)| < ε.

Let ‖g‖ = maxt∈[0,1]

|g(t)|. Take x ∈ [0, 1]. Then

|Pn(x)− g(x)| =∣∣∣∣∫ ∞

−∞g(x− t) kn(t) dt− g(x)

∫ ∞

−∞kn(t) dt

∣∣∣∣=

∣∣∣∣∫ 1

−1

(g(x− t)− g(x)) kn(t) dt

∣∣∣∣≤

∫ 1

−1

|g(x− t)− g(x)| kn(t) dt

≤∫ −δ

−1

2‖g‖ kn(t) dt +∫ δ

−δ

ε kn(t) dt +∫ 1

δ

2‖g‖ kn(t) dt

≤ 4‖g‖∫ 1

δ

kn(t) dt + ε.

The reader may verify that∫ 1

δkn(t) dt →n→∞ 0 for every δ > 0. Hence

‖Qn − f‖ →n→∞ 0, where Qn(x) = Pn(x) + f(0) + (f(1)− f(0))x. �

Exercise A.14.2. Show that∫ 1

δkn(t) dt →n→∞ 0 in the proof of the Weierstrass

Theorem A.14.1.

Definition A.14.3 (Involutive subalgebras). For f : X → C let us define f∗ : X →C by f∗(x) := f(x), and define |f | : X → C by |f |(x) := |f(x)|. A subalgebraA ⊂ F(X) is called involutive if f∗ ∈ A whenever f ∈ A.

Theorem A.14.4 (Stone–Weierstrass Theorem (1937)). Let X be a compact space.Let A ⊂ C(X) be an involutive subalgebra separating the points of X. Then A isdense in C(X).


Proof. If f ∈ A then f∗ ∈ A, so that the real part Ref =f + f∗

2belongs to A.

Let us defineAR := {Ref | f ∈ A};

this is a R-subalgebra of the R-algebra C(X, R) of continuous real-valued functionson X. Then

A = {f + ig | f, g ∈ AR},so that AR separates the points of X. If we can show that AR is dense in C(X, R)then A would be dense in C(X).

First we have to show that AR is closed under taking maximums and mini-mums. For f, g ∈ C(X, R) we define

max(f, g)(x) := max(f(x), g(x)), min(f, g)(x) := min(f(x), g(x)).

Notice that AR is an algebra over the field R. Since

max(f, g) =f + g

2+|f − g|

2, min(f, g) =

f + g

2− |f − g|

2,

it is enough to prove that |h| ∈ AR whenever h ∈ AR. Let h ∈ AR. By theWeierstrass Theorem A.14.1 there is a sequence of polynomials Pn : R → R suchthat

Pn(x)→n→∞ |x|uniformly on the interval [−‖h‖, ‖h‖]. Thereby

‖|h| − Pn(h)‖ →n→∞ 0,

where Pn(h)(x) := Pn(h(x)). Since Pn(h) ∈ AR for every n, this implies that|h| ∈ AR. Now we know that max(f, g),min(f, g) ∈ AR whenever f, g ∈ AR.

Now we are ready to prove that f ∈ C(X, R) can be approximated by ele-ments of AR. Take ε > 0 and x, y ∈ X, x = y. Since AR separates the points ofX, we may pick h ∈ AR such that h(x) = h(y). Let gxx = f(x)1, and let

gxy(z) :=h(z)− h(y)h(x)− h(y)

f(x) +h(z)− h(x)h(y)− h(x)

f(y).

Here gxx, gxy ∈ AR, since AR is an algebra. Furthermore,

gxy(x) = f(x), gxy(y) = f(y).

Due to the continuity of gxy, there is an open set Vxy ∈ V(y) such that

z ∈ Vxy ⇒ f(z)− ε < gxy(z).

Now {Vxy | y ∈ X} is an open cover of the compact space X, so that there is afinite subcover {Vxyj

| 1 ≤ j ≤ n}. Define

gx := max1≤j≤n

gxyj;

A.15. Manifolds 65

gx ∈ AR, because AR is closed under taking maximums. Moreover,

∀z ∈ X : f(z)− ε < gx(z).

Due to the continuity of gx (and since gx(x) = f(x)), there is an open set Ux ∈ V(x)such that

z ∈ Ux ⇒ gx(z) < f(z) + ε.

Now {Ux | x ∈ X} is an open cover of the compact space X, so that there is afinite subcover {Uxi

| 1 ≤ i ≤ m}. Define

g := min1≤i≤m

gxi ;

g ∈ AR, because AR is closed under taking minimums. Moreover,

∀z ∈ X : g(z) < f(z) + ε.

Thusf(z)− ε < min

1≤i≤mgxi

(z) = g(z) < f(z) + ε,

that is |g(z) − f(z)| < ε for every z ∈ X, i.e., ‖g − f‖ < ε. Hence AR is dense inC(X, R) implying that A is dense in C(X). �

Remark A.14.5. Notice that under the assumptions of the Stone–Weierstrass The-orem, the compact space is actually a compact Hausdorff space, since continuousfunctions separate the points.

A.15 Manifolds

We now give an example of Hausdorff spaces which is a starting point of thegeometric analysis. We will come back to this topic with more details in Section 5.2.

Definition A.15.1 (Manifold). A topological space (X, τ) is called an n-dimensional(topological) manifold if it is second countable, Hausdorff and each of its pointshas a neighbourhood homeomorphic to an open set of the Euclidean space Rn. Ifφ : U → U ′ is a homeomorphism, where U ∈ τ and U ′ ⊂ Rn is open then the pair(U, φ) is called a chart on X.

Exercise A.15.2. Show that the sphere Sn ={

x ∈ Rn+1 :∑n+1

j=1 x2j = 1

}is an

n-dimensional manifold.

Exercise A.15.3. Let X and Y be manifolds of respective dimensions m,n. Showthat X × Y is a manifold of dimension m + n.


Definition A.15.4 (Differentiable manifold). Let (X, τ) be an n-dimensional man-ifold. A collection A = {(Ui, φi) : i ∈ I} of charts on X is called a Ck-atlas if{Ui : i ∈ I} is a cover of X and if the mappings(

x �→ φj(φ−1i (x))

): φi(Ui ∩ Uj)→ φj(Ui ∩ Uj)

are Ck-smooth whenever Ui ∩ Uj = ∅. If there is a Ck-atlas then X is called aCk-manifold (differentiable manifold).

A.16 Connectedness and path-connectedness

In this section we discuss notions of connected and path-connected topologicalspaces and a relation between them.

Proposition A.16.1. Let (X, τ) be a topological space. Then the following state-ments are equivalent

(i) There exist non-empty open subsets U, V of X such that U ∩ V = ∅ andU ∪ V = X.

(ii) There exists a non-empty subset U of X such that U is open and closed andsuch that U = X.

(iii) There exists a continuous surjective mapping from X to the set {0, 1}equipped with the discrete topology.

Proof. Statements (i) and (ii) are equivalent if we take V = X\U . Let us showthat (i) implies (iii). Define a mapping f by f(x) = 0 for x ∈ U and f(x) = 1for x ∈ V . Since U and V are non-empty, the mapping f is surjective. If W isany subset of {0, 1}, its preimage f−1(W ) is one of the sets ∅, U, V, X. Since all ofthem are open, f is continuous.

Finally, to show that (iii) implies (i), we set U = f−1(0) and V = f−1(1).Since f is continuous, both sets are open. Moreover, clearly they are disjoint,U ∪ V = X, and they are non-empty because f is surjective. �

Definition A.16.2 (Connected topological space). A topological space (X, τ) issaid to be disconnected if it satisfies any of the equivalent properties of Proposi-tion A.16.1. Otherwise, it is said to be connected.

Proposition A.16.3 (“Connectedness” is a topological property). Let X and Y betopological spaces and let f : X → Y be continuous. If X is connected, then f(X)is also connected. Consequently, “connectedness” is a topological property.

Proof. Suppose that f(X) = U ∪ V with U, V as in Proposition A.16.1, (i). ThenX = f−1(U) ∪ f−1(V ) and sets f−1(U), f−1(V ) satisfy conditions of Proposi-tion A.16.1, (i), yielding a contradiction. �

A.16. Connectedness and path-connectedness 67

Exercise A.16.4. Prove that a subset A of a topological space X is disconnected(in the relative topology) if and only if there are open sets U, V in X such thatU ∩A = ∅, V ∩A = ∅, A ⊂ U ∪ V and U ∩ V ∩A = ∅.Proposition A.16.5 (Closures are connected). Let X be a topological space and letA ⊂ X. If A is connected, then its closure A is also connected.

Proof. Let U and V be open sets in X such that A ⊂ U ∪ V and U ∩ V ∩A = ∅.Since A ⊂ A, we have A ⊂ U ∪ V and U ∩ V ∩ A = ∅. Since A is connected, byExercise A.16.4 we must then have either U ∩ A = ∅ or V ∩ A = ∅, which meansthat either A ⊂ X\U or A ⊂ X\V . Since the sets X\U and X\V are closedin X, it follows that we have either A ⊂ X\U or A ⊂ X\V , which means thateither U ∩ A = ∅ or V ∩ A = ∅. By Exercise A.16.4 again, it means that A isconnected. �Definition A.16.6 (Path-connected topological spaces). A topological space X issaid to be path-connected if for any two points a, b ∈ X there is a path from a tob, i.e., a continuous mapping γ : [0, 1]→ X such that γ(0) = a and γ(1) = b.

Theorem A.16.7 (Path-connected =⇒ connected). A path-connected topologicalspace is connected.

Exercise A.16.8. Show that the converse is not true. For example, prove that theset X = {(0, t) : −1 ≤ t ≤ 1} ∪ {(t, sin 1

t ), t > 0} in the relative topology of theEuclidean space R2 is connected but not path-connected.

We first prove a special case of Theorem A.16.7, namely we show that inter-vals in R are connected. We then reduce the general case to this one. By an intervalin R we understand any open or closed or half-open, finite or infinite interval.

Theorem A.16.9 (Interval in R =⇒ connected). Every interval I in R with theEuclidean topology is connected.

Proof. We will prove it by contradiction. Suppose I = U ∪ V , where U and V arenon-empty, disjoint, and open in the relative topology of I. Let u ∈ U , v ∈ V , andassume u < v. Since I is an interval we have [u, v] ⊂ I, and we write

A = {x ∈ I : u ≤ x and [u, x] ⊂ U}.

Since u ∈ A, A is non-empty, and since v ∈ U , A is bounded above. Thus, wecan define w = sup A, and we have [u, w) ⊂ U . Since w ∈ [u, v], we also havew ∈ I = U ∪ V , so that either w ∈ U or w ∈ V .

We will now show that both choices are impossible. Suppose w ∈ U . Thenw < v and since U is open, there is some δ > 0 such that (w − δ, w + δ) ∩A ⊂ U .Now, if we take some z ∈ (w,w+δ)∩A, we have [w, z] ⊂ U , so that also [u, z] ∈ U ,contradicting w = sup A.

Suppose now w ∈ V . Then u < w and since V is open, there is some δ > 0such that (w − δ, w + δ) ∩ A ⊂ V . Now, if we take some z ∈ (w − δ, w) ∩ A, we


have (z, w] ⊂ V , so that for all x ∈ (z, w] we have that [u, x] ⊂ U , contradictingw = sup A again. �

Proof of Theorem A.16.7. Let X be a path-connected topological space and let fbe a continuous mapping from X to {0, 1} equipped with the discrete topology.By Proposition A.16.1 it is enough to show that f must be constant. Withoutloss of generality, suppose that f(x) = 0 for some x ∈ X. Let y ∈ X and let γbe a path from x to y. Then the composition mapping f ◦ γ : [0, 1] → {0, 1} iscontinuous. Since [0, 1] is connected by Theorem A.16.9 it follows that f ◦ γ cannot be surjective, so that f(y) = f(γ(1)) = f(γ(0)) = f(x) = 0. Thus, f(y) = 0for all y ∈ X, which means that f can not be surjective. �

Theorem A.16.9 has the converse:

Theorem A.16.10 (Connected in R =⇒ interval). If I is a connected subset of Rit must be an interval.

Proof. First we show that I ⊂ R is an interval if and only if for any a, c ∈ I andany b ∈ R with a < b < c we must have b ∈ I.

If I is an interval the implication is trivial. Conversely, we will prove thatif a ∈ I, then I ∩ [a,∞) is [a,∞) or [a, e] or [a, e) for some e ∈ R. If I is notbounded above, then for any b > a there is c ∈ I such that c > b. Hence b ∈ I byassumption. In this case I ∩ [a,∞) = [a,∞). So we may assume that I is boundedabove and let e = sup I. If e = a, then I ∩ [a,∞) = [a, a], so we may assume e > a.Then for any b with a < b < e there is some c ∈ I such that a < b < c, and henceb ∈ I by assumption. Therefore, I ∩ [a,∞) is [a, e] or [a, e) depending on whethere ∈ I or not. Arguing in a similar way for I ∩ (−∞, a] we get that I must be aninterval.

Now, suppose I is not an interval. By the above claim, there exits somea, c ∈ I and b ∈ I such that a < b < c. But then U = I∩(−∞, b) and V = I∩(b,∞)is a decomposition of I into a union of non-empty, open disjoint sets with U∪V = I,contradicting the assumption that I is connected. �

We will now show a converse to Theorem A.16.7, provided that we are dealingwith subsets of Rn.

Theorem A.16.11 (Open connected in Rn =⇒ path-connected). Every open con-nected subset of Rn with the Euclidean topology is path-connected.

Proof. First we note that if we have a path γ1 from a to b and a path γ2 fromb to c, we can glue them together to obtain a path from a to c, e.g., by settingγ(t) = γ1(2t) for 0 ≤ t ≤ 1

2 , and γ(t) = γ2(2t− 1) for 12 ≤ t ≤ 1.

Let A be a non-empty open connected set in Rn (the statement is trivial forthe empty set). Take some a ∈ A and define

U = {b ∈ A : there is a path from a to b in A}.

A.17. Co-induction and quotient spaces 69

We claim that U is open and closed. Indeed, if b ∈ U , we have Bε(b) ⊂ A for someε > 0. Consequently, for any c ∈ Bε(b) we have a path from a to b in A by thedefinition of U , and we obviously also have a path from b to c in Bε(b) (e.g., justa straight line). Glueing these paths together, we obtain a path from a to c in A,which means that Bε(b) ⊂ U and hence U is open. To show that U is also closed,take some b ∈ A\U . Then we have Bε(b) ⊂ A for some ε > 0. If now c ∈ Bε(b),there is a path from c to b in Bε(b). Consequently, there can be no path from ato c because otherwise there would be a path from a to b in A. Thus, c ∈ A\U ,implying that A\U is open.

Finally, writing A = U ∪ (A\U) as a union of two disjoint open sets, andobserving that U contains a and is, therefore, non-empty, it follows that A\U =∅ because A is connected. But this means that A = U and hence A is path-connected. �

A.17 Co-induction and quotient spaces

Definition A.17.1 (Co-induced topology). Let X and J be sets, (Xj , τj) be topo-logical spaces for every j ∈ J , and F = {fj : Xj → X | j ∈ J} be a family ofmappings. The F-co-induced topology of X is the strongest topology τ on X suchthat the mappings fj are continuous for every j ∈ J .

Exercise A.17.2. Let τ be the co-induced topology from Definition A.17.1. Showthat

τ ={U ⊂ X | ∀j ∈ J : f−1

j (U) ∈ τj

}.

Definition A.17.3 (Quotient topology). Let (X, τX) be a topological space, and let∼ be an equivalence relation on X. Let

[x] := {y ∈ X | x ∼ y},

X/ ∼:= {[x] | x ∈ X},

and define the quotient map π : X → X/ ∼ by π(x) := [x]. The quotient topologyof the quotient space X/ ∼ is the {π}-co-induced topology on X/ ∼.

Exercise A.17.4. Show that X/ ∼ is compact if X is compact.

Example. Let A be a topological vector space and J its subspace. Let us write[x] := x +J for x ∈ A. Then the quotient topology of A/J = {[x] | x ∈ A} is thetopology co-induced by the family {(x �→ [x]) : A → A/J }.Remark A.17.5. The message of the following Exercise A.17.6 is that if our com-pact space X is not Hausdorff, we can factor out the inessential information thatthe continuous functions f : X → C do not see, to obtain a compact Hausdorffspace related nicely to X.


Exercise A.17.6. Let X be a topological space, and let us define a relation R ⊂X ×X by

(x, y) ∈ Rdefinition

⇐⇒ ∀f ∈ C(X) : f(x) = f(y).

Prove:

(a) R is an equivalence relation on X.(b) There is a natural bijection between the sets C(X) and C(X/R).(c) X/R is a Hausdorff space.(d) If X is a compact Hausdorff space then X ∼= X/R.

Exercise A.17.7. For A ⊂ X, let us define the equivalence relation RA by

(x, y) ∈ RA

definition

⇐⇒ x = y or {x, y} ⊂ A.

Let X be a topological space, and let ∞ ⊂ X be a closed subset. Prove that themapping

X \∞ → (X/R∞) \ {∞}, x �→ [x],

is a homeomorphism.

Finally, let us state a basic property of co-induced topologies:

Proposition A.17.8. Let X have the F-co-induced topology, and Y be a topologicalspace. A mapping g : X → Y is continuous if and only if g ◦ f is continuous forevery f ∈ F .

Proof. If g is continuous then the composed mapping g ◦ f is continuous for everyf ∈ F .

Conversely, suppose g ◦ fj is continuous for every fj ∈ F , fj : Xj → X. LetV ⊂ Y be open. Then

f−1j (g−1(V )) = (g ◦ fj)−1(V ) ⊂ Xj is open;

thereby g−1(V ) = fj(f−1j (g−1(V ))) ⊂ X is open. �

Corollary A.17.9. Let X, Y be topological spaces, R be an equivalence relation onX, and endow X/R with the quotient topology. A mapping f : X/R → Y iscontinuous if and only if (x �→ f([x])) : X → Y is continuous. �

A.18 Induction and product spaces

The main theorem of this section is Tihonov’s theorem which is a generalisation ofTheorem A.11.14 to infinitely many sets. However, we also discuss other topologiesinduced by infinite families, and some of their properties.

A.18. Induction and product spaces 71

Definition A.18.1 (Induced topology). Let X and J be sets, (Xj , τj) be topolog-ical spaces for every j ∈ J and F = {fj : X → Xj | j ∈ J} be a family ofmappings. The F-induced topology of X is the weakest topology τ on X such thatthe mappings fj are continuous for every j ∈ J .

Example. Let (X, τX) be a topological space, A ⊂ X, and let ι : A→ X be definedby ι(a) = a. Then the {ι}-induced topology on A is

τX |A := {U ∩A | U ∈ τX}.

This is called the relative topology of A, see Definition A.7.18. Let f : X → Y .The restriction f |A = f ◦ ι : A→ Y satisfies f |A(a) = f(a) for every a ∈ A ⊂ X.

Exercise A.18.2. Prove Tietze’s Extension Theorem: Let X be a compact Hausdorffspace, K ⊂ X closed and f ∈ C(K). Then there exists F ∈ C(X) such thatF |K = f . (Hint: approximate F by continuous functions that would exist byUrysohn’s lemma.)

Example. Let (X, τ) be a topological space. Let σ be the C(X) = C(X, τ)-inducedtopology, i.e., the weakest topology on X making the all τ -continuous functionscontinuous. Obviously, σ ⊂ τ , and C(X, σ) = C(X, τ). If (X, τ) is a compactHausdorff space it is easy to check that σ = τ .

Example. Let X, Y be topological spaces with bases BX ,BY , respectively. Recallthat the product topology for X × Y = {(x, y) | x ∈ X, y ∈ Y } has a base

{U × V | U ∈ BX , V ∈ BY }.

This topology is actually induced by the family

{pX : X × Y → X, pY : X × Y → Y },

where the coordinate projections pX and pY are defined by pX((x, y)) = x andpY ((x, y)) = y.

Definition A.18.3 (Product topology). Let Xj be a set for every j ∈ J . TheCartesian product

X =∏j∈J

Xj

is the set of the mappings

x : J →⋃j∈J

Xj such that ∀j ∈ J : x(j) ∈ Xj .

Due to the Axiom of Choice, X is non-empty if all Xj are non-empty. The mapping

pj : X → Xj , x �→ xj := x(j),


is called the jth coordinate projection. Let (Xj , τj) be topological spaces. LetX :=

∏j∈J Xj be the Cartesian product. Then the {pj | j ∈ J}-induced topology

on X is called the product topology of X.If Xj = Y for all j ∈ J , it is customary to write∏

j∈J

Xj = Y J = {f | f : J → Y }.

Let us state a basic property of induced topologies:

Proposition A.18.4. Let X have the F-induced topology, and Y be a topologicalspace. A mapping g : Y → X is continuous if and only if f ◦ g is continuous forevery f ∈ F .

Proof. If g is continuous then the composed mapping f ◦ g is continuous for everyf ∈ F , by Proposition A.10.10.

Conversely, suppose fj ◦ g is continuous for every fj ∈ F , f : X → Xj . Lety ∈ Y , V ⊂ X be open, g(y) ∈ V . Then there exist {fjk

}nk=1 ⊂ F and open sets

Wjk⊂ Xjk

such that

g(y) ∈n⋂

k=1

f−1jk

(Wjk) ⊂ V.

Let

U :=n⋂

k=1

(fjk◦ g)−1(Wjk

).

Then U ⊂ Y is open, y ∈ U , and g(U) ⊂ V ; hence g : Y → X is continuous at anarbitrary point y ∈ Y , i.e., g ∈ C(Y,X). �Remark A.18.5 (Hausdorff preserved in products). It is easy to see that a Carte-sian product of Hausdorff spaces is always Hausdorff: if X =

∏j∈J Xj and x, y ∈

X, x = y, then there exists j ∈ J such that xj = yj . Therefore there are open setsUj , Vj ⊂ Xj such that

xj ∈ Uj , yj ∈ Vj , Uj ∩ Vj = ∅.

Let U := p−1j (Uj) and V := p−1(Vj). Then U, V ⊂ X are open,

x ∈ U, y ∈ V, U ∩ V = ∅.

Also compactness is preserved in products; this is stated in Tihonov’s Theorem(Tychonoff’s Theorem). Before proving this we introduce a tool, which can becompared with Proposition A.11.5:

Definition A.18.6 (Non-Empty Finite InterSection (NEFIS) property). Let X bea set. Let NEFIS(X) be the set of those families F ⊂ P(X) such that every finitesubfamily of F has a non-empty intersection. In other words, a family F ⊂ P(X)belongs to NEFIS(X) if and only if

⋂F ′ = ∅ for every finite subfamily F ′ ⊂ F .

A.18. Induction and product spaces 73

Lemma A.18.7. A topological space X is compact if and only if F ∈ NEFIS(X)whenever F ⊂ P(X) is a family of closed sets satisfying

⋂F = ∅.

Proof. Let X be a set, U ⊂ P(X), and F := {X \ U | U ∈ U}. Then⋂F =

⋂U∈U

(X \ U) = X \⋃U ,

so that U is a cover of X if and only if⋂F = ∅. Now the claim follows the

definition of compactness. �

Theorem A.18.8 (Tihonov’s Theorem (1935)). Let Xj be a compact space for everyj ∈ J . Then X =

∏j∈J

Xj is compact.

Proof. To avoid the trivial case, suppose Xj = ∅ for every j ∈ J . Let F ∈NEFIS(X) be a family of closed sets. In order to prove the compactness of Xwe have to show that

⋂F = ∅.Let

P := {G ∈ NEFIS(X) | F ⊂ G}.Let us equip the set P with a partial order relation ≤:

G ≤ H definition⇐⇒ G ⊂ H.

The Hausdorff Maximal Principle A.4.9 says that the chain {F} ⊂ P belongs toa maximal chain C ⊂ P . The reader may verify that G :=

⋃C ∈ P is a maximal

element of P .Notice that the maximal element G may contain non-closed sets. For every

j ∈ J the family{pj(G) | G ∈ G}

belongs to NEFIS(Xj). Define

Gj := {pj(G) | G ∈ G}.

Clearly also Gj ∈ NEFIS(Xj), and the elements of Gj are closed sets in Xj . SinceXj is compact, we have

⋂Gj = ∅. Hence, by the Axiom of Choice A.4.2, there isan element x := (xj)j∈J ∈ X such that

xj ∈⋂Gj .

We shall show that x ∈ ⋂F , which proves Tihonov’s Theorem.If Vj ⊂ Xj is a neighbourhood of xj and G ∈ G then

pj(G) ∩ Vj = ∅,


because xj ∈ pj(G). ThusG ∩ p−1

j (Vj) = ∅

for every G ∈ G, so that G ∪ {p−1j (Vj)} belongs to P ; the maximality of G implies

thatp−1

j (Vj) ∈ G.

Let V ∈ τX be a neighbourhood of x. Due to the definition of the producttopology,

x ∈n⋂

k=1

p−1jk

(Vjk) ⊂ V

for some finite index set {jk}nk=1 ⊂ J , where Vjk

⊂ Xjkis a neighbourhood of xjk

.Due to the maximality of G, any finite intersection of members of G belongs to G,so that

n⋂k=1

p−1jk

(Vjk) ∈ G.

Therefore for every G ∈ G and V ∈ VτX(x) we have

G ∩ V = ∅.

Hence x ∈ G for every G ∈ G, yielding

x ∈⋂

G∈GGF⊂G⊂

⋂F∈F

F =⋂

F∈FF =

⋂F ,

so that⋂F = ∅. �

Remark A.18.9. Actually, Tihonov’s Theorem A.18.8 is equivalent to the Axiomof Choice A.4.2; we shall not prove this.

A.19 Metrisable topologies

It is often very useful to know whether a topology on a space comes from somemetric. Here we try to construct metrics on compact spaces. We shall learn thata compact space X is metrisable if and only if the corresponding normed algebraC(X) is separable. Metrisability is equivalent to the existence of a countable familyof continuous functions separating the points of the space. As a vague analogy tomanifolds, the reader may view such a countable family as a set of coordinatefunctions on the space.

Definition A.19.1 (Metrisable topology). A topological space (X, τ) is calledmetrisable if there exists a metric d on X such that the topology τ is the canonicalmetric topology of (X, d), i.e., if there exists a metric d on X such that τ = τd.

A.19. Metrisable topologies 75

Example (Discrete topology). The discrete topology on the set X is the collectionτ of all subsets of X. This is a metric topology corresponding to the discretemetric.

Exercise A.19.2. Let X, Y be metrisable. Prove that X×Y is metrisable, and that

(xn, yn) X×Y→ (x, y) ⇔ xnX→ x and yn

Y→ y.

Remark A.19.3. There are plenty of non-metrisable topological spaces, the easiestexample being X with more than one point and with τ = {∅, X}. If X is aninfinite-dimensional Banach space then the weak∗-topology1 of X ′ := L(X, C)is not metrisable. The distribution spaces D′(Rn), S ′(Rn) and E ′(Rn) are non-metrisable topological spaces. We shall later prove that for the compact Hausdorffspaces metrisability is equivalent to the existence of a countable base.

Exercise A.19.4. Show that (X, τ) is a topological space, where

τ = {U ⊂ X | U = ∅, or X \ U is finite} .

When is this topology metrisable?

Theorem A.19.5. Let (X, τ) be compact. Assume that there exists a countablefamily F ⊂ C(X) separating the points of X. Then (X, τ) is metrisable.

Proof. Let F = {fn}∞n=0 ⊂ C(X) separate the points of X. We can assume that|fn| ≤ 1 for every n ∈ N; otherwise consider for instance functions x �→ fn(x)/(1+|fn(x)|). Let us define

d(x, y) := supn∈N

2−n|fn(x)− fn(y)|

for every x, y ∈ X. Next we prove that d : X ×X → [0,∞) is a metric: d(x, y) =0 ⇔ x = y, because {fn}∞n=0 is a separating family. Clearly also d(x, y) = d(y, x)for every x, y ∈ X. Let x, y, z ∈ X. We have the triangle inequality:

d(x, z) = supn∈N

2−n|fn(x)− fn(z)|

≤ supn∈N

(2−n|fn(x)− fn(y)|+ 2−n|fn(y)− fn(z)|)

≤ supm∈N

2−m|fm(x)− fm(y)|+ supn∈N

2−n|fn(y)− fn(z)|

= d(x, y) + d(y, z).

Hence d is a metric on X.Finally, let us prove that the metric topology coincides with the original

topology, τd = τ . Let x ∈ X, ε > 0. Take N ∈ N such that 2−N < ε. Define

Un := f−1n (Bε(fn(x))) ∈ Vτ (x), U :=

N⋂n=0

Un ∈ Vτ (x).

1see Definition B.4.35


If y ∈ U thend(x, y) = sup

n∈N

2−n|fn(x)− fn(y)| < ε.

Thus x ∈ U ⊂ Bε(x) = {y ∈ X | d(x, y) < ε}. This proves that the originaltopology τ is finer than the metric topology τd, i.e., τd ⊂ τ . Combined with thefacts that (X, τ) is compact and (X, τd) is Hausdorff, this implies that we musthave τd = τ , by Corollary A.12.8. �Corollary A.19.6. Let X be a compact Hausdorff space. Then X is metrisable ifand only if it has a countable basis.

Proof. Suppose X is a compact space, metrisable with a metric d. Let r > 0.Then Br = {Bd(x, r) | x ∈ X} is an open cover of X, thus having a finite subcover

B′r ⊂ Br. Then B :=∞⋃

n=1

B′1/n is a countable basis for X.

Conversely, suppose X is a compact Hausdorff space with a countable basisB. Then the family

C := {(B1, B2) ∈ B × B | B1 ⊂ B2}is countable. For each (B1, B2) ∈ C, Urysohn’s Lemma (Theorem A.12.11) providesa function fB1B2 ∈ C(X) satisfying

fB1B2(B1) = {0} and fB1B2(X \B2) = {1}.Next we show that the countable family

F = {fB1B2 : (B1, B2) ∈ C} ⊂ C(X)

separates the points of X. Indeed, Take x, y ∈ X, x = y. Then W := X \ {y} ∈V(x). Since X is a compact Hausdorff space, by Corollary A.12.6 there existsU ∈ V(x) such that U ⊂ W . Take B′, B ∈ B such that x ∈ B′ ⊂ B′ ⊂ B ⊂ U .Then fB′B(x) = 0 = 1 = fB′B(y). Thus X is metrisable. �Corollary A.19.7. Let X be a compact Hausdorff space. Then X is metrisable ifand only if C(X) is separable.

Proof. Suppose X is a metrisable compact space. Let F ⊂ C(X) be a countablefamily separating the points of X (as in the proof of the previous corollary). LetG be the set of finite products of functions f for which f ∈ F ∪ F∗ ∪ {1}; theset G = {gj}∞j=0 is countable. The linear span A of G is the involutive algebragenerated by F (the smallest ∗-algebra containing F , see Definition D.5.1); dueto the Stone–Weierstrass Theorem (see Theorem A.14.4), A is dense in C(X). IfS ⊂ C is a countable dense set then

{λ01 +n∑

j=1

λjgj | n ∈ Z+, (λj)nj=0 ⊂ S}

is a countable dense subset of A, thereby dense in C(X).

A.20. Topology via generalised sequences 77

Conversely, assume that F = {fn}∞n=0 ⊂ C(X) is a dense subset. Take x, y ∈X, x = y. By Urysohn’s Lemma (Theorem A.12.11) there exists f ∈ C(X) suchthat f(x) = 0 = 1 = f(y). Take fn ∈ F such that ‖f − fn‖ < 1/2. Then

|fn(x)| < 1/2 and |fn(y)| > 1/2,

so that fn(x) = fn(y); F separates the points of X. �Exercise A.19.8. Prove that a topological space with a countable basis is separable.Prove that a metric space has a countable basis if and only if it is separable.

Exercise A.19.9. There are non-metrisable separable compact Hausdorff spaces!Prove that X,

X = {f : [0, 1]→ [0, 1] | x ≤ y ⇒ f(x) ≤ f(y)},

endowed with a relative topology, is such a space. Hint: Tihonov’s Theorem.

A.20 Topology via generalised sequences

Definition A.20.1 (Directed set). A non-empty set J is directed if there exists arelation “≤” such that ≤⊂ J × J (where (x, y) ∈≤ is usually denoted by x ≤ y)such that for every x, y, z ∈ J it holds that

1. x ≤ x,2. if x ≤ y and y ≤ z then x ≤ z,3. there exists w ∈ J such that x ≤ w and y ≤ w.

Definition A.20.2 (Nets and convergence). A net (or a generalised sequence) in atopological space (X, τ) is a mapping (j �→ xj) : J → X, denoted also by (xj)j∈J ,where J is a directed set. If K ⊂ J is a directed set (with respect to the naturalinherited relation ≤) then the net (xj)j∈K is called a subnet of the net (xj)j∈J .A net (xj)j∈J converges to a point p ∈ X, denoted by

xj → p or xj −−→j∈J

p or limj∈J

xj = lim xj = p,

if for every neighbourhood U of p there exists jU ∈ J such that xj ∈ U wheneverjU ≤ j.

Example. A sequence (xj)j∈Z+ is a net, where Z+ is directed by the usual partial or-der; sequences characterise topology in spaces of countable local bases, for instancemetric spaces. But there are more complicated topologies, where sequences are notenough; for instance, the weak∗-topology for the dual of an infinite-dimensionaltopological vector space.

Exercise A.20.3 (Nets and closure). Let X be a topological space. Show that p ∈ Xbelongs to the closure of S ⊂ X if and only if there exists a net (xj)j∈J : J → Ssuch that xj → p.


Exercise A.20.4 (Nets and continuity). Show that a function f : X → Y is contin-uous at p ∈ X if and only if f(xj)→ f(p) whenever xj → p for nets (xj)j∈J in X.

Exercise A.20.5 (Nets and compactness). Show that a topological space X iscompact if and only if its every net has a converging subnet.

Exercise A.20.6. In the spirit of Exercises A.20.3, A.20.4 and A.20.5, express othertopological concepts via nets.

Chapter B

Elementary Functional Analysis

We assume that the reader already has knowledge of (complex) matrices, deter-minants, etc. In this chapter, we shall present basic machinery for dealing withvector spaces, especially Banach and Hilbert spaces. We do not go into depth inthis direction as there are plenty of excellent specialised monographs available de-voted to various aspects of the subject, see, e.g., [11, 35, 53, 59, 63, 70, 87, 89,90, 116, 134, 146, 153]. However, we still make an independent presentation of acollection of results which are indispensable for anyone working in analysis, andwhich are useful for other parts of this book.

B.1 Vector spaces

Definition B.1.1 (Vector space). Let K ∈ {R, C}. A K-vector space (or a vectorspace over the field K, or a vector space if K is implicitly known) is a set V endowedwith mappings

((x, y) �→ x + y) : V × V → V,

((λ, x) �→ λx) : K× V → V

such that there exists an origin 0 ∈ V and such that the following properties hold:

(x + y) + z = x + (y + z),x + 0 = x,

x + (−1)x = 0,

x + y = y + x,

1x = x,

λ(μx) = (λμ)x,

λ(x + y) = λx + λy,

(λ + μ)x = λx + μx

80 Chapter B. Elementary Functional Analysis

for all x, y, z ∈ V and λ, μ ∈ K. We may write x + y + z := (x + y) + z and−x := (−1)x. Elements of a vector space are called vectors.

Definition B.1.2 (Convex and balanced sets). A subset C of a vector space isconvex if tx + (1− t)y ∈ C for every x, y ∈ C whenever 0 < t < 1. A subset B ofa vector space is balanced if λx ∈ B for every x ∈ B whenever |λ| ≤ 1.

Example. K ∈ {R, C} is itself a vector space over K, likewise Kn with operations(xk)n

k=1 + (yk)nk=1 := (xk + yk)n

k=1 and λ(xk)nk=1 := (λxk)n

k=1.Example. Let V be a K-vector space and X = ∅. The set V X of mappings f :X → V is a K-vector space with pointwise operations (f + g)(x) := f(x) + g(x)and (λf)(x) := λ f(x). The vector space Kn can be naturally identified with KX ,where X = {k ∈ Z+ : k ≤ n}.Example. Let V be a vector space such that its vector operations restricted toW ⊂ V endow this subset with the vector space structure. Then W is called avector subspace. A vector space V has always trivial subspaces {0} and V . Thevector space V X has, e.g., the subspace {f : X → V | ∀x ∈ K : f(x) = 0}, whereK ⊂ X is a fixed subset.

Definition B.1.3 (Algebraic basis). Let V be a vector space and S ⊂ V . Let uswrite ∑

x∈S

λ(x)x =∑

x∈S: λ(x)�=0

λ(x)x,

when λ : S → K is finitely supported, i.e., {x ∈ S : λ(x) = 0} is finite. The spanof a subset S of a vector space V is

span(S) :=

{∑x∈S

λ(x)x

∣∣∣∣∣ λ : S → K finitely supported

}.

Thus span(S) is the smallest subspace containing S ⊂ V . A subset S of a K-vectorspace is said to be linearly independent if∑

x∈S

λ(x)x = 0 ⇒ λ ≡ 0.

A subset is linearly dependent if it is not linearly independent. A subset S ⊂ V iscalled an algebraic basis (or a Hamel basis) of V if S is linearly independent andV = span(S).

Remark B.1.4. Let B be an algebraic basis for V . Then there exists a unique setof functions (x �→ 〈x, b〉B) : V → K such that

x =∑b∈B〈x, b〉B b

for every x ∈ V . Notice that 〈x, b〉B = 0 for at most finitely many b ∈ B. Considerthis, e.g., with respect to the vector space KX in the example before.

B.1. Vector spaces 81

Example. The canonical algebraic basis for Kn is {ek}nk=1, where ek = (δjk)n

j=1

and δkk = 1 and δjk = 0 otherwise.

Lemma B.1.5. Every vector space V = {0} has an algebraic basis. Moreover, anytwo algebraic bases have the same cardinality1.

Proof. Let F be the family of all linearly independent subsets of V . Now F = ∅,because {x} ∈ F for every x ∈ V \{0}. Endow F with a partial order by inclusion.Let C ⊂ F be a chain and let F :=

⋃ C. It is easy to check that F ∈ F is an upperbound for C. Thereby there is a maximal element M ∈ F . Obviously, M is analgebraic basis for V .

Let A,B be algebraic bases for V . The reader may prove (by induction) thatcard(A) = card(B) when A is finite. So suppose card(A) ≤ card(B), where A isinfinite. Now card(A) = card(S), where

S := {(a, b) ∈ A× B : 〈a, b〉B = ∅} .

Assume card(A) < card(B). Thus

∃b0 ∈ B ∀a ∈ A : 〈a, b0〉B = 0.

But then

b0 =∑a∈A〈b0, a〉A a

=∑a∈A〈b0, a〉A

∑b∈B〈a, b〉B b

=∑b∈B

(∑a∈A〈b0, a〉A 〈a, b〉B

)b

=∑

b∈B\{b0}

(∑a∈A〈b0, a〉A 〈a, b〉B

)b

∈ span(B \ {b0}),

contradicting the linear independence of B. Thus card(A) = card(B). �

Definition B.1.6 (Algebraic dimension). By Lemma B.1.5, we may define the al-gebraic dimension dim(V ) of a vector space V to be the cardinality of any of itsalgebraic bases. The vector space V is said to be finite-dimensional if dim(V ) isfinite, and infinite-dimensional otherwise.

1Here we will use the notation card(A) for the cardinality of A to avoid any confusion with thenotation for norms; see Definition A.4.13.


Definition B.1.7 (Linear operators and functionals). Let V,W be K-vector spaces.A mapping A : V → W is called a linear operator (or a linear mapping), denotedA ∈ L(V,W ), if {

A(u + v) = A(u) + A(v),A(λv) = λ A(v)

for every u, v ∈ V and λ ∈ K. Then it is customary to write Av := A(v), andL(V ) := L(V, V ). A linear mapping f : V → K is called a linear functional.

Definition B.1.8 (Kernel and image). The kernel Ker(A) ⊂ V of a linear operatorA : V →W is defined by

Ker(A) := {u ∈ V : Au = 0},where 0 is the origin of the vector space W . The image Im(A) ⊂ W of A isdefined by

Im(A) := {Au : u ∈ V }.Exercise B.1.9. Show that Ker(A) is a vector subspace of V and that Im(A) is avector subspace of W .

Exercise B.1.10. Let C ⊂ V be convex and A ∈ L(V,W ). Show that A(C) ⊂ Wis convex.

Definition B.1.11 (Spectrum of an operator). Let V be a K-vector space. LetI ∈ L(V ) denote the identity operator (x �→ x) : V → V . The spectrum ofA ∈ L(V ) is

σ(A) := {λ ∈ K : λI −A is not bijective} .

Exercise B.1.12. Appealing to the Fundamental Theorem of Algebra, show thatσ(A) = ∅ for A ∈ L(Cn).

Exercise B.1.13. Give an example, where σ(A) = ∅ = σ(A2).

Exercise B.1.14. Show that σ(A) = {0} if A is nilpotent, i.e., if Ak = 0 for somek ∈ Z+.

Exercise B.1.15. Show that σ(AB) ∪ {0} = σ(BA) ∪ {0} in general, and thatσ(AB) = σ(BA) if A is bijective.

Definition B.1.16 (Quotient vector space). Let M be a subspace of a K-vectorspace V . Let us endow the quotient set V/M := {x + M | x ∈ V } with the oper-ations

((x + M,y + M) �→ x + y + M) : V/M × V/M → V/M,

(λ, (y + M)) �→ λy + M) : K× V/M → V/M.

Then it is easy to show that with these operations, this so-called quotient vectorspace is indeed a vector space.

Remark B.1.17. In the case of a topological vector space V (see Definition B.2.1),the quotient V/M is endowed with the quotient topology, and then V/M is atopological vector space if and only if the original subspace M ⊂ V was closed.

B.1. Vector spaces 83

B.1.1 Tensor products

The basic idea in multilinear algebra is to linearise multilinear operators. Thefunctional analytic foundation is provided by tensor products that we conciselyreview here. We also introduce locally convex spaces and Frechet spaces as well asMontel and nuclear spaces.

Definition B.1.18 (Bilinear mappings). Let Xj (1 ≤ j ≤ r) and V be K-vectorspaces (that is, vector spaces over the field K). A mapping A : X1 ×X2 → V is2-linear (or bilinear) if x �→ A(x, x2) and x �→ A(x1, x) are linear mappings foreach xj ∈ Xj . The reader may guess what conditions an r-linear mapping

X1 × · · · ×Xr → V

satisfies.

Definition B.1.19 (Tensor product of spaces). The algebraic tensor product of K-vector spaces X1, . . . , Xr is a K-vector space V endowed with an r-linear mappingi such that for every K-vector space W and for every r-linear mapping

A : X1 × · · · ×Xr →W

there exists a (unique) linear mapping A : V → W satisfying Ai = A. (Thereader is encouraged to draw a commutative diagram involving the vector spacesand mappings i, A, A!) Any two tensor products for X1, . . . , Xr can easily be seenisomorphic, so that we may denote the tensor product of these vector spaces by

X1 ⊗ · · · ⊗Xr.

In fact, such a tensor product always exists. Indeed, let X, Y be K-vectorspaces. We may formally define the set B := {x ⊗ y | x ∈ X, y ∈ Y }, wherex ⊗ y = a ⊗ b if and only if x = a and y = b. Let Z be the K-vector space withbasis B, i.e.,

Z = span {x⊗ y | x ∈ X, y ∈ Y }

=

⎧⎨⎩n∑

j=0

λj(xj ⊗ yj) : n ∈ N, λj ∈ K, xj ∈ X, yj ∈ Y

⎫⎬⎭ .

Let

[0⊗ 0] := span{α1β1(x1 ⊗ y1) + α1β2(x1 ⊗ y2)

+α2β1(x2 ⊗ y1) + α2β2(x2 ⊗ y2)−(α1x1 + α2x2)⊗ (β1y1 + β2y2) :

αj , βj ∈ K, xj ∈ X, yj ∈ Y}.


For z ∈ Z, let [z] := z + [0⊗ 0]. The tensor product of X, Y is the K-vector space

X ⊗ Y := Z/[0⊗ 0] = {[z] | z ∈ Z} ,

where ([z1], [z2]) �→ [z1 + z2] and (λ, [z]) �→ [λz] are well defined mappings (X ⊗Y )× (X ⊗ Y )→ X ⊗ Y and K× (X ⊗ Y )→ X ⊗ Y , respectively.

Definition B.1.20 (Tensor product of operators). Let X, Y , V , W be K-vectorspaces, and let A : X → V and B : Y → W be linear operators. The tensorproduct of A,B is the linear operator A ⊗ B : X ⊗ Y → V ⊗W , which is theunique linear extension of the mapping x⊗y �→ Ax⊗By, where x ∈ X and y ∈ Y .

Example. Let X and Y be finite-dimensional K-vector spaces with bases

{xi}dim(X)i=1 and {yj}dim(Y )

j=1 ,

respectively. Then X ⊗ Y has a basis

{xi ⊗ yj | 1 ≤ i ≤ dim(X), 1 ≤ j ≤ dim(Y )} .

Let S be a finite set. Let F(S) be the K-vector space of functions S → K; it hasa basis {δx | x ∈ S}, where δx(y) = 1 if x = y, and δx(y) = 0 otherwise. Nowit is easy to see that for finite sets S1, S2 the vector spaces F(S1) ⊗ F(S2) andF(S1×S2) are isomorphic; for fj ∈ F(Sj), we may regard f1⊗f2 ∈ F(S1)⊗F(S2)as a function f1 ⊗ f2 ∈ F(S1 × S2) by

(f1 ⊗ f2)(x1, x2) := f1(x1) f2(x2).

Definition B.1.21 (Inner product on V ⊗W ). Suppose V,W are finite-dimensionalinner product spaces over K. The natural inner product for V ⊗W is obtained byextending

〈v1 ⊗ w1, v2 ⊗ w2〉V⊗W := 〈v1, v2〉V 〈w1, w2〉W .

Definition B.1.22 (Duals of tensor product spaces). The dual (V ⊗W )′ of a tensorproduct space V ⊗W is naturally identified with V ′ ⊗W ′.

Alternative approach to tensor products. Now we briefly describe another ap-proach to tensor products.

Definition B.1.23 (Algebraic tensor product). Let K ∈ {R, C}. Let X, Y be K-vector spaces, and X ′, Y ′ their respective algebraic duals, i.e., the spaces of linearfunctionals X → K and Y → K. For x ∈ X and y ∈ Y , define the bilinearfunctional x⊗ y : X ′ × Y ′ → K by

(x⊗ y)(x′, y′) := x′(x) y′(y).

Let B(X ′, Y ′) denote the space of all bilinear functionals X ′ × Y ′ → K. Thealgebraic tensor product (or simply the tensor product) X⊗Y is the vector subspaceof B(X ′, Y ′) which is spanned by the set {x⊗ y : x ∈ X, y ∈ Y }.

B.2. Topological vector spaces 85

Exercise B.1.24. Show that a : (X ⊗ Y )′ → B(X, Y ) is a linear bijection, wherea(f)(x⊗ y) := f(x, y) for f ∈ (X ⊗ Y )′, x ∈ X and y ∈ Y .

Exercise B.1.25. Let X, Y, Z be K-vector spaces. Let B(X, Y ;Z) denote the vectorspace of bilinear mappings X × Y → Z. Find a linear bijection B(X, Y ;Z) →L(X ⊗ Y, Z), where L(V,Z) is the vector space of linear mappings V → Z.

B.2 Topological vector spaces

Vector spaces can be combined with topology. For reader’s convenience, if one hasnot encountered Banach and Hilbert spaces yet, we suggest skipping the sectionson topological vector spaces and locally convex spaces at this point, and returninghere later.

Definition B.2.1 (Topological vector space). A topological vector space V over afield K ∈ {R, C} is both a topological space and a vector space over K such that{0} ⊂ V is closed and such that the mappings

((λ, x) �→ λx) : K× V → V,

((x, y) �→ x + y) : V × V → V

are continuous. The dual space V ′ of a topological vector space V consists ofcontinuous linear functionals f : V → K.

Exercise B.2.2. Show that a topological vector space is a Hausdorff space.

Exercise B.2.3. Show that in a topological vector space every neighbourhood of 0contains a balanced neighbourhood of 0.

Exercise B.2.4. Prove that a topological vector space V is metrisable if and only ifit has a countable family {Uj}∞j=1 of neighbourhoods of 0 ∈ V such that

⋂∞j=1 Uj =

{0}. Moreover, show that in this case a compatible metric d : V ×V → [0,∞) canbe chosen translation-invariant in the sense that d(x+ z, y + z) = d(x, y) for everyx, y, z ∈ V .

Definition B.2.5 (Equicontinuity in vector space). Let X be a topological spaceand V a topological vector space. A family F of mappings f : X → V is calledequicontinuous at p ∈ X if for every neighbourhood W ⊂ V of f(p) there exists aneighbourhood U ⊂ X of p such that f(x) ∈W whenever f ∈ F and x ∈ U .

Remark B.2.6 (NEFIS property and compactness). Recall the Non-Empty Fi-nite Intersection property (NEFIS) from Definition A.18.6: that is, we denote byNEFIS(X) the set of those families F ⊂ P(X) such that every finite subfamily ofF has a non-empty intersection. Recall also that a topological space X is compactif and only if

⋂F = ∅ whenever F ∈ NEFIS(X) consists of closed sets.

Definition B.2.7 (Small sets property). Let X be a topological vector space. Afamily F ⊂ P(X) is said to contain small sets if for every neighbourhood U of0 ∈ X there exists x ∈ X and S ∈ F such that S ⊂ x + U .


Definition B.2.8 (Completeness of topological vector spaces). A subset S of atopological vector space X is called complete if

⋂F = ∅ whenever F ∈ NEFIS(X)consists of closed subsets of S and contains small sets.

Exercise B.2.9 (Completeness and Cauchy nets). A net (xj)j∈J in a topologicalvector space X is called a Cauchy net if for every neighbourhood V of 0 ∈ X thereexists k = kV ∈ J such that xi − xj ∈ V whenever k ≤ i, j. Prove that S ⊂ X iscomplete if and only if each Cauchy net in S converges to a point in S.

Exercise B.2.10. Show that a complete subset of a topological vector space isclosed, and that a closed subset of a complete topological vector space is complete.

Exercise B.2.11 (Completeness and Cartesian product). Let Xj be a topologicalvector space for each j ∈ J . Show that the product space X =

∏j∈J Xj is complete

if and only if Xj is complete for every j ∈ J .

Definition B.2.12 (Total boundedness in topological vector spaces). A subset S ofa topological vector space X is totally bounded if for every neighbourhood U of0 ∈ X there exists a finite set F ⊂ X such that S ⊂ F + U .

Exercise B.2.13 (Hausdorff Total Boundedness Theorem). Prove the followingHausdorff Total Boundedness Theorem: A subset of a topological vector space iscompact if and only if it is totally bounded and complete.

Definition B.2.14 (Completion of a topological vector space). A completion of atopological vector space X is an injective open continuous linear mapping ι : X →X, where ι(X) is a dense subset of the complete topological vector space X.

Exercise B.2.15 (Existence and uniqueness of completion). Let X be a topologicalvector space. Show that it has a completion ι : X → X, and that this completionis unique in the following sense: if κ : X → Z is another completion, then thelinear mapping (ι(x) �→ κ(x)) : ι(X) → Z has a unique continuous extension toan isomorphism X → Z of topological vector spaces.

Exercise B.2.16 (Extension of continuous linear operators). Let A : X → Y becontinuous and linear, where the topological vector spaces X, Y have respectivecompletions ιX : X → X and ιY : Y → Y . Show that there exists a uniquecontinuous linear mapping A : X → Y such that A ◦ ιX = ιY ◦ A, i.e., that thefollowing diagram is commutative:

XA−−−−→ Y

ιX

⏐⏐( ⏐⏐(ιY

XA−−−−→ Y

B.3. Locally convex spaces 87

B.3 Locally convex spaces

A locally convex space is a topological vector space where a local base for thetopology can be given by convex neighbourhoods. If the reader is not familiarwith Banach and Hilbert spaces yet, we suggest first examining those concepts,and returning to this section only afterwards. This is why in this section we willfreely refer to Section B.4 to illustrate the introduced concepts in a simpler settingof Banach spaces. In the sequel, we present some essential results for locally convexspaces in a series of exercises of widely varying difficulty, for which the reader mayfind help, e.g., from [63], [89] and [134].

Definition B.3.1 (Locally convex spaces). A topological vector space V (over K) iscalled locally convex if for every neighbourhood U of 0 ∈ V there exists a convexneighbourhood C such that 0 ∈ C ⊂ U .

Exercise B.3.2. Show that in a locally convex space each neighbourhood of 0contains a convex balanced neighbourhood of 0.

Exercise B.3.3. Let U be the family of all convex balanced neighbourhoods of 0 ina topological vector space V . For U ∈ U , define a so-called Minkowski functionalpU : V → [0,∞) by

pU (x) := inf{λ ∈ R+ : x/λ ∈ U

}.

Show that pU is a seminorm (see Definition B.4.1). Moreover, prove that V islocally convex if and only if its topology is induced by the family

{pU : V → [0,∞) | U ∈ U} .

Definition B.3.4 (Frechet spaces). A locally convex space having a complete (andtranslation-invariant) metric is called a Frechet space.

Exercise B.3.5. Show that a locally convex space V is metrisable if and only if it hasthe following property: there exists a countable collection {pk}∞k=1 of continuousseminorms pk : V → [0,∞) such that for every x ∈ V \ {0} there exists kx ∈ Z+

satisfying pkx(x) = 0 (i.e., the seminorm family separates the points of V ).

Exercise B.3.6. Let k ∈ Z+ ∪ {0,∞} and let U ⊂ Rn be an open non-empty set.Endow space Ck(U) with a Frechet space structure.

Exercise B.3.7. Let Ω ⊂ C be open and non-empty. Endow the space H(Ω) ⊂ C(Ω)of analytic functions f : Ω→ C with a structure of a Frechet space.

Definition B.3.8 (Schwartz space). For f ∈ C∞(Rn) and α, β ∈ Nn0 , let

pαβ(f) := supx∈Rn

∣∣xβ∂αx f(x)

∣∣ .

If pαβ(f) <∞ for every α, β, then f is called a rapidly decreasing smooth function.The collection of such functions is called the Schwartz space S(Rn).


Exercise B.3.9. Show that the Schwartz space S(Rn) is a Frechet space.

Definition B.3.10 (LF-space). A C-vector space X is called an LF-space (or alimit of Frechet spaces) if X =

⋃∞j=1 Xj , where each Xj ⊂ Xj+1 is a subspace of

X, having a Frechet space topology τj such that τj = {U ∩Xj : U ∈ τj+1}. Thetopology τ of the LF-space X is then generated by the set

τ := {x + V | x ∈ X, V ∈ C, V ∩Xj ∈ τj for every j} ,

where C is the family of those convex subsets of X that contain 0.

Exercise B.3.11. Let τ be the topology of an LF-space X =⋃∞

j=1 Xj as in Defini-tion B.3.10. Prove that

τj = {U ∩Xj | U ∈ τ} .

Moreover, show that a linear functional f : X → C is continuous if and only if therestriction f |Xj : Xj → C is continuous for every j.

Exercise B.3.12. Let U ⊂ Rn be an open non-empty set. Let D(U) consist ofcompactly supported C∞-smooth functions f : U → C. Endow D(U) with anLF-space structure; this is not a Frechet space anymore.

Definition B.3.13 (Test functions and distributions). The LF-space D(U) of Exer-cise B.3.12 is called the space of test functions, and a continuous linear functionalf : D(U)→ C is called a distribution on U ⊂ Rn. The space of distributions on Uis denoted by D′(U).

Exercise B.3.14 (Locally convex Hahn–Banach Theorem). Prove the following ana-logue of the Hahn–Banach Theorem B.4.25: Let X be a locally convex space (overK) and f : M → K be a continuous linear functional on a vector subspace M ⊂ X.Then there exists a continuous extension F : X → K such that F |M = f .

Exercise B.3.15. Let X be a K-vector space, and suppose V is a vector space oflinear functionals f : X → K that separates the points of X. Show that V inducesa locally convex topology on X, and that then the dual X ′ = V .

Definition B.3.16 (Weak topology). Let X be a topological vector space such thatthe dual X ′ = L(X, K) separates the points of X. The X ′-induced topology iscalled the weak topology of X.

Exercise B.3.17 (Closure of convex sets). Let X be locally convex and C ⊂ Xconvex. Show that the closure of C is the same in both the original topology andthe weak topology.

Definition B.3.18 (Weak∗-topology). Let X be a topological vector space. Theweak∗-topology of the dual X ′ is the topology induced by the family {x′ | x ∈ X},where x′ : X ′ → K is defined by x′(f) := f(x).

Exercise B.3.19. Let x ∈ X. Show that x′ = (f �→ f(x)) : X ′ → K is linear.Moreover, prove that if a linear functional f : X ′ → K is continuous with respectto the weak∗-topology, then f = x′ for some x ∈ X.


Exercise B.3.20 (Banach–Alaoglu Theorem in topological vector spaces). Provethe following generalisation of the Banach–Alaoglu Theorem B.4.36: Let X be atopological vector space. Let U ⊂ X be a neighbourhood of 0 ∈ X, and let

K := {f ∈ X ′ | ∀x ∈ U : |f(x)| ≤ 1} .

Then K ⊂ X ′ is compact in the weak∗-topology.

Definition B.3.21 (Convex hull). The convex hull of a subset S of a vector spaceX is the intersection of all convex sets that contain S. (Notice that at least X isa convex set containing S.)

Exercise B.3.22. Show that the convex hull of S is the smallest convex set thatcontains S.

Exercise B.3.23. Show that x ∈ X belongs to the convex hull of S if and only ifx =

∑nk=1 tkxk for some n ∈ Z+, where the vectors xk ∈ S, and tk > 0 are such

that∑n

k=1 tk = 1.

Definition B.3.24 (Extreme set). Let K be a subset of a vector space X. A non-empty set E ⊂ K is called an extreme set of K if the conditions{

x, y ∈ K,

tx + (1− t)y ∈ E for some t ∈ (0, 1)

imply that x, y ∈ E. A point z ∈ K is called an extreme point of K ⊂ X if {z} isan extreme set of K (alternative characterisation: if x, y ∈ K and z = tx+(1− t)yfor some 0 < t < 1 then x = y = z).

Exercise B.3.25 (Krein–Milman Theorem). Prove the following Krein–MilmanTheorem: Let X be a locally convex space and K ⊂ X compact. Then K is con-tained in the closure of the convex hull of the set of the extreme points of K.(Hint: The first problem is the very existence of extreme points. The family ofcompact extreme sets of K can be ordered by inclusion, and by the HausdorffMaximal Principle there is a maximal chain. Notice that X ′ separates the pointsof X. . . )

Exercise B.3.26. Let K be a compact subset of a Frechet space X. Show that theclosure of the convex hull of K is compact.

Exercise B.3.27. Let f : G → X be continuous, where X is a Frechet space andG is a compact Hausdorff space. Let μ be a finite positive Borel measure on G.Show that there exists a unique vector v ∈ X such that

φ(v) =∫

G

φ(f) dμ

for every φ ∈ X ′.


Definition B.3.28 (Pettis integral). Let f : G → X, μ and v be as in Exer-cise B.3.27. Then the vector v ∈ X is called the Pettis integral (or the weakintegral) of f with respect to μ, denoted by

v =∫

G

f dμ.

Exercise B.3.29. Let f : G→ X and μ be as in Definition B.3.28. Assume that Xis even a Banach space. Show that∥∥∥∥∫

G

f dμ

∥∥∥∥ ≤ ∫G

‖f‖ dμ.

Definition B.3.30 (Barreled space). A subset B of a topological vector space Xis called a barrel if it is closed, balanced, convex and X =

⋃t>0 tB. A topological

vector space is called barreled if its every barrel contains a neighbourhood of theorigin.

Remark B.3.31. Notice that a barreled space is not necessarily locally convex.

Exercise B.3.32 (LF-spaces are barreled). Show that LF-spaces are barreled.

Definition B.3.33 (Heine–Borel property). A metric space is said to satisfy theHeine–Borel property if its closed and bounded sets are compact.

Definition B.3.34 (Montel space). A barreled locally convex space with the Heine–Borel property is called a Montel space.

Exercise B.3.35. Prove that C∞(U) and D(U) are Montel spaces, where U ⊂ Rn

is open and non-empty.

Exercise B.3.36. Let Ω ⊂ C be open and non-empty. Show that the space H(Ω)of analytic functions on Ω is a Montel space.

Exercise B.3.37. Prove that the Schwartz space S(Rn) is a Montel space.

Exercise B.3.38. Let U ⊂ Rn be open and non-empty. Show that Ck(U) is not aMontel space if k ∈ N0.

B.3.1 Topological tensor products

In this section we review the topological tensor products. If the reader is interestedin more details on this subject we refer to [87] and to [134].

Definition B.3.39 (Projective tensor product). Let X ⊗ Y be the algebraic tensorproduct of locally convex spaces X, Y . The projective tensor topology or the π-topology of X⊗Y is the strongest topology for which the bilinear mapping ((x, y) �→x⊗y) : X×Y → X⊗Y is continuous. This topological space is denoted by X⊗πY ,and its completion by X⊗πY .


Exercise B.3.40. Let X, Y be locally convex spaces over C. Show that the dualof X ⊗π Y (and also of X⊗πY ) is isomorphic to the space of continuous bilinearmappings X × Y → C.

Exercise B.3.41. Let X, Y be locally convex metrisable spaces. Show that X⊗πYis a Frechet space. Moreover, if X, Y are barreled, show that X ⊗ Y is barreled.

Exercise B.3.42. Let X, Y be locally convex metrisable barreled spaces. Show thatX ⊗ Y is barreled.

Exercise B.3.43 (Projective Banach tensor product). Let X, Y be Banach spaces.For f ∈ X ⊗ Y , define

‖f‖π := inf

⎧⎨⎩∑j

‖xj‖ ‖yj‖ : f =∑

j

xj ⊗ yj

⎫⎬⎭ .

Show that f �→ ‖f‖π is a norm on X ⊗ Y , and that the corresponding normtopology is the projective tensor topology.

Exercise B.3.44. Let X, Y be locally convex spaces over C. Show that the algebraictensor product X ⊗ Y can be identified with the space B(X ′, Y ′) of continuousbilinear functionals X ′×Y ′ → C, where X ′ and Y ′ are the dual spaces with weaktopologies.

Definition B.3.45 (Injective tensor product). Let X, Y be locally convex spacesover C. Let B(X ′, Y ′) be the space of those bilinear functionals X ′×Y ′ → C thatare continuous separately in each variable. Endow B(X ′, Y ′) with the topology τof uniform convergence on the products of an equicontinuous subset of X ′ and anequicontinuous subset of Y ′. Interpreting X⊗Y ⊂ B(X ′, Y ′) as in Exercise B.3.44,let the injective tensor topology be the restriction of τ to X ⊗ Y . This topologicalspace is denoted by X ⊗ε Y , and its completion by X⊗εY .

Exercise B.3.46. Let X, Y be locally convex spaces over C. Show that the bilinearmapping ((x, y) �→ x ⊗ y) : X × Y → X ⊗ε Y is continuous. From this, deducethat the injective topology of X ⊗ Y is coarser than the projective topology (i.e.,is a subset of the projective topology).

Exercise B.3.47. Studying the mapping ((x, y) �→ x⊗y) : X×Y → X⊗Y , explainhow the inclusion X⊗πY ⊂ X⊗εY should be understood.

Exercise B.3.48 (Injective Banach tensor product). Let X, Y be Banach spaces.For f ∈ X ⊗ Y , define

‖f‖ε := sup {|x′ ⊗ y′(f)| : x′ ∈ X ′, y′ ∈ Y ′, ‖x′‖ = 1 = ‖y′‖} .

Show that f �→ ‖f‖ε is a norm on X ⊗ Y , and that the corresponding normtopology is the injective tensor topology.


Definition B.3.49 (Nuclear space). A locally convex space X is called nuclear ifX⊗πY = X⊗εY for every locally convex space Y (where the equality of sets isunderstood in the sense of Exercise B.3.47). In such a case, these completed tensorproducts are written simply X⊗Y .

Exercise B.3.50. Let X, Y be nuclear spaces, and let M,N ⊂ X be vector sub-spaces such that N is closed. Show that M , X/N , X × Y and X⊗Y are nuclearspaces.

Exercise B.3.51. Show that C∞(U), D(U), S(Rn), H(Ω) and their dual spaces (ofdistributions) are nuclear.

Exercise B.3.52. Let X, Y be Frechet spaces and X nuclear. Show that L(X ′, Y ) ∼=X⊗Y , L(X, Y ) ∼= X ′⊗Y , and that X ′⊗Y ′ ∼= (X⊗Y )′.

Exercise B.3.53. Prove the following Schwartz Kernel Theorem B.3.55:

Remark B.3.54. In the following Schwartz Kernel Theorem B.3.55, we denote〈ψ, Aφ〉 := (Aφ)(ψ), and 〈ψ ⊗ φ,KA〉 := KA(ψ ⊗ φ).

Theorem B.3.55 (Schwartz Kernel Theorem). Let U ⊂ Rm, V ⊂ Rn be open andnon-empty, and let A : D(U)→ D′(V ) be linear and continuous. Then there existsa unique distribution KA ∈ D′(V )⊗D′(U) ∼= D′(V × U) such that

〈ψ, Aφ〉 = 〈ψ ⊗ φ,KA〉

for every φ ∈ D(U) and ψ ∈ D(V ). Moreover, if A : D(U)→ C∞(V ) is continuousthen it can be interpreted that KA ∈ C∞(V,D′(U)).

Definition B.3.56 (Schwartz kernel). The distribution KA in Theorem B.3.55 iscalled the Schwartz kernel of A, written informally as

Aφ(x) =∫

V

KA(x, y) φ(y) dy.

Exercise B.3.57. Let A : D(U) → D′(V ) be continuous and linear as in The-orem B.3.55. Give necessary and sufficient conditions for A such that KA ∈C∞(V × U).

Exercise B.3.58. Find variants of the Schwartz Kernel Theorem B.3.55 forSchwartz functions and for tempered distributions.

B.4 Banach spaces

Definition B.4.1 (Seminorm and norm; normed spaces). Let X be a K-vectorspace. A mapping p : X → R is a seminorm if{

p(x + y) ≤ p(x) + p(y),p(λx) = |λ| p(x)

B.4. Banach spaces 93

for every x, y ∈ X and λ ∈ R. If p : X → R is a seminorm for which p(x) = 0implies x = 0, then it is called a norm. Typically, a norm on X is written asx �→ ‖x‖X or simply ‖x‖. A vector space with a norm is called a normed space.

Example. On the vector space K, the absolute value mapping x �→ |x| is a norm.

Exercise B.4.2. Let p : X → [0,∞) be a seminorm on a K-vector space X and

x ∼ ydefinition⇐⇒ p(x− y) = 0.

Prove the following claims:

(a) ∼ is an equivalence relation on X.(b) The set L := {[x] : x ∈ X}, with [x] := {y ∈ X : x ∼ y}, is an R-vector

space when endowed with operations

[x] + [y] := [x + y], λ[x] := [λx]

and the norm [x] �→ p(x).

Exercise B.4.3. Let wj ≥ 0 for every j ∈ J . Define

∑j∈J

wj := sup

{∑k∈K

wk : K ⊂ J finite

}.

Show that {j ∈ J : wj > 0} is at most countable if∑

j∈J wj <∞.

Exercise B.4.4. For x ∈ KJ define

‖x‖�p :=

⎧⎨⎩(∑

j∈J |xj |p)1/p

, if 1 ≤ p <∞,

supj∈J |xj |, if p =∞,

where xj := x(j). Show that �p(J) :={x ∈ KJ : ‖x‖�p <∞

}is a Banach space

with respect to the norm x �→ ‖x‖�p .

Exercise B.4.5. Norms p1, p2 on a vector space V are called (Lipschitz) equivalentif a−1p1(x) ≤ p2(x) ≤ ap1(x) for every x ∈ V , where a ≥ 1 is a constant. Showthat any two norms on a finite-dimensional space V are equivalent. Consequently,a finite-dimensional normed space is a Banach space.

Exercise B.4.6. Let K be a compact space. Show that

C(K) := {f : K → K | f continuous }

is a Banach space when endowed with the norm

f �→ ‖f‖C(K) := supx∈K

|f(x)|.


Remark B.4.7. The previous exercise deals with special cases of of Lp(μ), theLebesgue p-spaces. These Banach spaces are introduced in Definition C.4.6.

Definition B.4.8 (Normed and Banach spaces). Notice that the norm metric((x, y) �→ ‖x − y‖) : X × X → R is a metric on X. Let τX denote the corre-sponding metric topology, called the norm topology, where the open ball centeredat x ∈ X with radius r > 0 is

BX(x, r) = B(x, r) = {y ∈ X : ‖x− y‖ < r} .

Ball BX(0, 1) is called the open unit ball. The closed ball centered at x ∈ X withradius r > 0 is

B(x, r) := {y ∈ X : ‖x− y‖ ≤ r} .

Notice that here B(x, r) = B(x, r), where S refers to the norm closure of a setS ⊂ X. A Banach space is a normed space where the norm metric is complete.

Exercise B.4.9. Show that V := {x ∈ �p(J) : {j ∈ J : xj = 0} finite} is a densenormed vector subspace of �p(J).

Definition B.4.10 (Bounded linear operators). A linear mapping A : X → Ybetween normed spaces X, Y is called bounded, denoted A ∈ L(X, Y ), if

‖Ax‖ ≤ C ‖x‖

for every x ∈ X, where C <∞ is a constant. The norm of A ∈ L(X, Y ) is

‖A‖ := supx∈X: ‖x‖≤1

‖Ax‖.

This norm is also called the operator norm and is often denoted by ‖A‖op. Weoften abbreviate L(X) := L(X, X).

Exercise B.4.11. Let A : X → Y be a linear operator between normed spaces Xand Y . Show that A is bounded if and only if it is continuous.

Exercise B.4.12. Show that L(X, Y ) is really a normed space.

Exercise B.4.13. Show that ‖AB‖ ≤ ‖A‖‖B‖ if B ∈ L(X, Y ) and A ∈ L(Y, Z).

Exercise B.4.14. Show that L(X, Y ) is a Banach space if Y is a Banach space.

Definition B.4.15 (Duals). Let V be a Banach space over K. The dual of V is thespace

V ′ = L(V, K) := {A : V → K | A bounded and linear}endowed with the (operator) norm

A �→ ‖A‖ := supf∈V : ‖f‖V ≤1

|A(f)|.


Exercise B.4.16. Prove that V ′ is a Banach space.

Definition B.4.17 (Compact linear operator). Let X, Y be normed spaces, andlet B = B(0, 1) = {x ∈ X : ‖x‖ ≤ 1}. A linear mapping A : X → Y is calledcompact, written A ∈ LC(X, Y ), if the closure of A(B) ⊂ Y is compact. We alsowrite LC(X) := LC(X, X).

Exercise B.4.18. Show that LC(X, Y ) is a linear subspace of L(X, Y ), and it isclosed if Y is complete.

Exercise B.4.19. Let B0, C0 ∈ L(X, Y ) and B1, C1 ∈ L(Y, Z) such that C0, C1 arecompact. Show that C1B0, B1C0 are compact.

Lemma B.4.20. (Almost Orthogonality Lemma [F. Riesz]) Let X be a normedspace with closed subspace Y = X. For each ε > 0 there exists xε ∈ X such that‖xε‖ = 1 and dist(xε, Y ) ≥ 1− ε.

Proof. Let z ∈ X \ Y and r := dist(z, Y ) > 0. Take y = yε ∈ Y such thatr ≤ ‖z − y‖ < (1− ε)−1r. Let xε := (z − y)/‖z − y‖. If u ∈ Y then

‖xε − u‖ =∥∥∥∥ z − y

‖z − y‖ − u

∥∥∥∥=

‖z − (y + ‖z − y‖u)‖‖z − y‖

>r

(1− ε)−1r

= 1− ε,

showing that dist(xε, Y ) ≥ 1− ε. �

Theorem B.4.21 (Riesz’s Compactness Theorem). Let X be a normed space. ThenX is finite-dimensional if and only if B(0, 1) is compact.

Proof. By the Heine–Borel Theorem, a set in a finite-dimensional normed spaceis compact if and only if it is bounded.

Now let X be infinite-dimensional. Let 0 < ε < 1 and x1 ∈ X such that‖x1‖ = 1. Inductively, let Yk := span{xj}k

j=1 = X, and choose xk+1 ∈ X \ Yk = ∅such that ‖xk+1‖ = 1 and dist(xk+1, Yk) > 1−ε. Then it is clear that the sequence(xk)∞k=1 does not have a converging subsequence. Hence by Theorem A.13.4, B(0, 1)is not compact. �

Remark B.4.22 (Is identity compact?). Riesz’s Compactness Theorem B.4.21could also be stated: a normed space X is finite-dimensional if and only if theidentity mapping I = (x �→ x) : X → X is compact. This together with the resultsof Exercises B.4.18 and B.4.19 proves that LC(X) is a closed two-sided properideal of L(X), where X is a Banach space and X is not finite-dimensional.


Theorem B.4.23 (Baire’s Theorem). Let (X, d) be a complete metric space andUj ⊂ X be dense and open for each k ∈ Z+. Then G =

⋂∞k=1 Uk is dense.

Proof. We must show that G∩B(x0, r0) = ∅ for any x0 ∈ X and r0 > 0. AssumingX = ∅, take x1 and r1 such that

B(x1, r1) ⊂ U1 ∩ B(x0, r0).

Inductively, we choose xk+1 and rk+1 < 1/k so that

B(xk+1, rk+1) ⊂ Uk+1 ∩ B(xk, rk).

Then (xk)∞k=1 is a Cauchy sequence, thus converging to some x ∈ X by complete-ness. By construction, x ∈ G ∩ B(x0, r0). �Exercise B.4.24 (Baire’s Theorem and interior points). Clearly, Baire’s Theo-rem B.4.23 is equivalent to the following: in a complete metric space, a countableunion of sets without interior points is without interior points. Use this to provethat an algebraic basis of an infinite-dimensional Banach space must be uncount-able.

Theorem B.4.25 (Hahn–Banach Theorem). Let X be a real normed space andf : Mf → R be bounded and linear on a vector subspace Mf ⊂ X. Then thereexists extension F : X → R such that F |Mf

= f and ‖F‖ = ‖f‖.

Proof. Let

S := {h : Mh → R | h linear on vector subspace Mh ⊂ X,

Mf ⊂Mh, ‖h‖ = ‖f‖} .

Then f ∈ S = ∅. Endow S with the partial order

g ≤ h ⇐⇒{

Mg ⊂Mh,

g = h|Mg.

Take a chain (fj)j∈J ⊂ S. Then fj ≤ h for each j ∈ J , where h ∈ S is definedso that Mh =

⋃j∈J Mfj , h|Mfj

= fj . Thereby, in view of Zorn’s lemma (TheoremA.4.10), there is a maximal element F : MF → R in S. Suppose MF = X. Thentake x0 ∈ X \MF . Given a ∈ R, define

Ga : MF + Rx0 → R, Ga(u + tx0) = F (u)− ta.

Then Ga is bounded, linear, and Ga|MF= F . Hence ‖Ga‖ ≥ ‖F‖ = ‖f‖. Could

it be that ‖Ga‖ = ‖F‖ (this would contradict the maximality of F )? For anyu, v ∈MF ,

|F (u)− F (v)| = |F (u− v)|≤ ‖F‖ ‖u− v‖≤ ‖F‖ (‖u + x0‖+ ‖v + x0‖) .


Hence there exists a0 ∈ R such that

F (u)− ‖F‖ ‖u + x0‖ ≤ a0 ≤ F (v) + ‖F‖ ‖v + x0‖

for every u, v ∈MF . Thus

|F (w)− a0| ≤ ‖F‖ ‖w + x0‖

for every w ∈MF . From this (assuming that non-trivially t = 0), we get

|Ga0(u + tx0)| = |t| |u/t− a0|≤ |t| ‖F‖ ‖u/t + x0‖= ‖F‖ ‖u + tx0‖;

but this means ‖Ga0‖ ≤ ‖F‖, a contradiction. �Exercise B.4.26 (Complex version of the Hahn–Banach Theorem). Prove the com-plex version of the Hahn–Banach Theorem: Let X be a complex normed space andf : Mf → C be bounded and linear on a vector subspace Mf ⊂ X. Then thereexists an extension F : X → C such that F |Mf

= f and ‖F‖ = ‖f‖.Corollary B.4.27. Let X be a normed space and x ∈ X. Then

‖x‖ = max {|F (x)| : F ∈ L(X, K), ‖F‖ ≤ 1} .

Corollary B.4.28 (Hahn–Banach =⇒ Riesz’ Compactness Theorem). Let X be anormed space. Then B(0, 1) is compact if and only if it is finite-dimensional.

Proof. By the Heine–Borel Theorem, a closed set in a finite-dimensional normedspace is compact if and only if it is bounded.

The proof for the converse follows [28]: Suppose X is locally compact andlet S1 := {x ∈ X : ‖x‖ = 1}. Then

{S1 ∩Ker(f) : f ∈ L(X, K)

}is a family of

compact sets, whose intersection is empty by the Hahn–Banach Theorem. Therebythere exists {fk}n

k=1 ⊂ L(X, K) such that

n⋂k=1

S1 ∩Ker(fk) = ∅, i.e.,n⋂

k=1

Ker(fk) = {0}.

Since the co-dimension of Ker(fk) ≤ 1, this implies that dim(X) ≤ n. �Theorem B.4.29 (Banach–Steinhaus Theorem, or Uniform Boundedness Princi-ple). Let X be a Banach space, let Y be a normed space, and let {Aj}j∈J ⊂L(X, Y ) be such that

supj∈J‖Ajx‖ <∞

for every x ∈ X. Then supj∈J‖Aj‖ <∞.


Proof. Let pj(x) := ‖Ajx‖ and p(x) := sup{pj(x) | j ∈ J}. Clearly, p, pj : X → Rare seminorms. Moreover, pj is continuous for every j ∈ J , and we must show thatalso p is continuous. Since pj is continuous for every j ∈ J , set

Uk := {x ∈ X : p(x) > k} =⋃j∈J

{x ∈ X : pj(x) > k}

is open. Now⋂∞

k=1 Uk = ∅, so that by Baire’s Theorem B.4.23 there exists k0 ∈ Z+

for which Uk0 = X; actually, here U1 = X, because U1 = k−1Uk. Choose x0 ∈ Xand r0 > 0 such that

B(x0, r0) ⊂ X \ U1.

If z ∈ B(0, 1) then

r0 p(z) = p(r0z)≤ p(x0 + r0z) + p(−x0)≤ 2.

Thus ‖Aj‖ ≤ 2/r0 for every j ∈ J . �

Definition B.4.30 (Open mappings). A mapping f : X → Y between topologicalspaces X, Y is said to be open, if f(U) ⊂ Y is open for every open U ⊂ X.

Theorem B.4.31 (Open Mapping Theorem). Let A ∈ L(X, Y ) be surjective, whereX, Y are Banach spaces. Then A is open.

Proof. It is sufficient to show that BY (0, r) ⊂ A(BX(0, 1)) for some r > 0. Foreach k ∈ Z+, set Uk := Y \ A(BX(0, k)) is open. Now

⋂∞k=1 Uk = ∅, because A

is surjective. By Baire’s Theorem B.4.23, Uk0 = Y for some k0 ∈ Z+; actually,U1 = Y , because A(BX(0, 1)) = k−1A(BX(0, k)). Take y0 ∈ Y and r0 > 0 suchthat

BY (y0, r0) ⊂ Y \ U1.

NowBY (y0, r0) ⊂ Y \ U1 = A(BX(0, 1)).

Let ε > 0 and y ∈ BY (0, r0). Take w1, w2 ∈ BX(0, 1) such that

‖y0 −Aw1‖ < ε/2,

‖(y0 + y)−Aw2‖ < ε/2.

Then w1 − w2 ∈ BX(0, 2) and ‖y −A(w1 − w2)‖ < ε. By linearity, this yields

∀ε > 0 ∀y ∈ BY (0, r0) ∃x ∈ BX(0, 2‖y‖/r0) : ‖y −Ax‖ < ε.

Thus if z ∈ BY (0, r0), take x0 ∈ BX(0, 2) such that ‖z−Ax0‖ < r0/2. Inductively,choose xk ∈ BX(0, 21−k) such that ‖z − A

∑kj=0 xj‖ < 21−kr0. Now

∑kj=0 xj →k


x ∈ BX(0, 4) ⊂ BX(0, 5), because X is complete. We have z = Ax by continuityof A. Thereby

BY (0, r0) ⊂ A(BX(0, 5)),

implying BY (0, r0/5) ⊂ A(BX(0, 1)). �Corollary B.4.32 (Bounded Inverse Theorem). Let B ∈ L(X, Y ) be bijective -between Banach spaces X, Y . Then B−1 is continuous.

Definition B.4.33 (Graph). The graph of a mapping f : X → Y is

Γ(f) := {(x, f(x)) | x ∈ X}⊂ X × Y.

Theorem B.4.34 (Closed Graph Theorem). Let A : X → Y be a linear mappingbetween Banach spaces X, Y . Then A is continuous if and only if its graph is closedin X × Y .

Proof. Suppose A is continuous. Take a Cauchy sequence ((xj , Axj))∞j=1 of Γ(A) ⊂X×Y . Then (xj)∞j=1 is a Cauchy sequence of X, thereby converging to some x ∈ Xby completeness. Then Axj → Ax by the continuity of A. Hence (xj , Axj) →(x,Ax) ∈ Γ(A); the graph is closed.

Now assume that Γ(A) ⊂ X × Y is closed. Thus the graph is a Banachsubspace of X × Y . Define a mapping B := (x �→ (x,Ax)) : X → Γ(A). It iseasy to see that B is a linear bijection. By the Open Mapping Theorem, B iscontinuous. This implies the continuity of A. �Definition B.4.35 (Weak∗-topology). Let x �→ ‖x‖ be the norm of a normed vectorspace X over a field K ∈ {R, C}. The dual space X ′ = L(X, K) of X is a set ofbounded linear functionals f : X → K, having a norm

‖f‖ := supx∈X: ‖x‖≤1

|f(x)|.

This endows X ′ with a Banach space structure. However, it is often better to usea weaker topology for the dual: let us define x(f) := f(x) for every x ∈ X andf ∈ X ′; this gives the interpretation X ⊂ X ′′ := L(X ′, K), because

|x(f)| = |f(x)| ≤ ‖f‖‖x‖.

So we may treat X as a set of functions X ′ → K, and we define the weak∗-topologyof X ′ to be the X-induced2 topology of X ′.

Theorem B.4.36 (Banach–Alaoglu Theorem). Let X be a Banach space. Then theclosed unit ball

K := BX′(0, 1) = {φ ∈ X ′ : ‖φ‖X′ ≤ 1}of X ′ is weak∗-compact.2see Definition A.18.1


Proof. Due to Tihonov’s Theorem A.18.8,

P :=∏x∈X

{λ ∈ C : |λ| ≤ ‖x‖} = D(0, ‖x‖)X

is compact in the product topology τP . Any element f ∈ P is a mapping

f : X → C such that f(x) ≤ ‖x‖.Hence K = X ′ ∩ P . Let τ1 and τ2 be the relative topologies of K inherited fromthe weak∗-topology τX′ of X ′ and the product topology τP of P , respectively. Weshall prove that τ1 = τ2 and that K ⊂ P is closed; this would show that K is acompact Hausdorff space.

First, let φ ∈ X ′, f ∈ P , S ⊂ X, and δ > 0. Define

U(φ, S, δ) := {ψ ∈ X ′ : x ∈ S ⇒ |ψx− φx| < δ},V (f, S, δ) := {g ∈ P : x ∈ S ⇒ |g(x)− f(x)| < δ}.

Then

U := {U(φ, S, δ) | φ ∈ X ′, S ⊂ X finite, δ > 0},V := {V (f, S, δ) | f ∈ P, S ⊂ X finite, δ > 0}

are bases for the topologies τX′ and τP , respectively. Clearly

K ∩ U(φ, S, δ) = K ∩ V (φ, S, δ),

so that the topologies τX′ and τP agree on K, i.e., τ1 = τ2.Still we have to show that K ⊂ P is closed. Let f ∈ K ⊂ P . First we show

that f is linear. Take x, y ∈ X, λ, μ ∈ C and δ > 0. Choose φδ ∈ K such that

f ∈ V (φδ, {x, y, λx + μy}, δ).Then

|f(λx + μy)− (λf(x) + μf(y))|≤ |f(λx + μy)− φδ(λx + μy)|+ |φδ(λx + μy)− (λf(x) + μf(y))|= |f(λx + μy)− φδ(λx + μy)|+ |λ(φδx− f(x)) + μ(φδy − f(y))|≤ |f(λx + μy)− φδ(λx + μy)|+ |λ| |φδx− f(x)|+ |μ| |φδy − f(y)|≤ δ (1 + |λ|+ |μ|).

This holds for every δ > 0, so that actually

f(λx + μy) = λf(x) + μf(y),

f is linear! Moreover, ‖f‖ ≤ 1, because

|f(x)| ≤ |f(x)− φδx|+ |φδx| ≤ δ + ‖x‖.Hence f ∈ K, K is closed. �


Remark B.4.37. The Banach–Alaoglu Theorem B.4.36 implies that a boundedweak∗-closed subset of the dual space is a compact Hausdorff space in the relativeweak∗-topology. However, in a normed space norm-closed balls are compact if andonly if the dimension is finite!

B.4.1 Banach space adjoint

We now come back to the adjoints of Banach spaces and of operators introducedin Definition B.4.15. Here we give a condensed treatment to acquaint the readerwith the topic.

Definition B.4.38 (Duality). Let X be a Banach space and X ′ = L(X, K) its dual.For x ∈ X and x′ ∈ X ′ let us write

〈x, x′〉 := x′(x).

We endow X ′ with the norm x′ �→ ‖x′‖ given by

‖x′‖ := sup {|〈x, x′〉| : x ∈ X, ‖x‖ ≤ 1} .

Exercise B.4.39. Let X be a Banach space and x ∈ X. Show that

‖x‖ = sup {|〈x, x′〉| : x′ ∈ X ′, ‖x′‖ ≤ 1} .

Exercise B.4.40. Let X, Y be Banach spaces with respective duals X ′, Y ′. LetA ∈ L(X, Y ). Show that there exists a unique A′ ∈ L(Y ′, X ′) such that

〈Ax, y′〉 = 〈x,A′(y′)〉 (B.1)

for every x ∈ X and y′ ∈ Y ′. Prove also that

‖A′‖ = ‖A‖.

Definition B.4.41 (Adjoint operator). Let A ∈ L(X, Y ) be as in Exercise B.4.40.Then A′ ∈ L(Y ′, X ′) defined by (B.1) is called the (Banach) adjoint of A.

Exercise B.4.42. Show that A ∈ L(X, Y ) is compact if and only if A′ ∈ L(Y ′, X ′)is compact.

Definition B.4.43 (Complemented subspace). A closed subspace V of a topologicalvector space X is said to be complemented in X by a subspace W ⊂ X if{

V + W = X andV ∩W = {0}.

Then we write X = V ⊕W , saying that X is the direct sum of V and W .


Exercise B.4.44. Show that a closed subspace V is complemented in X if X/V isfinite-dimensional.

Exercise B.4.45. Show that a finite-dimensional subspace of a locally convex spaceis complemented. (Hint: Hahn–Banach.)

Exercise B.4.46. Let A ∈ L(X) be compact, where X is a Banach space. Let λ bea non-zero scalar. Show that the range set

(λI −A)(X) = {λx−Ax : x ∈ X}

is closed, Ker(λI −A) = {x ∈ X : Ax = λx} is finite-dimensional, and that

dim (Ker(λI −A))= dim (Ker(λI −A′))= dim (X/((λI −A)(X)))= dim (X ′/((λI −A′)(X ′))) .

Definition B.4.47 (Reflexive space). Let X be a Banach space and X ′ = L(X, K)its dual Banach space. The second dual of X is X ′′ := (X ′)′ = L(X ′, K). It is theneasy to show that we can define a linear isometry (x �→ x′′) : X → X ′′ onto aclosed subspace of X ′′ by

x′′(f) := f(x).

Thus X can be regarded as a subspace of X ′′. If X ′′ = {x′′ : x ∈ X} then X iscalled reflexive.

Exercise B.4.48. Show that (x �→ x′′) : X → X ′′ in Definition B.4.47 has theclaimed properties.

Exercise B.4.49. Let 1 < p <∞. Show that �p = �p(Z+) is reflexive. What about�1 and �∞?

Exercise B.4.50. Show that C([0, 1]) is not reflexive.

Exercise B.4.51. Let X be a Banach space. Prove that X is reflexive if and onlyif its closed unit ball is compact in the weak topology. (Hint: Hahn–Banach andBanach–Alaoglu).

Exercise B.4.52. Let V be a closed subspace of a reflexive Banach space X. Showthat V and X/V are reflexive.

Exercise B.4.53. Show that X is reflexive if and only if X ′ is reflexive.

B.5. Hilbert spaces 103

B.5 Hilbert spaces

Definition B.5.1 (Inner product and Hilbert spaces). Let H be a C-vector space.A mapping ((x, y) �→ 〈x, y〉) : H×H → C is an inner product if

〈x + y, z〉 = 〈x, z〉+ 〈y, z〉,〈λx, y〉 = λ〈x, y〉,〈y, x〉 = 〈x, y〉,〈x, x〉 ≥ 0,

〈x, x〉 = 0 ⇒ x = 0

for every x, y ∈ H and λ, μ ∈ C. Then H endowed with the inner product is calledan inner product space. An inner product defines the canonical norm

‖x‖ := 〈x, x〉1/2;

we shall soon prove that this is a norm in the usual sense. H is called a Hilbertspace (or a complete inner product space) if it is a Banach space with respect tothe canonical norm.

Exercise B.5.2. Show that �2(J) is a Hilbert space, where

〈x, y〉 =∑j∈J

xjyj .

Definition B.5.3 (Orthogonality). Vectors x, y ∈ H are said to be orthogonal in aninner product space H, denoted x⊥y, if 〈x, y〉 = 0. For S ⊂ H, let

S⊥ := {x ∈ H | ∀y ∈ S : x⊥y} .

Subspaces M,N ⊂ H are called orthogonal, denoted by M⊥N , if 〈x, y〉 = 0 forevery x ∈ M and y ∈ N . A collection {xα}α∈I is called orthonormal if ‖xα‖ = 1for all α ∈ I and if 〈xα, xβ〉 = 0 for all α = β, α, β ∈ I.

Exercise B.5.4. Show that S⊥ ⊂ H is a closed vector subspace, and that S ⊂(S⊥)⊥. Show that if V is a closed vector subspace of H then V = (V ⊥)⊥.

Exercise B.5.5 (Pythagoras’ Theorem). Let x1, x2, . . ., xn ∈ H be mutuallyorthogonal, i.e., assume that xi⊥xj for all i = j. Prove that ‖∑n

j=1 xj‖2 =∑nj=1 ‖xj‖2. (This generalised the famous theorem of Pythagoras of Samos on

the triangles in the plane.)

Proposition B.5.6 (Cauchy–Schwarz inequality). Let H be an inner product space.Then

|〈x, y〉| ≤ ‖x‖ ‖y‖ (B.2)

for every x, y ∈ H.


Proof. We may assume that x = 0 and y = 0, otherwise the statement is trivial.For t ∈ R,

0 ≤ ‖x− ty‖2

= 〈x− ty, x− ty〉= 〈x, x〉 − t〈x, y〉 − t〈y, x〉+ t2〈y, y〉= ‖y‖2t2 − 2tRe〈x, y〉+ ‖x‖2

= ‖y‖2(

t− Re〈x, y〉‖y‖2

)2

+

(‖x‖2 −

(Re〈x, y〉‖y‖

)2)

.

Taking t = Re〈x,y〉‖y‖2 , we get

|Re〈x, y〉| ≤ ‖x‖ ‖y‖

for every x, y ∈ H. Now 〈x, y〉 = |〈x, y〉| eiφ for some φ ∈ R, and

|〈x, y〉| = 〈e−iφx, y〉 =∣∣Re〈e−iφx, y〉

∣∣ ≤ ‖e−iφx‖ ‖y‖ = ‖x‖ ‖y‖.

This completes the proof. �

Corollary B.5.7 (Triangle inequality). Let H be an inner product space. Then

‖x + y‖ ≤ ‖x‖+ ‖y‖.

Consequently, the canonical norm of an inner product space is a norm in the usualsense.

Proof. Now

‖x + y‖2 = 〈x + y, x + y〉= 〈x, x〉+ 〈x, y〉+ 〈y, x〉+ 〈y, y〉

(B.2)

≤ ‖x‖2 + 2 ‖x‖ ‖y‖+ ‖y‖2

= (‖x‖+ ‖y‖)2,


Remark B.5.8. One may naturally study R-Hilbert spaces, where the scalar fieldis R and the inner product takes real values. Then

〈x, y〉 =‖x‖2 + ‖y‖2 − ‖x− y‖2

2.

Thus the inner product can be recovered from the norm here.


Exercise B.5.9. Prove this remark. In (C-) Hilbert spaces, prove that

〈x, y〉 =‖x + y‖2 − ‖x− y‖2 + i‖x + iy‖2 − i‖x− iy‖2

4.

Exercise B.5.10. Every Hilbert space is canonically a Banach space, but not viceversa: in a real Banach space, (x, y) �→ (‖x‖2 +‖y‖2−‖x−y‖2)/2 does not alwaysdefine an inner product. Present some examples.

Lemma B.5.11. Let H be a Hilbert space. Suppose C ⊂ H is closed, convex andnon-empty. Then there exists unique z ∈ C such that ‖z‖ = inf{‖x‖ : x ∈ C}.

Proof. Let r := inf{‖x‖ : x ∈ C}. For any x, y ∈ H, the parallelogram identity

‖x + y‖2 + ‖x− y‖2 = 2(‖x‖2 + ‖y‖2) (B.3)

holds. Take a sequence (xk)∞k=1 in C such that ‖xk‖ →k→∞ r. Now (xj+xk)/2 ∈ Cdue to convexity, so that 4r2 ≤ ‖xj + xk‖2. Hence

4r2 + ‖xj − xk‖2 ≤ ‖xj + xk‖2 + ‖xj − xk‖2(B.3)= 2(‖xj‖2 + ‖xk‖2)

−−−−−→j,k→∞

4r2,

implying ‖xj − xk‖ →j,k→∞ 0. Thus (xk)∞k=1 is a Cauchy sequence, convergingto some z ∈ C with ‖z‖ = r (recall that H is complete and C ⊂ H is closed). Ifz′ ∈ C satisfies ‖z′‖ = d then the alternating sequence (z, z′, z, z′, . . .) would be aCauchy sequence, by the reasoning above: hence z = z′. �

Exercise B.5.12 (Parallelogram identity). Show that the parallelogram identity(B.3):

‖x + y‖2 + ‖x− y‖2 = 2(‖x‖2 + ‖y‖2)holds for all x, y ∈ H.

Lemma B.5.13. Let M be a vector subspace in a Hilbert space H. Let ‖z‖ ≤ ‖z+u‖for every u ∈M . Then z ∈M⊥.

Proof. To get a contradiction, assume 〈z, v〉 = 0 for some v ∈ M . Multiplying vby a scalar, we may assume that Re〈z, v〉 = 0. If r ∈ R then

0 ≤ ‖z − rv‖2 − ‖z‖2 = r2‖v‖2 − 2rRe〈z, v〉 = r(r‖v‖2 − 2Re〈z, v〉),

but this inequality fails when r is between 0 and 2Re〈z, v〉/‖v‖2. �

Definition B.5.14 (Orthogonal projection). Let M be a closed subspace of a Hilbertspace H. Then we may define PM : H → H so that PM (x) ∈M is the point in Mclosest to x ∈ H. Mapping PM is called the orthogonal projection onto M .


Proposition B.5.15. Operator PM : H → H defined above is linear, and ‖PM‖ = 1(unless M = {0}). Moreover, PM⊥ = I − PM .

Proof. Let x ∈ H, P := PM and Q = I − P . By Definition B.5.14, P (x) ∈M and

‖Q(x)‖ ≤ ‖Q(x) + u‖

for every u ∈ M . This implies Q(x) ∈ M⊥ by Lemma B.5.13. Let x, y ∈ H andλ, μ ∈ C. Since

λx = λ (P (x) + Q(x)) ,

μy = μ (P (y) + Q(y)) ,

λx + μy = P (λx + μy) + Q(λx + μy),

we get

M � P (λx + μy)− λP (x)− μP (y) = λQ(x) + μQ(y)−Q(λx + μy) ∈M⊥.

This implies the linearity of P , because M ∩M⊥ = {0}. Finally,

‖x‖2 = ‖Px + Qx‖2 = ‖Px‖2 + ‖Qx‖2 + 2%〈Px, Qx〉 = ‖Px‖2 + ‖Qx‖2;

in particular, ‖Px‖ ≤ ‖x‖. �

Remark B.5.16. We have proven that

H = M ⊕M⊥.

This means that M,M⊥ are closed subspaces of the Hilbert space H such thatM⊥M⊥ and that M + M⊥ = H.

Definition B.5.17 (Direct sum). Let {Hj : j ∈ J} be a family of pair-wise orthog-onal closed subspaces of H. If the span of

⋃j∈J Hj is dense in H then H is said

to be a direct sum of {Hj : j ∈ J}, denoted by

H =⊕j∈J

Hj .

If H is a direct sum of {Mj}kj=1, we write H =

⊕kj=1 Mj . Especially, M1 ⊕M2 =⊕2

j=1 Mj .

Remark B.5.18. If H is a Hilbert space, it is easy to see that f = (x �→ 〈x, y〉) :H → C is a linear functional, and ‖f‖ = ‖y‖ due to the Cauchy–Schwarz in-equality and to f(y) = ‖y‖2. Actually, there are no other kinds of bounded linearfunctionals on a Hilbert space:


Theorem B.5.19 (Riesz (Hilbert Space) Representation Theorem). Let f : H → Cbe a bounded linear functional on a Hilbert space H. Then there exists a uniquey ∈ H such that

f(x) = 〈x, y〉for every x ∈ H. Moreover, ‖f‖ = ‖y‖.

Sometimes this theorem is also called the Frechet–Riesz (representation) the-orem.

Proof. Assume the non-trivial case f = 0. Thus we may choose u ∈ Ker(f)⊥

for which ‖u‖ = 1. Pursuing for a suitable representative y ∈ H, we notice thatf(u) = 〈u, f(u)u〉, inspiring an investigation:

〈x, f(u)u〉 − f(x) = 〈f(u)x, u〉 − 〈f(x)u, u〉= 〈f(u)x− f(x)u, u〉= 0,

since f(u)x − f(x)u ∈ Ker(f). Thus f(x) = 〈x, f(u)u〉 for every x ∈ H. Further-more, if f(x) = 〈x, y〉 = 〈x, z〉 for every x ∈ H then

0 = f(x)− f(x) = 〈x, y〉 − 〈x, z〉 = 〈x, y − z〉 x=y−z= ‖y − z‖2,

so that y = z. �Definition B.5.20 (Adjoint operator). Let H be a Hilbert space, z ∈ H and A ∈L(H). Then a bounded linear functional on H is defined by x �→ 〈Ax, z〉, so thatby Theorem B.5.19 there exists a unique vector A∗z ∈ H satisfying

〈Ax, z〉 = 〈x, A∗z〉

for every x ∈ H. This defines a mapping A∗ : H → H, which is called the adjointof A ∈ L(H). If A∗ = A then A is called self-adjoint.

Exercise B.5.21. Let λ ∈ C and A,B ∈ L(H). Show that (λA)∗ = λA∗, (A+B)∗ =A∗ + B∗ and (AB)∗ = B∗A∗.

Exercise B.5.22. Show that the adjoint operator A∗ : H → H of A ∈ L(H) is linearand bounded. Moreover, show that (A∗)∗ = A, ‖A∗A‖ = ‖A‖2 and ‖A∗‖ = ‖A‖.Lemma B.5.23. Let A∗ = A ∈ L(H). Then

‖A‖ = supx: ‖x‖≤1

|〈Ax, x〉| .

Proof. Let r := sup {|〈Ax, x〉| : x ∈ H, ‖x‖ ≤ 1}. Then

r(B.2)

≤ supx: ‖x‖≤1

‖Ax‖ ‖x‖ ≤ ‖A‖.


Let us assume that Ax = 0 for ‖x‖ = 1, and let y := Ax/‖Ax‖. Since A∗ = A, wehave 〈Ax, y〉 = 〈x,Ay〉 = 〈Ay, x〉 ∈ R, so that

‖Ax‖ = 〈Ax, y〉A∗=A=

14

(〈A(x + y), x + y〉 − 〈A(x− y), x− y〉)

≤ 14

(|〈A(x + y), x + y〉|+ |〈A(x− y), x− y〉|)

≤ r

4(‖x + y‖2 + ‖x− y‖2

)(B.3)=

r

2(‖x‖2 + ‖y‖2

)= r.

This concludes the proof. �

Lemma B.5.24. Let H = {0}. Let A∗ = A ∈ L(H) be compact. Then there existsa non-zero x ∈ H such that Ax = +‖A‖x or Ax = −‖A‖x.

Proof. Assume the non-trivial case ‖A‖ > 0. By Lemma B.5.23, we may chooseλ ∈ {±‖A‖} to be an accumulation point of the set {〈Ax, x〉 : x ∈ H, ‖x‖ ≤ 1}.For each k ∈ Z+, take xk ∈ H such that ‖xk‖ ≤ 1 and 〈Axk, xk〉 →k→∞ λ.Since A is compact, by Theorem A.13.4 it follows that the sequence (Axk)∞k=1 hasa convergent subsequence; omitting elements from the sequence, we may assumethat z := limk Axk ∈ H exists. Now

0 ≤ ‖Axk − λxk‖2

= ‖Axk‖2 + λ2‖xk‖2 − 2λ〈Axk, xk〉≤ ‖A‖2 + λ2 − 2λ〈Axk, xk〉

−−−−→k→∞

0,

implying that limk λxk exists and is equal to limk Axk = z. Finally, let x := z/λ,so that by continuity Ax = limk Axk = λx. �

Theorem B.5.25 (Diagonalisation of compact self-adjoint operators). Let H beinfinite-dimensional and A∗ = A ∈ L(H) be compact. Then there exist {λk}∞k=1 ⊂R and an orthonormal set {xk}∞k=1 ⊂ H such that |λk+1| ≤ |λk|, limk λk = 0 and

Ax =∞∑

k=1

λk〈x, xk〉 xk

for every x ∈ H.

Proof. By Lemma B.5.24, take λ1 ∈ R and x1 ∈ H such that ‖x1‖ = 1, Ax1 = λ1x1

and ‖A1‖ = |λ1|. Then we proceed by induction as follows. Let Hk :=({xj}k−1

j=1

)⊥.


Then A∗k = Ak := A|Hk∈ L(Hk) is compact as it is a finite-dimensional operator,

so we may apply Lemma B.5.24 to choose λk ∈ R and xk ∈ Hk such that ‖xk‖ = 1,Axk = λkxk and ‖Ak‖ := |λk|. Since H is infinite-dimensional, we obtain anorthonormal family {xk}∞k=1 ⊂ H, and Axk = λkxk for each k ∈ Z+, where|λk+1| ≤ |λk|.

Since A is compact, (Axk)∞k=1 has a converging subsequence. Actually,(Ak)∞k=1 itself must converge and λk → 0, because

‖Axj −Axk‖ = ‖λjxj − λkxk‖ =√

λ2j + λ2

k ≥ |λk|

for every j, k ∈ Z+. If x ∈ H then zk := x−∑k−1j=1 〈x, xk〉 xk ∈ Hk, and

‖Azk‖ = ‖Akzk‖ ≤ ‖Ak‖‖zk‖ = |λk| ‖zk‖ ≤ |λk| ‖x‖ −−−−→k→∞

0,


Corollary B.5.26 (Hilbert–Schmidt Spectral Theorem). Let A∗ = A ∈ L(H) becompact. Then σ(A) is at most countable, and Ker(λI − A) is finite-dimensionalif 0 = λ ∈ σ(A). Moreover, σ(A) \ {0} is discrete, and

H =⊕

λ∈σ(A)

Ker(λI −A).

Exercise B.5.27. Prove the Hilbert–Schmidt Spectral Theorem using TheoremB.5.25.

Definition B.5.28 (Weak topology on a Hilbert space). The weak topology of aHilbert space H is the smallest topology for which mappings

(u �→ 〈u, v〉H) : H → C

are continuous for all v ∈ H.

Exercise B.5.29 (Weak = weak∗ in Hilbert spaces). Show that Hilbert spaces arereflexive. Prove that in a Hilbert space the weak topology is the same as theweak∗-topology, introduced in Definition B.4.35.

As a consequence of Exercise B.5.29 and the Banach–Alaoglu Theorem B.4.36we obtain:

Theorem B.5.30 (Banach–Alaoglu Theorem for Hilbert spaces). Let H be a Hilbertspace. Its closed unit ball

B = {v ∈ H : ‖v‖H ≤ 1}

is compact in the weak topology.


Exercise B.5.31. Let {eα}α∈I be an orthonormal collection in H and let x ∈ H.Show that ∑

α∈I

|〈x, eα〉|2 ≤ ‖x‖2. (B.4)

(Hint: Pythagoras’ theorem.) Consequently, deduce from Exercise B.4.3 that theset of α such that 〈x, eα〉 = 0 is at most countable.

We finish with the following theorem which is of importance, because it allowsone to decompose elements into “simpler ones”, which is particularly importantin applications.

Theorem B.5.32 (Orthonormal sets in a Hilbert space). Let {eα}α∈I be an or-thonormal set in the Hilbert space H. Then the following conditions are equivalent:

(i) For every x ∈ H there are only countably many α ∈ I such that 〈x, eα〉 = 0,and the equality

x =∑α∈I

〈x, eα〉eα

holds, where the series is converging in norm, independent of any orderingof its terms.

(ii) If 〈x, eα〉 = 0 for all α ∈ I, then x = 0.(iii) (Plancherel’s identity) For every x ∈ H it holds that ‖x‖2 =

∑α∈I |〈x, eα〉|2.

Proof. (i)⇒ (iii). This follows by enumerating countably many eα’s with 〈x, eα〉 =0 by {ej}∞j=1, and using the identity

‖x‖2 −n∑

j=1

|〈x, ej〉|2 = ‖x−n∑

j=1

〈x, ej〉ej‖2.

(iii) ⇒ (ii) is automatic. Finally, let us show (ii) ⇒ (i). It follows from the lastpart of Exercise B.5.31 that the collection of eα with 〈x, eα〉 = 0 is countable, soit can be enumerated by {ej}∞j=1. Now, the identity

‖j2∑

j=j1

〈x, ej〉ej‖2 =j2∑

j=j1

‖〈x, ej〉‖2

and (B.4) imply that the right-hand side → 0 as j1, j2 → ∞. This means thatthe series

∑∞j=1 〈x, ej〉ej converges. Setting y := x −∑∞

j=1 〈x, ej〉ej we see that〈y, eα〉 = 0 for all α ∈ I, which implies that y = 0. �Exercise B.5.33. Verify the identities stated in the proof.

Definition B.5.34 (Orthonormal basis). An orthonormal set satisfying conditionsof Theorem B.5.32 is called an orthonormal basis of the Hilbert space H. Then wehave the following properties


Theorem B.5.35 (Every Hilbert space has an orthonormal basis). Every Hilbertspace H has an orthonormal basis. An orthonormal basis is countable if and onlyif H is separable, in which case any other basis is also countable.

Exercise B.5.36. Prove Theorem B.5.35: the first part follows from Zorn’s lemmaif we order orthonormal collections by inclusion, since the maximal element wouldsatisfy property (ii) of Theorem B.5.32. The second part follows from the Gram–Schmidt process;

Exercise B.5.37 (Gram–Schmidt orthonormalisation process). Let {xk}∞k=1 be alinearly independent family of vectors in a Hilbert space H. Let⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

y1 := x1,

ek := yk/‖yk‖, and

yk+1 := xk+1 −k∑

j=1

〈xk+1, ej〉ej

for all k ∈ Z+. Show that {ek}∞k=1 is an orthonormal set in H, such that

span {ek}nk=1 = span {xk}n

k=1

for every n ∈ Z+ ∪ {∞}.

B.5.1 Trace class, Hilbert–Schmidt, and Schatten classes

Definition B.5.38 (Trace class operators). Let H be a Hilbert space with orthonor-mal basis {ej | j ∈ J}. Let A ∈ L(H). Let us write

‖A‖S1 :=∑j∈J

|〈Aej , ej〉H| ;

this is the trace norm of A, and the trace class is the (Banach) space

S1 = S1(H) := {A ∈ L(H) : ‖A‖S1 <∞} .

The trace is the linear functional Tr : S1(H)→ C, defined by

A �→∑j∈J

〈Aej , ej〉H.

Exercise B.5.39. Verify that the definition of the trace is independent of the choiceof the orthonormal basis for H. Consequently, if (aij)i,j∈J is the matrix represen-tation of A ∈ S1 with respect to the chosen basis, then Tr(A) =

∑j∈J ajj .


Exercise B.5.40 (Properties of trace). Prove the following properties of the tracefunctional:

Tr(AB) = Tr(BA),Tr(A∗) = Tr(A),

Tr(A∗A) ≥ 0,

Tr(A⊕B) = Tr(A) + Tr(B),

dim(H) <∞ ⇒{

Tr(IH) = dim(H),Tr(A⊗B) = Tr(A) Tr(B).

Exercise B.5.41 (Trace on a finite-dimensional space). Show that the trace on afinite-dimensional vector space is independent of the choice of inner product. Thus,the trace of a square matrix is defined to be the sum of its diagonal elements;moreover, the trace is the sum of the eigenvalues (with multiplicities counted).

Exercise B.5.42. Let H be finite-dimensional. Let f : L(H) → C be a linearfunctional satisfying ⎧⎪⎨⎪⎩

f(AB) = f(BA),f(A∗A) ≥ 0,

f(IH) = dim(H)

for all A,B ∈ L(H). Show that f = Tr.

Definition B.5.43 (Hilbert-Schmidt operators). The space of Hilbert–Schmidt op-erators is

S2 = S2(H) := {A ∈ L(H) : A∗A ∈ S1(H)} ,

and it can be endowed with a Hilbert space structure via the inner product

〈A,B〉S2 := Tr(AB∗).

The Hilbert–Schmidt norm is then

‖A‖HS = ‖A‖S2 := 〈A,A〉1/2S2

.

The case of the Hilbert–Schmidt norm on the finite-dimensional spaces willbe discussed in more detail in Section 12.6.Remark B.5.44. In general, there are inclusions S1 ⊂ S2 ⊂ K ⊂ S∞, whereS∞ := L(H) and K ⊂ S∞ is the subspace of compact linear operators. Moreover,

‖A‖S∞ ≤ ‖A‖S2 ≤ ‖A‖S1

for all A ∈ S∞. One can show that the dual K′ = L(K, C) is isometrically isomor-phic to S1, and that (S1)′ is isometrically isomorphic to S∞. In the latter case, itturns out that a bounded linear functional on S1 is of the form A �→ Tr(AB) forsome B ∈ S∞. These phenomena are related to properties of the sequence spaces�p = �p(Z+). In analogy to the operator spaces, �1 ⊂ �2 ⊂ c0 ⊂ �∞, where c0 isthe space of sequences converging to 0, playing the counterpart of space K.


Remark B.5.45 (Schatten classes). Trace class operators S1 and Hilbert–Schmidtoperators S2 turn out to be special cases of the Schatten classes Sp, 1 ≤ p < ∞.These classes can be introduced with the help of the singular values μ2 ∈ σ(A∗A).To avoid the technicalities we assume that all the operators below are compact.Thus, for A ∈ L(H) we set

‖A‖Sp:=

⎛⎝ ∑μ2∈σ(A∗A)

μp

⎞⎠1/p

.

We note that operators that satisfy ‖A‖Sp< ∞ must have at most countable

spectrum σ(A∗A) in view of Exercise B.4.3, but in our case this is automaticallysatisfied since we assumed that A is compact. Therefore, denoting the sequence ofsingular values μ2

j ∈ σ(A∗A), counted with multiplicities, we have

‖A‖Sp = ‖{μj}j‖�p .

The Schatten class Sp is then defined as the space

Sp = Sp(H) :={A ∈ L(H) : ‖A‖Sp

<∞}

.

With this norm, Sp(H) is a Banach space, and S2(H) is a Hilbert space. In analogyto the trace class and Hilbert–Schmidt operators, one can show that actually‖A‖p

Sp= Tr(|A∗A|p/2) = Tr(|A|p) for a compact operator A.

Exercise B.5.46. Show that the Schatten classes S1 and S2 coincide with thepreviously defined trace class and Hilbert–Schmidt class, respectively.

Exercise B.5.47 (Holder’s inequality for Schatten classes). Show that a Schattenclass Sp is an ideal in L(H). LetH be separable. Show that if 1 ≤ p ≤ ∞, 1

p + 1q = 1,

A ∈ Sp and B ∈ Sq, then

‖AB‖S1 ≤ ‖A‖Sp‖B‖Sq

.

(Hint: approximate operators by matrices.)

Chapter C

Measure Theory and Integration

This chapter provides sufficient general information about measures and integra-tion for the purposes of this book. The starting point is the concept of an outermeasure, which “measures weights of subsets of a space”. We should first considerhow to sum such weights, which are either infinite or non-negative real numbers.For a finite set K, notation ∑

j∈K

aj

abbreviates the usual sum of numbers aj ∈ [0,∞] over the index set K. Theconventions here are that a +∞ =∞ for all a ∈ [0,∞], and that∑

j∈∅aj = 0.

Infinite summations are defined by limits as follows:

Definition C.0.1. The sum of numbers aj ∈ [0,∞] over the index set J is

∑j∈J

aj := sup

⎧⎨⎩∑j∈K

aj : K ⊂ J is finite

⎫⎬⎭ .

Exercise C.0.2. Let 0 < aj <∞ for each j ∈ J . Suppose∑j∈J

aj <∞.

Show that J is at most countable.

The message of Exercise C.0.2 is that for positive numbers, only countablesummations are interesting. In measure theory, where summations are fundamen-tal, such a “restriction to countability” will be encountered recurrently.

116 Chapter C. Measure Theory and Integration

C.1 Measures and outer measures

Recall that for a set X, by P(X) := {E | E ⊂ X} we denote its power set, i.e.,the family of all subsets of X. Let us write Ec := X \ E = {x ∈ X : x ∈ E} forthe complement set, when the space X is implicitly known from the context.

C.1.1 Measuring sets

Definition C.1.1 (Outer measure). A mapping ψ : P(X) → [0,∞] is an outermeasure on a set X = ∅ if

ψ(∅) = 0,

E ⊂ F ⇒ ψ(E) ≤ ψ(F ),

ψ

⎛⎝ ∞⋃j=1

Ej

⎞⎠ ≤∞∑

j=1

ψ(Ej)

for every E,F ⊂ X and {Ej}∞j=1 ⊂ P(X).

Intuitively, an outer measure is weighs the subsets of a space.Example. Define ψ : P(X)→ [0,∞] by ψ(∅) = 0 and ψ(E) = 1, when ∅ = E ⊂ X.This is an outer measure.Example. Let ψ : P(X) → [0,∞], where ψ(E) is the number of points in the setE ⊂ X. Such an outer measure is called a counting measure for obvious reasons.

At first sight, constructing meaningful non-trivial outer measures may ap-pear difficult. However, there is an easy and useful method for generating outermeasures out of simpler set functions, which we call the measurelets:

Definition C.1.2 (Measurelets). Let A ⊂ P(X) cover X, i.e., X =⋃A. We call a

mapping m : A → [0,∞] a measurelet on X. Members of the family A are calledthe elementary sets. A measurelet m : A → [0,∞] on X generates a mappingm∗ : P(X)→ [0,∞] defined by

m∗(E) := inf

{∑A∈B

m(A) : B ⊂ A is countable, E ⊂⋃B

}.

Exercise C.1.3. Let

A := {∅, R2} ∪ {S ⊂ R2 : S a finite union of polygons}.Let us define a measurelet A : A → [0,∞] by the following informal demands:

(1) A(rectangle) = base · height.(2) A(S1 ∪ S2) = A(S1) + A(S2), if the interiors of the sets S1, S2 are disjoint.(3) The measurelet A does not change in translations nor rotations of sets.

Using these rules, calculate the measurelets of a parallelogram and a triangle.

C.1. Measures and outer measures 117

Apparently, there are plenty of measurelets: almost anything goes. Especially,outer measures are measurelets.

Theorem C.1.4. Let m : A → [0,∞] be a measurelet on a set X. Then m∗ :P(X)→ [0,∞] is an outer measure for which m∗(A) ≤ m(A) for every A ∈ A.

Proof. Clearly, m∗ : P(X) → [0,∞] is well defined, and m∗(A) ≤ m(A) for everyA ∈ A. We see that m∗(∅) = 0, because

∑A∈∅m(A) = 0, ∅ ⊂ A is countable, and

∅ ⊂ ⋃ ∅. Next, if E ⊂ F ⊂ X then m∗(E) ≤ m∗(F ), because any cover {Aj}∞j=1

of F is also a cover of E. Lastly, let {Ej}∞j=1 ⊂ P(X). Take ε > 0. For each j ≥ 1,choose {Ajk}∞k=1 ⊂ A such that

Ej ⊂∞⋃

k=1

Ajk and m∗(Ej) + 2−jε ≥∞∑

k=1

m(Ajk).

Then {Ajk}∞j,k=1 ⊂ A is a countable cover of⋃∞

j=1 Ej ⊂ X, and

m∗

⎛⎝ ∞⋃j=1

Ej

⎞⎠ ≤∞∑

j=1

∞∑k=1

m(Ajk)

≤∞∑

j=1

m∗(Ej) + ε.

Thus m∗(∞⋃

j=1

Ej) ≤∞∑

j=1

m∗(Ej); the proof is complete. �

Definition C.1.5 (Lebesgue’s outer measure). On the Euclidean space X = Rn,let us define the partial order ≤ by

a ≤ bdefinition⇐⇒ ∀j ∈ {1, . . . , n} : aj ≤ bj .

When a ≤ b, let the n-interval be

[a, b] := [a1, b1]× · · · × [an, bn] = {x ∈ Rn : a ≤ x ≤ b} .

For A = {[a, b] : a, b ∈ X, a ≤ b} let us define the Lebesgue measurelet m : A →[0,∞] by

m([a, b]) := volume([a, b]) =n∏

j=1

|aj − bj |.

Then the generated outer measure λ∗ = λ∗Rn := m∗ : P(Rn)→ [0,∞] is called theLebesgue outer measure of Rn.

Exercise C.1.6. Give an example of an outer measure that cannot be generatedby a measurelet.


Definition C.1.7 (Outer measure measurability). Let ψ : P(X) → [0,∞] be anouter measure. A set E ⊂ X is called ψ-measurable if

ψ(S) = ψ(E ∩ S) + ψ(Ec ∩ S)

for every S ⊂ X, where Ec = X\E. The family of ψ-measurable sets is denoted by

M(ψ) ⊂ P(X).

Remark C.1.8. Notice that trivially

ψ(S) ≤ ψ(E ∩ S) + ψ(Ec ∩ S)

by the properties of the outer measure. Intuitively, a measurable set E “sharplycuts” “rough” sets S ⊂ X into two disjoint pieces, E ∩ S and Ec ∩ S.Remark C.1.9 (Non-measurability). The Axiom of Choice can be used to “con-struct” a subset E ⊂ Rn which is not Lebesgue measurable. We will discuss thistopic in Section C.1.4.

Exercise C.1.10. Let ψ : P(X) → [0,∞] be an outer measure and E ⊂ X. DefineψE : P(X)→ [0,∞] by ψE(S) := ψ(E ∩S). Show that ψE is an outer measure forwhich M(ψ) ⊂M(ψE).

Lemma C.1.11. Let ψ : P(X) → [0,∞] be an outer measure and ψ(E) = 0. ThenE ∈M(ψ).

Proof. Let S ⊂ X. Then

ψ(S) ≤ ψ(E ∩ S) + ψ(Ec ∩ S)≤ ψ(E) + ψ(S)= ψ(S),

so that ψ(S) = ψ(E ∩ S) + ψ(Ec ∩ S); set E is ψ-measurable. �Lemma C.1.12. Let E,F ∈M(ψ). Then Ec, E ∩ F,E ∪ F ∈M(ψ).

Proof. The definition of ψ-measurability is clearly complement symmetric, so thatE ∈ M(ψ) ⇐⇒ Ec ∈ M(ψ). Next, it is sufficient to deal with E ∪ F , sinceE ∩ F = (Ec ∪ F c)c. Take S ⊂ X. Then

ψ(S) ≤ ψ((E ∪ F ) ∩ S) + ψ((E ∪ F )c ∩ S)= ψ((E ∪ F ) ∩ S) + ψ(Ec ∩ F c ∩ S)

E∈M(ψ)= ψ(E ∩ S) + ψ(Ec ∩ F ∩ S) + ψ(Ec ∩ F c ∩ S)

F∈M(ψ)= ψ(E ∩ S) + ψ(Ec ∩ S)

E∈M(ψ)= ψ(S).

Hence ψ(S) = ψ((E∪F )∩S)+ψ((E∪F )c∩S), so that E∪F is ψ-measurable. �


Exercise C.1.13. Let ψ : P(X) → [0,∞] be an outer measure. Let E ⊂ S ⊂ X,E ∈M(ψ) and ψ(E) <∞. Show that ψ(S \ E) = ψ(S)− ψ(E).

Definition C.1.14 (σ-algebras). A family M ⊂ P(X) is called a σ-algebra on X(pronounced: sigma-algebra) if

1.⋃ E ∈ M for every countable collection E ⊂ M, and

2. Ec ∈M for every E ∈M.

Remark C.1.15. Here, recall the conventions for the union and the intersection ofthe empty family: for A = ∅ ⊂ P(X), we naturally define

⋃A = ∅, but noticethat

⋂A = X (this is not as surprising as it might first seem). Thereby M is aσ-algebra on X if and only if

1.⋃∞

j=1 Ej ∈M whenever {Ej}∞j=1 ⊂M,2. Ec ∈M for every E ∈M, and3. ∅ ∈ M.

Thus, a σ-algebra on X contains always at least subsets ∅ ⊂ X and X ⊂ X.

Proposition C.1.16. Let A ⊂ P(X). There exists the smallest σ-algebra Σ(A) onX containing A, called the σ-algebra generated by A.

A word of warning: there is no summation in this σ-algebra business here,even though we have used the capital sigma symbol Σ.

Exercise C.1.17. Prove Proposition C.1.16.

Definition C.1.18 (Borel σ-algebra). The Borel σ-algebra of a topological space(X, τ) is Σ(τ) ⊂ P(X). The members of Σ(τ) are called Borel sets.

Definition C.1.19 (Disjoint family). A familyA ⊂ P(X) is called disjoint if A∩B =∅ for every A,B ∈ A for which A = B.

Remark C.1.20 (Disjointisation). In measure theory, the following “disjointisa-tion” process often comes in handy. Let M be a σ-algebra and {Ej}∞j=1 ⊂M. LetF1 := E1 and

Fk+1 := Ek+1 \k⋃

j=1

Ej .

Now {Fk}∞k=1 ⊂M is a disjoint family satisfying Fk ⊂ Ek and∞⋃

j=1

Ej =∞⋃

k=1

Fk.

Proposition C.1.21. Let ψ : P(X) → [0,∞] be an outer measure. Let {Fk}∞k=1 ⊂M(ψ) be disjoint. Then

⋃∞k=1 Fk ∈M(ψ) and

ψ(∞⋃

k=1

Fk ∩ S) =∞∑

k=1

ψ(Fk ∩ S) (C.1)


for every S ⊂ X, especially

ψ

( ∞⋃k=1

Fk

)=

∞∑k=1

ψ(Fk).

Proof. Let E :=⋃∞

k=1 Fk. Take S ⊂ X. By Lemma C.1.12, Gn :=⋃n

k=1 Fk ∈M(ψ). Now

ψ(S) ≤ ψ(E ∩ S) + ψ(Ec ∩ S)

≤∞∑

k=1

ψ(Fk ∩ S) + ψ(Ec ∩ S)

= limn→∞

(n∑

k=1

ψ(Fk ∩ S) + ψ(Ec ∩ S)

){Fk}n

k=1⊂M(ψ) disjoint= lim

n→∞ (ψ(Gn ∩ S) + ψ(Ec ∩ S))

Ec⊂Gcn≤ lim

n→∞ (ψ(Gn ∩ S) + ψ(Gcn ∩ S))

Gn∈M(ψ)= ψ(S).

Hence ψ(S) = ψ(E ∩ S) + ψ(Ec ∩ S), meaning that E ∈ M(ψ). Moreover, (C.1)follows from the above chain of (in)equalities. �Corollary C.1.22. Let ψ : P(X)→ [0,∞] be an outer measure. For each k ≥ 1, letEk ∈M(ψ) be such that Ek ⊂ Ek+1. Then

ψ

( ∞⋃k=1

Ek

)= lim

k→∞ψ(Ek). (C.2)

For each k ≥ 1, let Fk ∈M(ψ) such that Fk ⊃ Fk+1 and ψ(F1) <∞. Then

ψ

( ∞⋂k=1

Fk

)= lim

k→∞ψ(Fk). (C.3)

Proof. Let us assume that ψ(Ek) < ∞ for every k ≥ 1, for otherwise the firstclaim is trivial. Thereby

ψ(∞⋃

k=1

Ek) = ψ

(E1 ∪

∞⋃k=1

(Ek+1 \ Ek)

)Prop. C.1.21

= ψ(E1) +∞∑

k=1

ψ(Ek+1 \ Ek)

Exercise C.1.13= ψ(E1) + limn→∞

n∑k=1

(ψ(Ek+1)− ψ(Ek))

= limn→∞ψ(En+1).


Now

ψ(F1) = ψ

⎛⎝( ∞⋂k=1

Fk

)∪∞⋃

j=1

(F1 \ Fj)

⎞⎠Prop. C.1.21

= ψ

( ∞⋂k=1

Fk

)+ ψ

⎛⎝ ∞⋃j=1

(F1 \ Fj)

⎞⎠(C.2)= ψ

( ∞⋂k=1

Fk

)+ lim

j→∞ψ(F1 \ Fj)

Exercise C.1.13= ψ

( ∞⋂k=1

Fk

)+ lim

j→∞(ψ(F1)− ψ(Fj)),

from which (C.3) follows, since ψ(F1) <∞. �Exercise C.1.23. Give an example of an outer measure ψ : P(X) → [0,∞] andsets Ek ⊂ X such that Ek ⊂ Ek+1 for all k ∈ Z+ and

ψ

( ∞⋃k=1

Ek

)= lim

k→∞ψ(Ek).

Exercise C.1.24. Give an example that shows the indispensability of the assump-tion ψ(F1) < ∞ in Corollary C.1.22. For instance, find an outer measure ϕ :P(Z)→ [0,∞] and a family {Fk}∞k=1 ⊂M(ϕ) for which

ϕ

( ∞⋂k=1

Fk

)= lim

k→∞ϕ(Fk),

even though Fk ⊃ Fk+1 for all k.

Theorem C.1.25. Let ψ : P(X) → [0,∞] be an outer measure. Then the ψ-measurable sets form a σ-algebra M(ψ).

Proof. ∅ ∈ M(ψ) due to Lemma C.1.11. By Lemma C.1.12, we know thatM(ψ) isclosed under taking complements. We must prove that it is closed also under takingcountable unions. Let {Ej}∞j=1 ⊂ M(ψ). Applying the disjointisation process ofRemark C.1.20, we obtain a disjoint family {Fk}∞k=1 ⊂M(ψ), for which

⋃∞j=1 Ej =⋃∞

k=1 Fk. Exploiting Proposition C.1.21, the proof is concluded. �Definition C.1.26 (Measures and measure spaces). Let M be a σ-algebra on X.A mapping μ :M→ [0,∞] is called a measure on X if

μ(∅) = 0,

μ

⎛⎝ ∞⋃j=1

Ej

⎞⎠ =∞∑

j=1

μ(Ej)


whenever {Ej}∞j=1 ⊂M is a disjoint family. Then the triple (X,M, μ) is called ameasure space; such a measure space and the corresponding measure μ are called:

• finite, if μ(X) <∞;

• probability, if μ is a finite measure with μ(X) = 1;

• complete, if F ∈ M whenever there exists E ∈ M such that F ⊂ E andμ(E) = 0;

• Borel, if M = Σ(τ), σ-algebra of the Borel sets in a topological space (X, τ).However, sometimes the Borel condition may mean Σ(τ) ⊂M (more on thislater).

Theorem C.1.27. Let ψ : P(X)→ [0,∞] be an outer measure. Then the restrictionψ|M(ψ) :M(ψ)→ [0,∞] is a complete measure.

Proof. This follows by Proposition C.1.21 and Lemma C.1.11. �

Exercise C.1.28. Let μk : M → [0,∞] be measures for which μk(E) ≤ μk+1(E)for every E ∈M (and all k ∈ Z+). Show that μ :M→ [0,∞], where

μ(E) := limk→∞

μk(E).

Exercise C.1.29 (Borel–Cantelli Lemma). Let (X,M, μ) be a measure space,{Ej}∞j=1 ⊂M and

E :={x ∈ X | {j ∈ Z+ : x ∈ Ej} is infinite

}.

Prove that μ(E) = 0 if∞∑

j=1

μ(Ej) <∞.

This is the so-called Borel–Cantelli Lemma.

Remark C.1.30. By Theorem C.1.4, any measure μ generates the outer measureμ∗, whose restriction μ∗|M(μ∗) is a complete measure, which generates an outermeasure, and so on. Fortunately, this back-and-forth-process terminates, as weshall see in Theorems C.1.35 and C.1.36.

Lemma C.1.31. Let μ : M → [0,∞] be a measure on X. Then for every S ⊂ Xthere exists A ∈M such that

S ⊂ A : μ∗(S) = μ(A).

Consequently,μ∗(S) = min {μ(A) : S ⊂ A ∈M} .


Remark C.1.32. An outer measure ψ : P(X) → [0,∞] is called M-regular ifM⊂M(ψ) and

∀S ⊂ X ∃A ∈M : S ⊂ A, ψ(S) = ψ(A);

according to Lemma C.1.31, the outer measure μ∗ generated by a measure μ :M→ [0,∞] is M-regular.

Proof. If S ⊂ X then

μ∗(S) = inf

⎧⎨⎩∞∑

j=1

μ(Aj) : S ⊂∞⋃

j=1

Aj , {Aj}∞j=1 ⊂M

⎫⎬⎭≥ inf

⎧⎨⎩μ

⎛⎝ ∞⋃j=1

Aj

⎞⎠ : S ⊂∞⋃

j=1

Aj , {Aj}∞j=1 ⊂M

⎫⎬⎭= inf {μ(A) : S ⊂ A, A ∈M}≥ μ∗(S).

Thus μ∗(S) = inf{μ(A) : S ⊂ A ∈ M}. For ε > 0, choose Aε ∈ M such thatS ⊂ Aε and μ∗(S) + ε ≥ μ(Aε). Let A0 :=

⋂∞k=1 A1/k ∈M. Then S ⊂ A0, and

μ∗(S) ≤ μ(A0) ≤ μ(Aε) ≤ μ∗(S) + ε

implies μ∗(S) = μ(A0). �

Exercise C.1.33. Let ψ : P(X)→ [0,∞] be an M-regular outer measure and E ∈M(ψ). Define ψE : P(X) → [0,∞] by ψE(S) := ψ(E ∩ S) as in Exercise C.1.10.Show that ψE is an M-regular outer measure.

Exercise C.1.34. Let (X,M, μ) be a measure space and Ek ⊂ X such that Ek ⊂Ek+1 for all k ∈ Z+. Show that

μ∗( ∞⋃

k=1

Ek

)= lim

k→∞μ∗(Ek).

Notice that this does not violate Exercise C.1.23.

Theorem C.1.35 (Caratheodory–Hahn extension). Let μ : M→ [0,∞] be a mea-sure. Then M⊂M(μ∗) and μ = μ∗|M.

Proof. Let E ∈ M. Then μ∗(E) = μ(E), because trivially μ∗(E) ≤ μ(E) andbecause

μ(E) ≤ μ

⎛⎝ ∞⋃j=1

Ej

⎞⎠ ≤∞∑

j=1

μ(Ej)


for any {Ej}∞j=1 ⊂M covering E. To prove M⊂M(μ∗), we must show that

μ∗(S) = μ∗(E ∩ S) + μ∗(Ec ∩ S)

for any S ⊂ X. This follows, because

μ∗(E ∩ S) + μ∗(Ec ∩ S)≥ μ∗(S)

Lemma C.1.31= inf {μ(A) : S ⊂ A ∈M}= inf {μ(A ∩ E) + μ(A ∩ Ec) : S ⊂ A ∈M}≥ μ∗(E ∩ S) + μ∗(Ec ∩ S).

This concludes the proof. �Theorem C.1.36. Let μ :M→ [0,∞] be a measure. Then μ∗ =

(μ∗|M(μ∗)

)∗.Proof. Let ν := μ∗|M(μ∗). We must show that ν∗ = μ∗. Since μ = μ∗|M andM⊂M(μ∗) by Theorem C.1.35, we see that μ is a restriction of ν, and thus theinvestigation of Definition C.1.2 yields ν∗ ≤ μ∗. Moreover,

μ∗(S) ≥ ν∗(S)Lemma C.1.31= inf {ν(A) : S ⊂ A ∈M(μ∗)}Lemma C.1.31= inf {μ(B) : S ⊂ A ∈M(μ∗), A ⊂ B ∈M}

≥ inf {μ(B) : S ⊂ B ∈M}≥ μ∗(S),

so that μ∗(S) = ν∗(S). �Remark C.1.37. In the sequel, measures are often required to be complete. Thisrestriction is not severe, as measures can always be completed, e.g., by the Cara-theodory–Hahn extension, whose naturality is proclaimed by Theorems C.1.35 andC.1.36: if (X,M, μ) is a measure space, N =M(μ∗) and ν = μ∗|N , then (X,N , ν)is a complete measure space such that M ⊂ N and μ = ν|M, with μ∗ = ν∗. So,from this point onwards, we may assume that a measure μ :M→ [0,∞] is alreadyCaratheodory–Hahn complete, i.e., that M =M(μ∗).

C.1.2 Borel regularity

Borel measures are particularly important, providing a link with topology on thespace. We will study such measures in this section.

Definition C.1.38 (Borel regular outer measures). Let (X, τ) be a topological spaceand Σ(τ) its Borel σ-algebra. An outer measure ψ : P(X)→ [0,∞] is Borel regularif it is Σ(τ)-regular.


Definition C.1.39 (Metric outer measure). An outer measure ψ : P(X) → [0,∞]on a metric space (X, d) is called a metric outer measure if it satisfies the followingCaratheodory condition:

dist(A,B) > 0 ⇒ ψ(A ∪B) = ψ(A) + ψ(B). (C.4)

This condition characterises measurability of Borel sets of a metric space:

Theorem C.1.40. Let τd be the metric topology of a metric space (X, d). An outermeasure ψ : P(X)→ [0,∞] is a metric outer measure if and only if τd ⊂M(ψ).

Proof. The “if” part of the proof is left for the reader as Exercise C.1.41. TakeU ∈ τd. To show that U ∈M(ψ), we need to prove ψ(A∪B) = ψ(A)+ψ(B) whenA ⊂ U and B ⊂ U c. We may assume that ψ(A), ψ(B) <∞. For each k ∈ Z+, let

Ak := {x ∈ A | dist(x, U c) ≥ 1/k} .

Then dist(Ak, B) ≥ 1/k, enabling the application of the Caratheodory condi-tion (C.4) in

ψ(A) + ψ(B)trivial≥ ψ(A ∪B)

A⊃Ak= ψ(Ak ∪B)(C.4)= ψ(Ak) + ψ(B).

Clearlyψ(Ak) ≤ ψ(A) ≤ ψ(Ak) + ψ(A \Ak),

so we have to show that ψ(A \ Ak) → 0. Here A =⋃∞

k=1 Ak, since U is open.Consequently

ψ(A \Ak) = ψ

( ∞⋃l=k

(Al+1 \Al)

)

≤∞∑

l=k

ψ(Al+1 \Al)

(C.4)= ψ

( ∞⋃m=1

(Ak+2m+1 \Ak+2m)

)+ ψ

( ∞⋃m=1

(Ak+2m \Ak+2m−1)

)≤ 2 ψ(A) < ∞.

Thus ψ(A \Ak) ≤∞∑

l=k

ψ(Al+1 \Al) −−−−→k→∞

0. �

Exercise C.1.41. Let (X, d) be a metric space. Complete the proof of Theo-rem C.1.40 by showing that if Σ(τd) ⊂M(ψ) then ψ is a metric outer measure.


Theorem C.1.42 (Topological approximation of measurable sets). Let (X, d) be ametric space and ψ : P(X) → [0,∞] be a Borel regular outer measure such thatψ(X) <∞. Let E ⊂ X. Then the following statements are equivalent:

1. E ∈M(ψ).2. E can be ψ-approximated topologically: more precisely, for each ε > 0 there

exist closed Fε ⊂ X and open Gε ⊂ X such that Gε ⊃ E ⊃ Fε and ψ(Gε \Fε) < ε.

Proof. Let us assume the second condition. Let E ⊂ X such that for each ε > 0there exists a closed set Fε ⊂ X such that

Fε ⊂ E and ψ(E \ Fε) < ε.

If F =⋃∞

k=1 F1/k then E ⊃ F ∈ Σ(τd) ⊂M(ψ), since we assume the measurabilityof the Borel sets. Moreover, E ∈M(ψ), because

E = F ∪ (E \ F ),

where E \ F ∈M(ψ) due to

0 ≤ ψ(E \ F ) \ ψ(E \ F1/k) <1k

0−−−−→k→∞

.

Thus the second condition of the theorem implies the first one. Notice that herewe did not even need the assumption ψ(X) <∞ nor the sets Gε!

Conversely, we must show that ψ-measurable sets can be ψ-approximatedtopologically. This can be done by showing that

D := {A ∈M(ψ) | A can be ψ-approximated topologically}

is a σ-algebra containing τd; then the Borel regularity will imply D = M(ψ).Trivially, ∅ ∈ D, and if A ∈ D then also Ac ∈ D. Let {Ak}∞k=1 ⊂ D; now D is aσ-algebra if A :=

⋂∞k=1 Ak ∈ D. Clearly, A ∈ M(ψ), because each Ak ∈ M(ψ).

By the topological ψ-approximation, for each k ∈ Z+ we can take closed Fk ⊂ Xand open Gk ⊂ X such that

Gk ⊃ Ak ⊃ Fk and

{ψ(Gk \Ak) ≤ 2−kε,

ψ(Ak \ Fk) ≤ 2−kε.

Then the closed set⋂∞

k=1 Fk ψ-approximates the set⋂∞

k=1 Ak from inside:

ψ

( ∞⋂k=1

Ak \∞⋂

k=1

Fk

)≤ ψ

( ∞⋃k=1

(Ak \ Fk)

)

≤∞∑

k=1

ψ(Ak \ Fk)

≤ ε.


On the other hand, for large enough n ∈ Z+, the open set⋂n

k=1 Gk ψ-approximatesthe set

⋂∞k=1 Ak from outside:

ψ

(n⋂

k=1

Gk \∞⋂

k=1

Ak

)ψ(X)<∞−−−−−−→

n→∞ ψ

( ∞⋂k=1

Gk \∞⋂

k=1

Ak

)≤ · · · ≤ ε.

Thus we have seen that D is a σ-algebra, so in proving that τd ⊂ D, it sufficesto show that F ∈ D when F ⊂ X is closed. First, F ∈ M(ψ), because the Borelsets are measurable. Clearly, the closed set F ψ-approximates itself from inside, asψ(F \F ) = ψ(∅) = 0. Let Uε :=

⋃x∈F Bε(x), where Br(x) = {y ∈ X : d(x, y) < r}

is an open ball. Thus Uε ⊂ X is open, F ⊂ Uε, and

ψ(U1/k \ F )ψ(X)<∞−−−−−−→

k→∞ψ

( ∞⋂k=1

(U1/k \ F )

)F closed= ψ(∅) = 0.

Hence F can be ψ-approximated by open sets Uε from outside.Now we know that D ⊃ Σ(τd) is a σ-algebra. Take E ∈M(ψ). By the Borel

regularity, there exist Borel sets F,G ⊂ X such that

G ⊃ E, ψ(G) = ψ(E),F c ⊃ Ec, ψ(F c) = ψ(Ec).

By the topological ψ-approximation, take closed Fε ⊂ X and open Gε ⊂ X suchthat

Gε ⊃ G ⊃ E ⊃ F ⊃ Fε,

ψ(Gε \G) < ε, ψ(F \ Fε) < ε.

Then

ψ(Gε \ E) ≤ ψ(Gε \G) + ψ(G \ E) < ε,

ψ(E \ Fε) ≤ ψ(E \ F ) + ψ(F \ Fε) < ε,


Remark C.1.43. From the proof of Theorem C.1.42 we see that E ∈M(ψ) if andonly if E = B ∪N , where B is a Borel set and ψ(N) = 0.

Exercise C.1.44. In Theorem C.1.42 we assumed that ψ(X) <∞. Prove an anal-ogous result assuming that ψ(Br(x)) < ∞ whenever 0 < r < ∞; prove also thatRemark C.1.43 holds in this generalisation (hint: Exercises C.1.10 and C.1.33).Notice that this new assumption is satisfied by the Lebesgue outer measure λ∗Rn .


C.1.3 On Lebesgue measure

Recall the Lebesgue outer measure λRn : P(Rn) → [0,∞] from Definition C.1.5:first, we set the volume of an n-interval

Iab := [a, b] = [a1, b1]× · · · × [an, bn] ⊂ Rn

to bem(Iab) = volume(Iab)

= (b1 − a1) · · · (bn − an),

and define

λ∗Rn(E) = inf

⎧⎨⎩∞∑

k=1

m(Ik) : E ⊂∞⋃

j=1

Ik, Ik ⊂ Rn an n-interval

⎫⎬⎭ .

Definition C.1.45 (Lebesgue measure). The Lebesgue measure λRn :M→ [0,∞] isthe restriction of the Lebesgue outer measure λ∗Rn to the σ-algebraM =M(λ∗Rn).

For a measurable set E ⊂ Rn, the number λRn(E) ∈ [0,∞] can be thoughtas an “n-dimensional volume”. Next we try to justify this claim.

Proposition C.1.46. For any n-interval Iab ⊂ Rn,

λ∗Rn(Iab) = volume(Iab).

Proof. Trivially λ∗Rn(Iab) ≤ volume(Iab) by the definition of the Lebesgue outermeasure. Conversely, take ε > 0. Take a family {Aj}∞j=1 of n-intervals such thatIab ⊂

⋃∞j=1 Aj and

∞∑j=1

(Aj) < λ∗Rn(Iab) + ε.

Take a family {Bj}∞j=1 of n-intervals such that

Aj ⊂ int(Bj) and volume(Bj) ≤ volume(Aj) + 2−jε.

Then {int(Bj)}∞j=1 is an open cover of the compact set Iab ⊂ Rn, thus having

a finite subcover {int(Bjk)}l

k=1. Take a family {Ki}mi=1 of n-intervals such that

Iab =⋃m

i=1 Ki, {int(Ki)}mi=1 is disjoint and that for each i there exists jk such

that Ki ⊂ int(Bjk); that is, the idea is to chop the n-interval Iab into small


enough n-intervals Ki. Then

volume(Iab) =m∑

i=1

volume(Ki)

≤l∑

k=1

volume(Bjk)

≤∞∑

j=1

volume(Bj)

≤∞∑

j=1

volume(Aj) + ε

≤ λ∗Rn(Iab) + 2ε.

Hence volume(Iab) ≤ λ∗Rn(Iab). �Exercise C.1.47. Let t ∈ R, x ∈ Rn E ⊂ Rn,

tE := {ty | y ∈ E} andx + E := {x + y | y ∈ E} .

Show that

λ∗Rn(tE) = |t|n λ∗Rn(E),λ∗Rn(x + E) = λ∗Rn(E).

Moreover, show that tE, x + E ∈M if E ∈M =M(λ∗Rn).

Remark C.1.48 (Translation and rotation invariance of Lebesgue measure). Thetranslation invariance of the Lebesgue (outer) measure refers to the invarianceunder mapping E �→ x + E in Exercise C.1.47. The Lebesgue measure behaveswell also under linear mappings: for a linear mapping A : Rn → Rn and E ⊂ Rn,let AE := {Ay | y ∈ E}. Then

λ∗Rn(AE) = |det(A)| λ∗Rn(E),

where det(A) ∈ R is the determinant of A. Moreover, AE ∈ M if E ∈ M =M(λ∗Rn). Especially, the invariance under the orthogonal mappings is called therotation invariance.

Lemma C.1.49. Let us define the half-space E = {x ∈ Rn | xi ≥ 0}, where i ∈{1, . . . , n}. Then E ∈M =M(λ∗Rn).

Proof. Clearly, it is sufficient to deal with the case i = 1. Take S ⊂ Rn. Let ε > 0.Take a family {Aj}n

j=1 of n-intervals such that S ⊂ ⋃∞j=1 Aj and

n∑j=1

volume(Aj) ≤ λ∗Rn(S) + ε.


Notice that E ∩Aj and Ec ∩Aj are n-intervals, so that

λ∗Rn(S) ≤ λ∗Rn(E ∩ S) + λ∗Rn(Ec ∩ S)≤ λ∗Rn(E ∩ S) + λ∗Rn(Ec ∩ S)

≤ λ∗Rn

⎛⎝ ∞⋃j=1

(E ∩Aj)

⎞⎠ + λ∗Rn

⎛⎝ ∞⋃j=1

(Ec ∩Aj)

⎞⎠≤

∞∑j=1

(volume(E ∩Aj) + volume(Ec ∩Aj)

)=

∞∑j=1

volume(Aj)

≤ λ∗Rn(S) + ε.

Thus λ∗Rn(S) = λ∗Rn(E∩S)+λ∗Rn(Ec∩S). This proves the Lebesgue measurabilityof the half-space E ⊂ Rn. �

Corollary C.1.50. The closed n-interval [a, b] ⊂ Rn is Lebesgue measurable, andso is its interior.

Proof. First,

[a, b] =n⋂

k=1

({x ∈ Rn : ak ≤ xk} ∩ {x ∈ Rn : xk ≤ bk}) ,

so it is measurable, as a finite intersection of measurable sets. Finally, if c =(1, . . . , 1) ∈ Rn then the interior

int([a, b]) =∞⋃

k=1

[a + c/k, b− c/k].

Being a countable union of measurable sets, the interior is measurable. �

Definition C.1.51. For x ∈ Rn and r > 0, let the open cube be

Q(x, r) := x + (−r,+r)n

= {y ∈ Rn | ∀i ∈ {1, . . . , n} : |xi − yi| < r} .

This is a Lebesgue measurable set, as it is the interior of the closed n-intervalQ(x, r) = [a, b], where a = x− (r, . . . , r) and b = x + (r, . . . , r).

Corollary C.1.52 (Lebesgue outer measure is Borel regular). Lebesgue outer mea-sure λ∗ : P(Rn)→ [0,∞] is Borel regular.


Proof. Let U ⊂ Rn be open. It is easy to check that

Q(x, r/√

n) ⊂ Br(x) ⊂ Q(x, r).

Thus x ∈ U if and only if Q(x, r) ⊂ U for some r > 0. If Q(x, r) ⊂ U , takez ∈ Qn ∩Br/2(x); then x ∈ Q(z, r/2) ⊂ Q(x, r) ⊂ U . Thus

U =⋃ {

Q(z, 1/m) : z ∈ Qn, m ∈ Z+, Q(z, 1/m) ⊂ U}

,

which is measurable as a countable union of measurable sets. �Remark C.1.53. It now turns out that Lebesgue measurable sets are nearly openor closed sets:

Theorem C.1.54 (Topological approximation of Lebesgue measurable sets). LetE ⊂ Rn. The following three conditions are equivalent:

1. E ∈M(λ∗Rn).2. For every ε > 0 there exists an open set U ⊂ Rn such that E ⊂ U and

λ∗Rn(U \ E) < ε.3. For every ε > 0 there exists a closed set S ⊂ Rn such that S ⊂ E and

λ∗Rn(E \ S) < ε.

Proof. Let us show that the first condition implies the second one. Suppose E ⊂ Rn

is Lebesgue measurable. Let ε > 0. For a moment, assume that

λRn(E) <∞. (C.5)

Take a family {Aj}∞j=1 of n-intervals such E ⊂ ⋃∞j=1 Aj and

∞∑j=1

volume(Aj) < λRn(E) + ε. (C.6)

We may think that this is an ε-tight cover of E, and we may loosen it a bit bytaking a family {Bj}∞j=1 on n-intervals such that Aj ⊂ int(Bj) and

λRn(Bj) ≤ λRn(Aj) + 2−jε. (C.7)

Let U :=⋃∞

j=1 int(Bj). Then U ⊂ Rn is open, E ⊂ U and

λRn(U) ≤∞∑

j=1

λRn(Bj)

(C.7)

≤∞∑

j=1

λRn(Aj) + ε

(C.6)< λRn(E) + 2ε.


From this we get (as E,U are measurable and E ⊂ U)

λ∗Rn(U \ E) = λRn(U \ E)(C.5)= λRn(U)− λRn(E) < 2ε.

Thus the case of (C.5) is completely solved. Now let us forget the restriction (C.5),and let Ek := E ∩Bd(0, k), where d is the Euclidean distance. Then

Ek ∈M(λ∗Rn), λRn(Ek) <∞, E =∞⋃

k=1

Ek.

By the earlier part of the proof, for each k there exists an open set Uk ⊂ Rn suchthat Ek ⊂ Uk and

λRn(Uk \ Ek) < 2−kε.

Let U :=⋃∞

k=1 Uk. Then U is open, E ⊂ U and

U \ E =

( ∞⋃k=1

Uk

)\ E =

∞⋃k=1

(Uk \ E) ⊂∞⋃

k=1

(Uk \ Ek),

implying

λ∗Rn(U \ E) ≤∞∑

k=1

λ∗Rn(Uk \ Ek) <∞∑

k=1

2−kε = ε.

Thus the first condition in Theorem C.1.54 implies the second one.Let us now assume the second condition, about approximation by open sets

from outside: thereby for each k ∈ Z+ there exists an open set Uk ⊂ Rn such thatE ⊂ Uk and

λ∗Rn(Uk \ E) <1k

.

Let G :=⋂∞

k=1 Uk. Then E ⊂ G ∈M(λ∗Rn), and G \E ⊂ Uj \E for every j ∈ Z+.Hence

λ∗Rn(G \ E) ≤ λ∗Rn(Uj \ E) <1j−−−→j→∞

0,

so that λ∗Rn(G \ E) = 0. Thus G \ E ∈ M(λ∗Rn) by Lemma C.1.11, so that E =G \ (G \E) ∈M(λ∗Rn). This shows that the second condition implies the first onein Theorem C.1.54.

Let us now show that the first and the second conditions imply the thirdcondition. Let E ∈ M(λ∗Rn). Take ε > 0. Since Ec ∈ M(λ∗Rn), there exists anopen set U ⊂ Rn such that Ec ⊂ U and

λRn(U \ Ec) < ε.

Now S := U c ⊂ Rn is closed, S ⊂ E, and E \ S = U \ Ec. This establishes thethird condition, about approximation by closed sets from inside.

The rest of the proof is left for the reader as an exercise. Naturally, thereasoning can be made similar to the case where the second condition implied thefirst one. �


Exercise C.1.55. Complete the proof of Theorem C.1.54 by showing that the thirdcondition implies the first one.

Remark C.1.56 (Lebesgue is “almost” Borel). From the proof of Theorem C.1.54and from a solution to Exercise C.1.55, we may notice that a set E ⊂ Rn isLebesgue measurable if and only if there exist Borel sets F,G ⊂ Rn such thatF ⊂ E ⊂ G and

λRn(G \ F ) = 0.

Moreover, closer examination reveals that G can be taken as a countable intersec-tion of open sets, and correspondingly F as a countable union of closed sets. Inthis sense, a Lebesgue measurable set is almost Borel (up to measure zero), andit looks nearly as if open from outside, and nearly as if closed from inside.

C.1.4 Lebesgue non-measurable sets

The Axiom of Choice (Axiom A.4.2) can be used to “construct” a Lebesgue non-measurable subset S ⊂ Rn. Let f : P(Rn)→ Rn be a choice function. Let

S := {f(x + Qn) | x ∈ Rn} .

Let us show that this set is non-measurable. Now λ∗Rn(S) > 0, because Rn = Qn+S is the union of a countable family {q + S | q ∈ Qn}, where λ∗Rn(r+S) = λ∗Rn(S).Moreover, if 0 = q ∈ Qn then S ∩ (q + S) = ∅. By the following result, this provesthe non-measurability of S:

Proposition C.1.57. Let S ⊂ Rn be Lebesgue measurable and λRn(S) > 0. Thenthere exists δ > 0 such that λRn (S ∩ (x + S)) > 0 whenever ‖x‖Rn < δ.

Proof. Let 0 < ε < 1. Since λ(S) > 0, there exists an n-interval I = [a, b] ⊂ Rn

such thatλ(S ∩ I) = (1− ε) λ(I) > 0.

Let E = S ∩ I. Then λ(I \E) = λ(I)− λ(E) = ε λ(I) due to the measurability ofE. For any x ∈ Rn,

I ∩ (x + I) = (E ∪ (x + E)) ∪ (I \ E) ∪ ((x + I) \ (x + E)) ,

so that

λ (I ∩ (x + I)) ≤ λ (E ∩ (x + E)) + λ (I \ E) + λ ((x + I) \ (x + E))= λ (E ∩ (x + E)) + 2ε λ (I) ,

where the last equality follows by the translation invariance of the Lebesgue mea-sure. The reader easily verifies that limx→0 λ(I + (x + I)) = λ(I). Thus the claimfollows if we choose ε small enough. �


Exercise C.1.58. Let I = [a, b] ⊂ Rn be an n-interval. Show that

λRn (I ∩ (x + I)) −−−−−−→‖x‖Rn→0

λRn(I).

Actually, it can be shown that in the Zermelo–Fraenkel set theory without theAxiom of Choice, one cannot prove the existence of Lebesgue non-measurable sets:see [114]. In practice, we do not have to worry about non-Lebesgue-measurabilitymuch.

C.2 Measurable functions

In topology, continuous functions were essential; in measure theory, the nice func-tions are the measurable ones. Before going into details, let us sketch the commonframework behind both continuity and measurability. Let us say that f : X → Yinduces (or pulls back) from a family B ⊂ P(Y ) a new family f∗(B) ⊂ P(X)defined by

f∗(B) :={f−1(B) ⊂ X | B ∈ B

},

and f : X → Y co-induces (or pushes forward) from a family A ⊂ P(X) a newfamily f∗(A) ⊂ P(Y ) defined by

f∗(A) :={B ⊂ Y | f−1(B) ∈ A

}.

Here if A,B are topologies (or respectively σ-algebras) then f∗(A), f∗(B) arealso topologies (or respectively σ-algebras), since f−1 : P(Y ) → P(X) preservesunions, intersections and complementations.

Exercise C.2.1. Let A,B be σ-algebras. Check that f∗(A), f∗(B) are indeed σ-algebras.

C.2.1 Well-behaving functions

Definition C.2.2 (Measurability). Let MX ,MY be σ-algebras on X and Y , re-spectively. A function f : X → Y is called (MX ,MY )-measurable if

f−1(V ) ∈MX

for every V ∈MY ; that is, if f∗(MY ) ⊂MX .

Remark C.2.3. We see that the measurability behaves well in compositions pro-vided that the involved σ-algebras naturally match: if

Xf−−−−−−−−−−−−→

(M,N )-measurableY

g−−−−−−−−−−−→(N ,O)-measurable

Z

then g ◦ f : X → Z is (M,O)-measurable. For us, a most important case isY = Z = [−∞,+∞] = R ∪ {−∞,+∞}, for which the canonical σ-algebra willbe the collection Σ(τ∞) of Borel sets, where τ∞ ⊂ P([−∞,+∞]) is the smallesttopology for which all the intervals [a, b] ⊂ [−∞,+∞] are closed.

C.2. Measurable functions 135

Definition C.2.4 (Borel/Lebesgue mesurability). LetM be a σ-algebra on X, andlet τX be a topology of X. A function f : X → [−∞,+∞] is called

• M-measurable if it is (M,Σ(τ∞))-measurable, and

• Borel measurable if it is Σ(τX)-measurable.

A function f : Rn → [−∞,+∞] is called Lebesgue measurable if it is M(λ∗Rn)-measurable.

Definition C.2.5. The characteristic function χE : X → R of a subset E ⊂ X isdefined by

χE(x) :=

{1, if x ∈ E,

0, if x ∈ Ec.

Notice that χE is M-measurable if and only if E ∈M.

Definition C.2.6. Let a ∈ R and f, g : X → [−∞,+∞]. We write

{f > a} := {x ∈ X | f(x) > a},{f > g} := {x ∈ X | f(x) > g(x)}.

In an analogous manner one defines sets

{f < a}, {f ≥ a}, {f ≤ a}, {f = a}, {f = a},

{f < g}, {f ≥ g}, {f ≤ g}, {f = g}, {f = g},

and so on.

Theorem C.2.7. Let M be a σ-algebra on X and f : X → [−∞,+∞]. Then thefollowing conditions are equivalent:

1. f is M-measurable.

2. {f > a} is measurable for each a ∈ R.

3. {f ≥ a} is measurable for each a ∈ R.

4. {f < a} is measurable for each a ∈ R.

5. {f ≤ a} is measurable for each a ∈ R.

Proof. If f is M-measurable then {f > a} = f−1((a,+∞]) ∈ M, because(a,+∞] ⊂ [−∞,+∞] is a Borel set.

Now suppose {f > a} ∈ M for every a ∈ R: we have to show that f isM-measurable. We notice that f is (M,D)-measurable, where

D := f∗(M) ={B ⊂ [−∞,+∞] | f−1(B) ∈M

}.


Furthermore, f is M-measurable, because Σ(τ∞) ⊂ D, because for every [a, b] ⊂[−∞,+∞] we have

f−1([a, b]) = {f ≥ a} ∩ {f ≤ b}

=∞⋂

k=1

{f > a− 1/k} ∩ {f > b}c ∈ M;

recall that Σ(τ∞) is the smallest σ-algebra containing every interval. Thus f isM-measurable. All the other claims have essentially similar proofs. �Remark C.2.8. Let f, g : X → [−∞,+∞] be M-measurable. By Theorem C.2.7,then {f > g} ∈ M, because

{f > g} =⋃r∈Q

({f > r} ∩ {g < r}) ;

notice that here the union is countable! Similarly, also

{f ≥ g}, {f < g}, {f ≤ g}, {f = g}, {f = g} ∈ M.

Example. A continuous function f : X → [−∞,+∞] is Borel measurable, because{f ≥ a} ⊂ X is closed for each a ∈ R. Therefore a continuous function f :Rn → [−∞,+∞] is Lebesgue measurable, because Borel sets in Rn are Lebesguemeasurable.

Theorem C.2.9. Let λ ∈ R and 0 < p < ∞. Let f, g : X → R be M-measurable.Then

λf, f + g, fg, |f |p,min(f, g),max(f, g) : X → R

are M-measurable. Moreover, if 0 ∈ f(X) = {f(x) : x ∈ X} then 1/f is M-measurable.

Proof. The reader may easily show that λf is M-measurable. If a ∈ R then

{f + g > a} =⋃

q∈Q: q>a

{f + g > q}

=⋃

r,s∈Q: r+s>a

({f > r} ∩ {g > s}) ∈ M,

showing that f + g is M-measurable. If a ≥ 0 then{f2 > a

}=

{f >

√a}∪

{f < −

√a}

,

so that f2 is M-measurable. Thereby also

fg =(f + g)2 − (f − g)2

4


is M-measurable. If 0 ∈ f(X) then 1/f : X → R is M-measurable, since it is acomposition of

• the M-measurable mapping (x �→ f(x)) : X → R \ {0}, and

• the continuous mapping (t �→ 1/t) : R \ {0} → R.

The rest of the proof is left as Exercise C.2.10. �

Exercise C.2.10. Complete the proof of Theorem C.2.9.

Remark C.2.11. Notice that f2 can be M-measurable even if f is not: consider,e.g., f = χE − 1/2, where E ∈ M.

Definition C.2.12 (μ-almost everywhere). Let (X,M, μ) be a complete measurespace. We say that a property holds μ-almost everywhere (abbreviated μ-a.e.) ifit holds in a set N c = X \N , where N ∈M and μ(N) = 0.

Theorem C.2.13. Let (X,M, μ) be complete and f, g : X → [−∞,+∞]. Let f beM-measurable and f = g μ-a.e. Then g is M-measurable.

Proof. Let N := {f = g} ∈ M. We have to show that {g > a} ∈ M for any a ∈ R.Notice that

{g > a} = (N ∩ {g > a}) ∪ (Nc ∩ {g > a})= (N ∩ {g > a}) ∪ (N c ∩ {f > a}) .

Clearly, N c ∩ {f > a} ∈ M. Moreover, N ∩ {g > a} ∈ M, because μ is completeand μ∗(N ∩ {g > a}) ≤ μ(N) = 0. �

Definition C.2.14 (Distinguishing functions?). Let (X,M, μ) be complete. Writef ∼μ g, if f = g μ-almost everywhere: we may identify those functions that μ “doesnot distinguish”. Especially, if f : X → [−∞,+∞] such that μ({|f | = ∞}) = 0,we may identify f with g : X → R defined by

g(x) :=

{f(x), when f(x) ∈ R,

0, otherwise.

C.2.2 Sequences of measurable functions

Theorem C.2.15. Let fj : X → [−∞,+∞] be M-measurable for each j ∈ Z+.Then

supj∈Z+

fj , infj∈Z+

fj , lim supj→∞

fj , lim infj→∞

fj

are also M-measurable.


Proof. First, {supj∈Z+

fj > a

}=

∞⋃j=1

{fj > a} ∈ M.

Second, the case of the infimum is handled analogously. Third, these previous casesimply the results for lim sup and lim inf. �Definition C.2.16 (Convergences). Let fj , f : X → R, where j ∈ Z+. Let usdefine various convergences fj → f in the following manner: We say that fj → fpointwise (word “pointwise” often omitted) if

∀x ∈ X : |fj(x)− f(x)| −−−→j→∞

0.

Saying that fj → f uniformly means

supx∈X

: |fj(x)− f(x)| −−−→j→∞

0.

Let (X,M, μ) be complete, fj : X → R beM-measurable and f : X → [−∞,+∞].We say that fj → f μ-a.e. if

fj → f pointwise μ-a.e. on X.

Saying that fj → f μ-almost uniformly means that{∀ε > 0 ∃Aε ∈M : (fj − f)|Aε −−−→

j→∞0 uniformly,

μ(Acε) < ε.

Saying that fj → f in measure μ means

∀ε > 0 : μ∗ ({|fj − f | ≥ ε}) −−−→j→∞

0.

Exercise C.2.17. Let functions fj : X → R be M-measurable for every j ∈ Z+.Show that E ∈M, where

E :={

x ∈ X : limj→∞

fj(x) ∈ R exists}

.

Exercise C.2.18. Let (X, τ) be a topological space, fj ∈ C(X) for each j ∈ Z+

and fj → f uniformly. Show that f : X → R is also continuous. This extendsTheorem A.9.7.

Remark C.2.19. Let (X,M, μ) be as above. By Theorems C.2.13 and C.2.15, iffj → f μ-a.e. then f : X → [−∞,+∞] is M-measurable. Moreover, if fj → f inmeasure or fj → f almost uniformly then f is M-measurable, and f(x) ∈ R forμ-a.e. x ∈ X (by Theorem C.2.24 and Exercise C.2.20, respectively).


Exercise C.2.20. Let fj → f μ-almost uniformly.

a) Show that fj → f in measure μ.b) Show that fj → f μ-almost everywhere.

These implications cannot be reversed: give examples.

Exercise C.2.21. For each j ∈ Z+, let fj : X → R be M-measurable. Let (fj)∞j=1

be a Cauchy sequence in measure μ, that is

∀ε > 0 : μ({|fi − fj | ≥ ε}) −−−−→i,j→∞

0.

Show that there exists f : X → [−∞,+∞] such that fj → f in measure μ.

Exercise C.2.22. Let fj → f μ-almost everywhere.

a) Show that fj → f in measure μ, if μ(X) <∞.b) Give an example where μ(X) =∞ and fj → f in measure μ;

consequently, here also fj → f μ-almost uniformly, by Exercise C.2.20.

For finite measure spaces, almost everywhere convergence implies almost uni-form convergence:

Theorem C.2.23 (Egorov: “finite pointwise is almost uniform”). Let (X,M, μ) bea complete finite measure space. Let fj → f μ-almost everywhere. Then fj → falmost uniformly.

Proof. Take ε > 0. We want to find Aε ∈ M such that μ(Acε) < ε and (fj −

f)|Aε−−−→j→∞

0 uniformly. Let

E := {|fj − f | → 0} .

Now E ∈M and μ(Ec) = 0, because fj → f μ-almost everywhere. Moreover,

Ajk :=∞⋂

i=j

{|fi − f | < 1

k

}∈M.

We may choose jk ∈ Z+ such that

μ(Acjkk) < 2−kε, (C.8)

because

limj→∞

μ(Acjk)

μ(X)<∞, Acjk⊃Ac

(j+1)k∈M= μ

⎛⎝ ∞⋂j=1

Acjk

⎞⎠E⊂⋃∞

j=1 Ajk

≤ μ(Ec) = 0.


Now Aε :=∞⋂

k=1

Ajkk ∈M is the desired set:

μ(Acε) = μ

( ∞⋃k=1

Acjkk

)≤

∞∑k=1

μ(Acjkk)

(C.8)< ε,

and (fi − f)|Aε→ 0 uniformly, because

Aε =∞⋂

k=1

∞⋂i=jk

{|fi − f | < 1

k

},

so that |fi(x)− f(x)| < 1k

for all x ∈ Aε whenever i ≥ jk. �

Theorem C.2.24. Let (X,M, μ) be complete and fj → f in measure μ. Then thereexists a subsequence {fjk

}∞k=1 ⊂ {fj}∞j=1 such that fjk→ f μ-almost everywhere.

Proof. Since fj → f in measure, for each k ∈ Z+ we may take jk ∈ Z+ such thatj1 = 1, jk+1 > jk and

μ∗({|fj − f | ≥ 1

k

})< 2k (C.9)

whenever j ≥ jk. Let Nk :={|fjk

− f | ≥ 1k

}and

N := lim supk→∞

Nk =∞⋂

j=1

∞⋃k=j

Nk.

Then μ∗(N) = 0 (thus N ∈M), because

μ∗(N) ≤ μ∗

⎛⎝ ∞⋃k=j

Nk

⎞⎠ ≤∞∑

k=j

μ∗(Nk)(C.9)−−−→j→∞

0.

Now

N c =∞⋃

j=1

∞⋂k=j

{|fjk

− f | < 1k

},

so that |fjk(x) − f(x)| <

1k

for all x ∈ N c whenever k is large enough. Thus

fjk(x) −−−−→

k→∞f(x) for all x ∈ N c. �

Corollary C.2.25. If fj → f in measure then f is measurable. �Exercise C.2.26. Draw a clear diagram about logical implications between differenttypes of convergences fj → f .


C.2.3 Approximating measurable functions

Definition C.2.27 (Simple functions). A function f : X → R is called simple if itsrange f(X) = {f(x) : x ∈ X} ⊂ R is finite. Its normal form is then

f =∑

a∈f(X)

a χf−1({a}).

Definition C.2.28 (Positive and negative parts). The positive and negative partsof a function f : X → [−∞,+∞] are f+, f− : X → [0,∞], respectively, where

f+(x) := max{0, f(x)},f− := (− f)+.

Theorem C.2.29. Let f : X → [−∞,+∞]. Then there exist simple functionsfj : X → R such that

fj → f pointwise.

Moreover, fj can be chosen so that

if 0 ≤ f then 0 ≤ fj ≤ fj+1 ≤ f,

if f bounded then fj → f uniformly ,

if f measurable then fj measurable.

Proof. Since f = f+ − f−, we may approximate f+ and f− separately. Thusassume f ≥ 0. Define

fi(x) :=

{k−12i , when k−1

2i ≤ f(x) < k2i and 1 ≤ k ≤ 2ii,

i, when f(x) ≥ i.(C.10)

We leave the further details for the reader. �Exercise C.2.30. Check that the functions fi defined in (C.10) have the desiredproperties.

Theorem C.2.31 (Luzin: “measurable is almost continuous”). Let (X, d) be ametric space, (X,M, μ) be a complete finite measure space. Let τd ⊂ M andf : X → R be M-measurable. Then for every ε > 0 there exists a closed setFε ⊂ X such that μ(F c

ε ) < ε and f |Fε : Fε → R is continuous.

Remark C.2.32. Notice that if f : X → R is continuous and E ⊂ X then f |E :E → R is continuous; this implication may not be reversible!

Proof. Since f = f+−f−, it suffices to assume that f ≥ 0. For each i ∈ Z+, f(X)has a disjoint Borel cover {[(j − 1)/i, j/i) : j ∈ Z+}, and{

Aij := f−1([(j − 1)/i, j/i))}∞

j=1⊂M


is a disjoint cover of X. Take closed sets Fij ⊂ X such that Fij ⊂ Aij and

μ(Aij \ Fij) < 2−(i+j)ε.

Then

limk→∞

μ(X \k⋃

j=1

Fij)μ(X)<∞

= μ

⎛⎝X \∞⋃

j=1

Fij

⎞⎠= μ

⎛⎝ ∞⋃j=1

(Aij \ Fij)

⎞⎠ =∞∑

j=1

μ(Aij \ Fij)

< 2−iε.

Thereby let Bi :=ki⋃

j=1

Fij , where ki is so large that

μ(X \Bi) < 2−iε.

Now Fε :=∞⋂

i=1

Bi ⊂ X is closed, and

μ(F cε ) = μ

( ∞⋃i=1

Bci

)≤

∞∑i=1

μ(Bci ) < ε.

Let us define gi : Bi → R such that

gi(x) := j/i, when x ∈ Fij .

Then gi : Bi → R is continuous, because gi|Fijis constant and because the closed

sets Fij are disjoint. Next, gi|Fε→ f |Fε

uniformly, because |f(x) − gi(x)| < 1/jwhenever x ∈ Fij ⊂ Aij ; the proof is complete, since continuity is preserved inuniform convergence. �Exercise C.2.33. Let (X,M, μ), where X = [0, 1] ⊂ R and μ is the restriction ofthe Lebesgue measure λR to X. Consider the characteristic function f = χQ∩X :X → R in the light of Luzin’s Theorem C.2.31.

Remark C.2.34 (Littlewood’s principles). With a pinch of salt, measure theorymay be crystallised in Littlewood’s principles:

1. “A measurable set is almost open”(see Topological Approximation Theorem C.1.42).

2. “A measurable function is almost continuous”(see Luzin’s Theorem C.2.31).

3. “Pointwise convergence is almost uniform convergence”(see Egorov’s Theorem C.2.23).

C.3. Integration 143

C.3 Integration

In this section, let (X,M, μ) be a complete measure space. The μ-integral∫f dμ

of an M-measurable function f : X → [−∞,+∞] is defined step-by-step:

1. first for a simple non-negative function;2. then for a non-negative function;3. finally, the general definition.

Definition C.3.1. Let s : X → [0,∞) be an M-measurable simple function. Itsintegral

∫s dμ ∈ [0,∞] is defined as∫

s dμ =∫ ∑

a∈s(X)

a χ{s=a} dμ :=∑

a∈s(X)

a · μ ({s = a}) ,

with the convention 0 · ∞ := 0. Especially,∫

χEdμ = μ(E) for E ∈M.

Definition C.3.2. Let f+ : X → [0,∞] be anM-measurable non-negative function.Its integral

∫f+dμ ∈ [0,∞] is defined as∫

f+dμ := sup{∫

s dμ : 0 ≤ s ≤ f+, s simple measurable}

.

Definition C.3.3 (Integral). Let f : X → [−∞,+∞] be anM-measurable function.Its integral

∫fdμ is defined as∫

f dμ :=∫

f+ dμ−∫

f− dμ

provided that∫

f+dμ <∞ or∫

f−dμ <∞: we want to avoid a situation ∞−∞here. If

∫f+dμ <∞ and

∫f−dμ <∞ then f is called μ-integrable. Let f : X → C

be M-measurable such that |f | : X → R is μ-integrable. The μ-integral of f isdefined by ∫

f dμ :=∫

Ref dμ + i∫

Imf dμ,

where Ref, Imf : X → R are the real and imaginary parts of f , respectively. If wewant to emphasize the variable in the integration, we may write∫

f dμ =∫

f(x) dμ(x),

or even∫

f(x) dx, if the measure is clear from the context. We shall also use theabbreviation ∫

E

f dμ :=∫

χEf dμ,

where E ∈ M; this is the integral of f over E ⊂ X. The Lebesgue integral is theintegral with respect to the Lebesgue measure.


C.3.1 Integrating simple non-negative functions

It is simple to integrate simple functions. We leave the details as an exercise forthe reader:

Exercise C.3.4. Let r, s : X → [0,∞) be M-measurable simple functions anda ∈ [0,∞). Show that∫

ar dμ = a

∫r dμ and

∫(r + s) dμ =

∫r dμ +

∫s dμ.

Moreover, if r ≤ s, show that∫

r dμ ≤∫

s dμ.

C.3.2 Integrating non-negative functions

Let us now concentrate on integrating measurable non-negative functions. Recallthat we are dealing with a complete measure space (X,M, μ).

Exercise C.3.5. Let S ∈M and

μS(E) := μ(E ∩ S).

Show that (X,M, μS) is complete and that∫f dμS =

∫S

f dμ

for all M-measurable f ≥ 0.

As an easy consequence of Exercise C.3.4, for M-measurable functionsf+, g+ : X → [0,∞] and a ∈ R+,∫

af+ dμ = a

∫f+ dμ,

if f+ ≤ g+ then∫

f+ dμ ≤∫

g+ dμ.

These observations will be used frequently. However, it is not evident whether∫(f+ + g+) dμ =

∫f+ dμ +

∫g+ dμ.

This will soon be obtained as a consequence of the following fundamental result:

Theorem C.3.6 (Monotone Convergence Theorem (Lebesgue, Levi)). For eachk ≥ 1, let fk : X → [0,∞] be M-measurable such that fk ≤ fk+1. Then

limk→∞

∫fk dμ =

∫lim

k→∞fk dμ.


The case when the limit f := limk→∞ fk is integrable was proved by Henri-Leon Lebesgue, with the integrability assumption removed by Beppo Levi. Ingeneral, the convergence here is meant in [0,∞], i.e., the limit may be infinite.

Proof of Theorem C.3.6. The function f := limk→∞

fk : X :→ [0,∞] is measurable

as a limit of measurable functions. Clearly, fk ≤ fk+1 ≤ f , so the increasingsequence of integrals

∫fkdμ ≤

∫fdμ converges to the limit

limk→∞

∫fk dμ ≤

∫f dμ.

Let 0 < ε < 1. Take a simple measurable function s such that s ≤ f and∫s dμ ≥ (1− ε)

∫f dμ.

Let Ek := {fk > (1− ε)s}. Since fk and s are measurable, Ek ∈M. Furthermore,∫fk dμ ≥

∫(1− ε)s χEk

dμ

=∑

a∈s(X)

(1− ε)a · μ (Ek ∩ {s = a})

−−−−→k→∞

(1− ε)∑

a∈s(X)

a · μ ({s = a})

= (1− ε)∫

s dμ

≥ (1− ε)2∫

f dμ,

where the limit is due to X =⋃∞

k=1 Ek, where Ek ⊂ Ek+1 ∈M. Thus

limk→∞

∫fk dμ ≥ (1− ε)2

∫f dμ.

Taking ε→ 0, the proof is complete. �Corollary C.3.7. Let f, g : X → [0,∞] be M-measurable. Then∫

(f + g) dμ =∫

f dμ +∫

g dμ.

Proof. Take measurable simple functions fk, gk : X → [0,∞) such that fk ≤ fk+1

and gk ≤ gk+1 for each k ∈ Z+, and fk → f and gk → g pointwise. Thenfk + gk : X → [0,∞) is measurable and simple, such that

fk + gk ≤ fk+1 + gk+1 −−−−→k→∞

f + g,


so that by the Monotone Convergence Theorem C.3.6,∫(f + g) dμ = lim

k→∞

∫(fk + gk) dμ

Exercise C.3.4= limk→∞

(∫fk dμ +

∫gk dμ

)=

∫f dμ +

∫g dμ,

establishing the result. �

Corollary C.3.8. Let gj : X → [0,∞] be M-measurable for each j ∈ Z+. Then∫ ∞∑j=1

gj dμ =∞∑

j=1

∫gj dμ.

Proof. For each k ∈ Z+, let us define functions fk, f : X → [0,∞] by

fk :=k∑

j=1

gj and f := limk→∞

fk =∞∑

j=1

gj .

These functions are measurable and fk ≤ fk+1 ≤ f , so

∫lim

k→∞

k∑j=1

gj dμ

MonotoneConvergence

= limk→∞

∫ k∑j=1

gj dμ

Corollary C.3.7= lim

k→∞

k∑j=1

∫gj dμ,


Exercise C.3.9. Let f ≥ 0 be M-measurable and∫

f dμ <∞. Prove that

∀ε > 0 ∃δ > 0 ∀A ∈M : μ(A) < δ ⇒∫

A

f dμ < ε.

Theorem C.3.10 (Fatou’s lemma). Let gk : X → [0,∞] be M-measurable for eachk ∈ Z+. Then ∫

lim infk→∞

gk dμ ≤ lim infk→∞

∫gk dμ.

Proof. Notice thatlim infk→∞

gk = supk≥1

infj≥k

gj .


Define fk := infj≥k

gj for each k ≥ 1. Now fk : X → [0,∞] is measurable and

fk ≤ fk+1, so that supk≥1

fk = limk→∞

fk, and

∫lim infk→∞

gk dμ =∫

supk≥1

fk dμ

=∫

limk→∞

fk dμ

Monotone Convergence= lim

k→∞

∫fk dμ

= lim infk→∞

∫fk dμ

≤ lim infk→∞

∫gk dμ.

The proof is complete. �

Exercise C.3.11. Sometimes∫

lim infk→∞

gk dμ < lim infk→∞

∫gk dμ happens in Fatou’s

Lemma C.3.10. Find an example.

Exercise C.3.12. Actually, the Monotone Convergence Theorem C.3.6 and Fatou’sLemma C.3.10 are logically equivalent: prove this.

Exercise C.3.13 (Reverse Fatou’s lemma). Prove the following reverse Fatou’slemma. Let gk : X → [0,∞] be M-measurable for each k ∈ Z+. Assume thatgk ≤ g for every k, where g is μ-integrable. Then∫

lim supk→∞

gk dμ ≥ lim supk→∞

∫gk dμ.

C.3.3 Integration in general

Let f : X → [−∞,+∞] be an M-measurable function. Recall that if

I+ =∫

f+ dμ <∞ or I− =∫

f− dμ <∞

then the μ-integral f is∫

fdμ = I+ − I−. Moreover, if both I+ and I− are finite,f is called μ-integrable. We shall be interested mainly in μ-integrable functions.

Theorem C.3.14. Let a ∈ R and f : X → [−∞,+∞] be μ-integrable. Then∫af dμ = a

∫f dμ.


Moreover, if g : X → [−∞,+∞] is μ-integrable such that f ≤ g, then∫f dμ ≤

∫g dμ.

Especially,∣∣∣∣∫ f dμ

∣∣∣∣ ≤ ∫|f | dμ.

Exercise C.3.15. Prove Theorem C.3.14.

Exercise C.3.16. Let E ∈ M and |f | ≤ g, where f is M-measurable and g isμ-integrable. Show that f and fχE are μ-integrable.

Exercise C.3.17 (Chebyshev’s inequality). Let 0 < a < ∞, and let f : X →[−∞,+∞] be M-measurable. Prove Chebyshev’s inequality

μ({|f | > a}) ≤ a−1

∫|f | dμ. (C.11)

We continue by noticing the short-sightedness of integrals:

Lemma C.3.18. Let f, g : X → [−∞,+∞] be μ-integrable. Then

1. Let E ∈M such that μ(E) = 0. Then∫

E

f dμ = 0.

2. Let f = g μ-almost everywhere. Then∫

f dμ =∫

g dμ.

3. Let∫|f | dμ = 0. Then f = 0 μ-almost everywhere.

Proof. First,∫E

f+ dμ =∫

f+χE dμ

= sup{∫

s dμ : s ≤ f+χE simple measurable}

μ(E)=0= 0,

proving the first result. Next, let us suppose f = g μ-almost everywhere. Then∫f+ dμ =

∫ (f+χ{f=g} + f+χ{f �=g}

)dμ

Corollary C.3.7=

∫{f=g}

f+ dμ +∫{f �=g}

f+ dμ

μ({f �=g})=0=

∫{f=g}

f+ dμ,


showing that∫

f+dμ =∫

g+dμ, establishing the second result. Finally,

μ ({f = 0}) = μ

( ∞⋃k=1

{|f | > 1/k})

≤∞∑

k=1

μ ({|f | > 1/k})

=∞∑

k=1

∫χ{|f |>1/k} dμ

≤∞∑

k=1

∫k|f | dμ

=∞∑

k=1

k

∫|f | dμ

so that if∫|f |dμ = 0, then μ({f = 0}) = 0. �

Proposition C.3.19. Let f : X → [−∞,+∞] be μ-integrable. Then f(x) ∈ R forμ-almost every x ∈ X.

Proof. First, {f+ =∞} =⋂∞

k=1 {f+ > k} ∈ M, because f+ is M-measurable.Thereby

μ({f+ =∞}

)=

1k

∫k · χ{f+=∞} dμ

≤ 1k

∫f+ dμ −−−−→

k→∞0,

so that μ ({f+ =∞}) = 0. Similarly, μ ({f− =∞}) = 0. �

Remark C.3.20. By Lemma C.3.18 and Proposition C.3.19, when it comes tointegration, we may identify a μ-integrable function f : X → [−∞,+∞] with thefunction f : X → R defined by

f(x) =

{f(x), when f(x) ∈ R,

0, when |f(x)| =∞.

We shall establish this identification without any further notice.

Theorem C.3.21 (Sum is integrable). Let f, g : X → [−∞,+∞] be μ-integrable.Then f + g is μ-integrable and∫

(f + g) dμ =∫

f dμ +∫

g dμ.


Proof. For integrable f, g : X → R, the function f + g : X → R is measurable.Notice that

f + g =

{(f+ − f−) + (g+ − g−),(f + g)+ − (f + g)−.

Since (f + g)+ ≤ f+ + g+, and (f + g)− ≤ f− + g−, the integrability of f + gfollows. Moreover, (f + g)+ + f−+ g− = (f + g)−+ f+ + g+. By Corollary C.3.7,∫

(f + g)+dμ +∫

f−dμ +∫

g−dμ =∫

(f + g)−dμ +∫

f+dμ +∫

g+dμ,

implying∫(f + g) dμ =

∫(f + g)+ dμ−

∫(f + g)− dμ

=∫

f+ dμ−∫

f− dμ +∫

g+ dμ−∫

g− dμ

=∫

f dμ +∫

g dμ.

The proof for the summation is thus complete. �

Theorem C.3.22 (Lebesgue’s Dominated Convergence Theorem). For each k ≥ 1,let fk : X → [−∞,+∞] be measurable and fk −−−−→

k→∞f pointwise. Assume that

|fk| ≤ g for every k ≥ 1, where g is μ-integrable. Then∫|fk − f | dμ −−−−→

k→∞0,∫

fk dμ −−−−→k→∞

∫f dμ.

Proof. The functions fk, f, |fk− f | are μ-integrable, because they are measurable,g is μ-integrable, |fk|, |f | ≤ g and |fk−f | ≤ 2g. For each k ≥ 1, we define functiongk := 2g − |fk − f |. Then the functions gk ≥ 0 satisfy the assumptions of Fatou’sLemma C.3.10, yielding∫

2g dμ =∫

lim infk→∞

gk dμ

Fatou≤ lim inf

k→∞

∫gk dμ

= lim infk→∞

(∫2g dμ−

∫|fk − f | dμ

)=

∫2g dμ− lim sup

k→∞

∫|fk − f | dμ.


Here we may cancel∫

2g dμ ∈ R, getting

lim supk→∞

∫|fk − f | dμ ≤ 0,

so that∫|fk − f | dμ −−−−→

k→∞0. Finally,

∣∣∣∣∫ fk dμ−∫

f dμ

∣∣∣∣ =∣∣∣∣∫ (fk − f) dμ

∣∣∣∣ ≤ ∫|fk − f | dμ −−−−→

k→∞0,

which completes the proof. �

Remark C.3.23. It is easy to slightly generalise Lebesgue’s Dominated Conver-gence Theorem C.3.22: the same conclusions hold even if we assume only thatfk → f almost everywhere, and that |fk| ≤ g almost everywhere, where g is inte-grable. This is because integrals are not affected if we change values of functionsin a set of measure zero.

Exercise C.3.24 (Indispensability of an integrable dominating function). Show thatin Theorem C.3.22 it is indispensable to require the μ-integrability of a dominatingfunction g. For this, consider X = [0, 1], μ the Lebesgue measure, and the sequence(fk)∞k=1 with fk(x) = k for x ∈ (0, 1/k], and fk(x) = 0 for x ∈ (1/k, 1]. Showthat the function h := supk fk ≥ 0 is not Lebesgue-integrable on [0, 1] (henceno dominating function here can be Lebesgue-integrable). Finally, show that theconclusion of Theorem C.3.22 fails for this sequence (fk)∞k=1.

Exercise C.3.25 (Fatou–Lebesgue Theorem). Prove the following Fatou–LebesgueTheorem: Let (fk)∞k=1 be a sequence of M-measurable functions fk : X → R ona measure space (X,M, μ). Assume that |fk| ≤ g for every k ≥ 1, where g isμ-integrable. Then lim infk→∞ fk and lim supk→∞ fk are μ-integrable and we have∫

lim infk→∞

fk dμ ≤ lim infk→∞

∫fk dμ ≤ lim sup

k→∞

∫fk dμ ≤

∫lim sup

k→∞fk dμ .

Proposition C.3.26 (Riemann vs Lebesgue). Let f : R→ R be Riemann-integrableon the closed interval [a, b] ⊂ R. Then fχ[a,b] is Lebesgue-integrable and theRiemann- and Lebesgue-integrals coincide:∫ b

a

f(x) dx =∫

[a,b]

f dλR.

Exercise C.3.27 (Riemann integration). Prove Proposition C.3.26. Recall the def-inition of the Riemann-integral: Let g : [a, b] → R be bounded. A finite sequencePn = (x0, . . . , xn) is called a partition of [a, b] if

a = x0 < x1 < x2 < · · · < xn−1 < xn = b,


for which the lower and upper Riemann sums L(g, Pn), U(g, Pn) are defined by

U(g, Pn) =n∑

k=1

(sup

xk−1≤x<xk

g(x)

)(xk − xk−1),

L(g, Pn) =n∑

k=1

(inf

xk−1≤x<xk

g(x))

(xk − xk−1).

Now L(g) ≤ U(g), where{U(g) := inf {U(g, P ) : P is a partition of [a, b]} ,

L(g) := sup {L(g, P ) : P is a partition of [a, b]} .

If L(g) = U(g), we say that g is Riemann-integrable with Riemann integral∫ b

a

g(x) dx = L(g).

Exercise C.3.28. Prove the following ε-criterion for Riemann integrability: if forany ε > 0 there is a partition P of [a, b] such that U(g, P ) − L(g, P ) < ε, then gis Riemann-integrable over [a, b].

Consequently, prove that if g is monotonic on [a, b] or if g is continuous on[a, b], then g is Riemann-integrable over [a, b].

C.4 Integral as a functional

C.4.1 Lebesgue spaces Lp(μ)

In the sequel, (X,M, μ) is a complete measure space. For instance, we may haveM =M(μ∗).

Definition C.4.1 (Lp(μ)-norms). For 1 ≤ p < ∞, the Lp(μ)-norm of an M-measurable function f : X → [−∞,+∞] is

‖f‖Lp(μ) :=(∫

|f |p dμ

)1/p

,

and let‖f‖L∞(μ) := inf {M ∈ [0,∞] : |f | ≤M μ-a.e.} ,

Here Lp is read “L-p” or “Lebesgue-p”. If μ is known from the context, notationsLp = Lp(X) = Lp(μ) are used: e.g., Lp(Rn) = Lp(λRn).

Remark C.4.2. The quantities ‖f‖Lp(μ) are not the norms because the non-degen-eracy fails: ‖f‖Lp(μ) = 0 only implies that f = 0 μ-a.e. In fact, clearly, ‖f‖Lp ∈

C.4. Integral as a functional 153

[0,∞], ‖f‖Lp = 0 if and only if f = 0 μ-almost everywhere, ‖λf‖Lp = |λ| ‖f‖Lp ,and ‖f‖L1(μ) =

∫|f | dμ. Also, in Theorem C.4.5 we will see that ||f + g||Lp ≤

||f ||Lp + ||g||Lp so that the triangle inequality would be satisfied. Thus, we willmodify the construction slightly (identifying functions equal μ-a.e.) in DefinitionC.4.6 to make them norms, so that the terminology “norm” will be justified.

Definition C.4.3 (Lebesgue conjugate). The Lebesgue conjugate of p ∈ [1,∞] isthe number p′ ∈ [1,∞] defined by

1p

+1p′

= 1

with the usual convention 1/∞ = 0.

The converse to the following theorem (the converse of Holder’s inequality)will be shown in Theorem C.4.56.

Theorem C.4.4 (Holder’s inequality). Let 1 ≤ p ≤ ∞ and q = p′. Let f, g : X →[−∞,+∞] be M-measurable. Then

‖fg‖L1 ≤ ‖f‖Lp‖g‖Lq .

Proof. For p = 1,

‖fg‖L1 =∫|f ||g| dμ

≤∫|f | dμ ‖g‖L∞

= ‖f‖L1 ‖g‖L∞ ;

the proof for p =∞ is symmetric. Finally, let us assume that 1 < p <∞. We mayassume the non-trivial case 0 < ‖f‖Lp <∞ and 0 < ‖g‖Lq <∞. Then

‖fg‖L1 = ‖f‖Lp ‖g‖Lq

∫ab dμ,

where a = |f |/‖f‖Lp and b = |g|/‖g‖Lq . The concavity of the logarithm gives

ln(ab) = ln(ap)/p + ln(bq)/q

1/q=1−1/p

≤ ln (ap/p + bq/q) ,

so∫

ab dμ ≤∫

(ap/p + bq/q) dμ = 1/p + 1/q = 1. �

Theorem C.4.5 (Minkowski’s inequality). Let 1 ≤ p ≤ ∞. Let f, g : X →[−∞,+∞] be M-measurable. Then

‖f + g‖Lp ≤ ‖f‖Lp + ‖f‖Lp . (C.12)


Proof. First,

‖f + g‖L1 =∫|f + g| dμ

≤∫|f | dμ +

∫|g| dμ

= ‖f‖L1 + ‖g‖L1 .

Also |f + g| ≤ |f | + |g|, so that |f + g| ≤ ‖f‖L∞ + ‖g‖L∞ almost everywhere,yielding

‖f + g‖L∞ ≤ ‖f‖L∞ + ‖g‖L∞ .

Finally, assume that 1 < p <∞. Then

‖f + g‖pLp =

∫|f + g|p dμ

≤∫

(|f |+ |g|) |f + g|p−1 dμ

=∥∥f |f + g|p−1

∥∥L1 +

∥∥g |f + g|p−1∥∥

L1

Holder≤ (‖f‖Lp + ‖g‖Lp)

∥∥|f + g|p−1∥∥

Lq ,

where

∥∥|f + g|p−1∥∥

Lq =(∫

|f + g|(p−1)q dμ

)1/q

=(∫

|f + g|p dμ

)(p−1)/p

= ‖f + g‖p−1Lp ,

concluding the proof. �

Definition C.4.6 (Lp(μ)-spaces). Let 1 ≤ p ≤ ∞ and

V p := {f : ‖f‖Lp <∞} .

Noticing that ‖λf‖Lp = |λ| ‖f‖Lp for any scalar λ, and recalling Minkowski’sinequality (C.12), we see that

(f �→ ‖f‖Lp) : V p → [0,∞)

is a seminorm on the vector space V p. Let us define an equivalence relation ∼ onV p by

f ∼ g ⇐⇒ ‖f − g‖Lp = 0;


i.e., f ∼ g ⇐⇒ f = g μ-almost everywhere. Let us denote the equivalenceclasses by

[f ] := {g ∈ V p : f ∼ g} .

We obtain the quotient vector space

Lp(μ) := V p/ ∼ = {[f ] : f ∈ V p}

with the usual vector space operations. Moreover,

([f ] �→ ‖f‖Lp) : Lp(μ)→ [0,∞)

is a norm on vector space Lp(μ). Customarily, �p(X) := Lp(μ), where μ is thecounting measure, i.e., μ(E) is the number of points in the set E.

Remark C.4.7. f ∈ [f ] is a function X → [−∞,+∞], but [f ] ∈ Lp(μ) is nota function, but an equivalence class of functions. However in practice, to avoidcumbersome notation, one often identifies f and [f ], e.g., writing briefly f ∈ Lp(μ).

Definition C.4.8 (Convergence in Lp(μ)). Let f ∈ Lp and {fj}∞j=1 ⊂ Lp. We saythat fj → f in Lp if

‖fj − f‖Lp −−−→j→∞

0.

Theorem C.4.9. Lp(μ) is a Banach space.

Proof. The case p = ∞ is left as Exercise C.4.10; let us consider the case 1 ≤p <∞. We already know that Lp(μ) is a normed space. Given a Cauchy sequence(fj)∞j=1 in Lp(μ), we need a candidate f for the limit of this sequence. Now (fj)∞j=1

is a Cauchy sequence in measure μ, because

μ ({|fi − fj | ≥ ε}) = μ ({|fi − fj |p ≥ εp})Chebyshev (C.11)

≤ ε−p

∫|fi − fj |p dμ

= ε−p‖fi − fj‖pLp

−−−−→i,j→∞

0.

Hence by Exercise C.2.21, fj → f in measure μ for an M-measurable function f .By Theorem C.2.24, fjk

→ f μ-almost everywhere for a subsequence (fjk)∞k=1 of

(fj)∞j=1. Here f ∈ Lp(μ), because

‖f‖pLp =

∫|f |p dμ

=∫

lim infk→∞

|fjk|p dμ

Fatou≤ lim inf

k→∞

∫|fjk|p dμ

≤ constant <∞,


because Cauchy sequences in a normed space are bounded. Finally, fi → f inLp(μ), because

‖fi − f‖pLp =

∫|fi − f |p dμ

=∫

lim infk→∞

|fi − fjk|p dμ

Fatou≤ lim inf

k→∞‖fi − fjk

‖pLp

(fi)∞i=1 Cauchy−−−−−−−−−−→i→∞

0.

Thus Lp(μ) is a Banach space for 1 ≤ p <∞. �Exercise C.4.10. Complete the previous proof by showing that L∞(μ) is a Banachspace.

Exercise C.4.11. Let 1 ≤ p <∞ and ‖fj−f‖Lp → 0, where f ∈ Lp and {fj}∞j=1 ⊂Lp. Show that

∀ε > 0 ∃δ > 0 ∀j ∈ Z+ : μ(E) < δ =⇒∫

E

|fj |p dμ < ε.

Why is p =∞ here?

Lemma C.4.12. Let g ∈ Lp(μ), where 1 ≤ p <∞. Show that

∀ε > 0 ∃Eg ∈M : μ(Eg) <∞ and∫

Ecg

|g|p dμ < ε.

Exercise C.4.13. Prove Lemma C.4.12.

Theorem C.4.14 (Vitali’s Convergence Theorem). Let 1 ≤ p < ∞. Let f, fj ∈Lp(μ) for each j ∈ Z+. Then properties (1,2,3) imply (0), and (0) implies proper-ties (2,3):

(0) fj → f in Lp.(1) fj → f μ-almost everywhere.(2) ∀ε > 0 ∃E ∈M ∀j ∈ Z+ : μ(E) <∞,

∫Ec |fj |p dμ < ε.

(3) ∀ε > 0 ∃δ > 0 ∀j ∈ Z+ ∀A ∈M : μ(A) < δ ⇒∫

A|fj |p dμ < ε.

Proof. First, let us show that (1, 2, 3) implies (0). Take ε > 0. Take δ > 0 as in(3). Take E ∈ M as in (2). Exploiting (1), Egorov’s Theorem C.2.23 says that(fj − f)|E → 0 μ-almost uniformly. Hence there exists B ∈M such that⎧⎪⎨⎪⎩

B ⊂ E,

μ(E \B) < δ,

(fj − f)|B → 0 uniformly.

(C.13)


We want to show that ‖fj → f‖Lp → 0:

‖fj − f‖pLp =

∫|fj − f |p dμ

=∫

B

|fj − f |p dμ +∫

Bc

|fj − f |p dμ,

and here the integral over B tends to 0 as j → ∞, by (C.13). What about theintegral over Bc? Since (t �→ tp) : R+ → R is a convex function, we have (a/2 +b/2)p ≤ ap/2 + bp/2, so that∫

Bc

|fj − f |p dμ

≤∫

Bc

2p−1 (|fj |p + |f |p) dμ

= 2p−1

(∫Ec

|fj |p dμ +∫

E\B|f |p dμ +

∫Bc

lim infj→∞

|fj |p dμ

)(2), (3), Fatou

< 2p−1 (ε + ε + 2ε) ;

thus ‖fj − f‖Lp → 0: we have proven that (0) follows from (1, 2, 3).Implication (0)⇒ (3) is left as Exercise C.4.15.Let us show that (0) ⇒ (2). Let fj → f in Lp(μ). Take ε > 0. Take jε ∈

Z+ such that ‖fj − f‖Lp < ε1/p whenever j > jε. Take Ef , Efj ∈ M as inLemma C.4.12. Let

E := Ef ∪jε⋃

j=1

Efj .

Then E ∈M and μ(E) <∞. If j ≤ jε then∫Ec

|fj |p dμ ≤∫

Ecfj

|fj |p dμ < ε.

If j > jε then∫Ec

|fj |p dμMinkowski≤ (‖χEc(fj − f)‖Lp + ‖χEcf‖Lp)p

≤(ε1/p + ε1/p

)p

,

so that∫

Ec

|fj |p dμ ≤ 2pε for every j ∈ Z+. We have shown that (0) ⇒ (2).


Finally, let us prove that (0)⇒ (1). We have

μ({|fj − f | ≥ ε}) = μ({|fj − f |p ≥ εp})Chebyshev

≤ ε−p

∫|fj − f |p dμ

(0)−−−→j→∞

0,

so that fj → f in measure μ. By Theorem C.2.24, there is a subsequence (fjk)∞k=1

such that fjk→ f μ-almost everywhere. We have shown that (0)⇒ (1). �

Exercise C.4.15. Complete the proof of Vitali’s Convergence Theorem C.4.14 byshowing that (0)⇒ (3).

Exercise C.4.16. Let 1 ≤ p ≤ ∞ and fj → f μ-a.e., where {fj}∞j=1 ⊂ Lp.(a) Let fj → g in Lp. Show that f = g μ-a.e.(b) Give an example where f ∈ Lp, but fj → f in Lp.

Finally, we give without proof a very useful interpolation theorem. But firstwe introduce

Definition C.4.17 (Semifinite measures). A measure μ is called semifinite if forevery E ∈ M with μ(E) = ∞ there exists F ∈ M such that F ⊂ E and 0 <μ(F ) <∞.

Theorem C.4.18 (M. Riesz–Thorin interpolation theorem). Let μ, ν be semifinitemeasures and let 1 ≤ p0, p1, q0, q1 ≤ ∞. For every 0 < t < 1 define pt and qt by

1pt

=1− t

p0+

t

p1,

1qt

=1− t

q0+

t

q1.

Assume that A is a linear operator such that

||Af ||Lq0 (ν) ≤ C0||f ||Lp0 (μ), ||Af ||Lq1 (ν) ≤ C1||f ||Lp1 (μ),

for all f ∈ Lp0(μ) and f ∈ Lp1(μ), respectively. Then for all 0 < t < 1, theoperator A extends to a bounded linear operator from Lpt(μ) to Lqt(ν) and wehave

||Af ||Lqt (ν) ≤ C1−t0 Ct

1||f ||Lpt (μ)

for all f ∈ Lpt(μ).

C.4.2 Signed measures

Definition C.4.19 (Signed measures). Let M be a σ-algebra on X. A mappingν :M→ R is called a signed measure on X if

ν

⎛⎝ ∞⋃j=1

Ej

⎞⎠ =∞∑

j=1

ν(Ej)

for any disjoint countable family {Ej}∞j=1 ⊂M.


Example. Let μ, ν : M→ [0,∞] be finite measures on X, that is μ(X) < ∞ andν(X) <∞. Then

μ− ν :M→ R

is a signed measure. It will turn out that there are no other types of signed measureson X, see the Jordan decomposition result in Corollary C.4.26

Remark C.4.20. For simplicity and in view of the planned applications of thisnotion we restrict the exposition to what may be called finite signed measures. Inprinciple, one can allow ν : M→ [−∞,+∞] assuming that only one of infinitiesmay be achieved. The statements and the proofs remain largely similar, so we mayleave this case as an exercise for an interested reader. For example, only one ofthe measures in Theorem C.4.25 would be finite, etc.

Exercise C.4.21. Let (X,M, μ) be a measure space and let f : X → [−∞,+∞]be μ-integrable. Define ν :M→ R by

ν(E) :=∫

E

f dμ. (C.14)

Show that ν is a signed measure. Moreover, prove that ν is a (finite) measure ifand only if f ≥ 0 μ-almost everywhere.

Definition C.4.22 (Variations of measures). Let ν :M→ R be a signed measure.Define mappings ν+, ν−, |ν| :M→ [0,∞] by

ν+(E) := supA∈M: A⊂E

ν(A),

ν− := (− ν)+,

|ν| := ν+ + ν−.

The mappings ν+, ν− are called the positive and negative variations (respectively)of ν, and the pair (ν+, ν−) is the Jordan decomposition of ν. The mapping |ν| isthe total variation of ν.

Exercise C.4.23. Show that ν+, ν−, |ν| :M→ [0,∞] are measures.

Exercise C.4.24. Let ν(E) =∫

E

f dμ as in (C.14). Show that

ν+(E) =∫

E

f+ dμ and ν−(E) =∫

E

f− dμ.

Hence here ν = ν+ − ν−, but this happens even generally:

Theorem C.4.25. Let ν :M→ R be a signed measure. Then the measures ν+, ν− :M→ [0,∞] are finite.


Proof. By Exercise C.4.23, ν+ and ν− are measures. Let us show that ν+ (andsimilarly ν−) is finite. To get a contradiction, assume that ν+(X) = ∞. TakeE0 ∈ M such that ν(E0) ≥ 0. Take A0 ∈ {E0, X \ E0} such that ν+(A0) = ∞.For k ∈ Z+, suppose Ek, Ak ∈ M have been chosen so that ν+(Ak) = ∞. TakeEk+1 ∈M such that

Ek+1 ⊂ Ak and ν(Ek+1) ≥ 1 + ν(Ek).

Take Ak+1 ∈ {Ek+1, Ak \ Ek+1} such that ν+(Ak+1) =∞. Then

1. either ∃k0 ∀k ≥ k0 : Ak+1 = Ek+1

2. or ∀k0 ∃k ≥ k0 : Ak+1 = Ak \ Ek+1.

Here in the first case, E ⊃ Ek ⊃ Ek+1 for every k ≥ k0, and

ν(Ek0) = ν

( ∞⋂k=k0

Ek

)+

∞∑k=k0

ν(Ek \ Ek+1)

= ν

( ∞⋂k=k0

Ek

)+

∞∑k=k0

(ν(Ek)− ν(Ek+1))

= −∞;

of course, this is a contradiction, excluding the first case. In the second case, takea disjoint family {Ekj}∞j=1 where kj+1 > kj ∈ Z+, so that

ν

⎛⎝ ∞⋃j=1

Ekj

⎞⎠ =∞∑

j=1

ν(Ekj ) = +∞,

again a contradiction; therefore ν+ and ν− must be finite measures. �

Corollary C.4.26 (Jordan Decomposition). Let ν : M → R be a signed measure.Then

ν = ν+ − ν−.

Proof. Let E ∈M. For any A ∈M we have

ν(E) = ν(A ∩ E) + ν(Ac ∩ E)≤ ν+(E)− (−ν)(Ac ∩ E),

yielding ν(E) ≤ ν+(E)− ν−(E). Similarly,

(−ν)(E) ≤ (−ν)+(E)− (−ν)−(E) = ν−(E)− ν+(E),

so that ν(E) ≥ ν+(E)− ν−(E). �


Exercise C.4.27. Let M be a σ-algebra on X. Let M(M) be the real vector spaceof all signed measures ν : M → R. For ν ∈ M(M), let ‖ν‖ = |ν|(X); show thatthis gives a Banach space norm on M(M).

Definition C.4.28 (Hahn decomposition). A pair (P, P c) is called a Hahn decom-position of a signed measure ν :M→ R if P ∈M and

∀E ∈M : ν(P ∩ E) ≥ 0 ≥ ν(P c ∩ E).

Then P is called a ν-positive set and P c is a ν-negative set.

Example. Let ν(E) =∫

E

f dμ as in (C.14). Then (P, P c) and (Q, Qc) are Hahn

decompositions of ν :M→ R, where

P := {f ≥ 0}, Q := {f > 0}.

Definition C.4.29 (Mutually singular measures). The measures μ, λ :M→ [0,∞]are mutually singular, denoted by μ⊥λ, if there exists P ∈M such that

μ(P ) = 0 = λ(P c).

Here, the zero-measure condition μ(P ) = 0 can be interpreted so that the measureμ does not see the set P ∈M.

Theorem C.4.30 (Hahn Decomposition). Let ν : M → R be a signed measure.Then ν has a Hahn decomposition (P, P c). More precisely,{

ν+(E) = +ν(P ∩ E),ν−(E) = −ν(P c ∩ E)

for each E ∈M. Especially, ν−⊥ν+ such that ν−(P ) = 0 = ν+(P c).

Proof. For each k ∈ Z+, take Ak ∈M such that ν+(X)− ν(Ak) < 2−k. Then

P := lim supk→∞

Ak =∞⋂

j=1

∞⋃k=j

Ak ∈M.

Moreover,

ν−(P ) ≤∞∑

k=j

ν−(Ak)

Corollary C.4.26=

∞∑k=j

(ν+(Ak)− ν(Ak)

)≤

∞∑k=j

2−k = 21−j ,


so that ν−(P ) = 0. On the other hand,

ν+(P c) = ν+(lim infk→∞

Ack)

= limj→∞

ν+

⎛⎝ ∞⋂k=j

Ack

⎞⎠≤ lim

j→∞ν+(Ac

j)

≤ limj→∞

2−j = 0,

so that ν+(P c) = 0. Thereby

ν(P ∩ E) Jordan= ν+(P ∩ E)− ν−(P ∩ E)= ν+(P ∩ E) + ν+(P c ∩ E)= ν+(E),

and similarly ν(P c ∩ E) = −ν−(E). �

Exercise C.4.31. Let (P, P c) and (Q,Qc) be two Hahn decompositions of a signedmeasure ν. Show that |ν|(P \Q) = 0. The moral here is that all the Hahn decom-positions are “essentially the same”.

Exercise C.4.32. Let ν = α− β, where α, β :M→ [0,∞] are finite measures andα⊥β. Show that

α = ν+ and β = ν−.

In this respect the Jordan decomposition is the most natural decomposition of νas a difference of two measures.

C.4.3 Derivatives of signed measures

In this section we study which signed measures ν :M→ R can be written in theintegral form as in (C.15). The key property is the absolute continuity of ν withrespect to μ, and the key result is the Radon–Nikodym Theorem C.4.38.

Definition C.4.33 (Radon–Nikodym derivative). Let (X,M, μ) be a measure spaceand f : X → [−∞,+∞] be μ-integrable. Let a signed measure ν : M → R bedefined by

ν(E) :=∫

E

f dμ. (C.15)

Thendν

dμ:= f is the Radon–Nikodym derivative of ν with respect to μ.


Remark C.4.34. Actually, the Radon–Nikodym derivative dν/dμ = f is not anintegrable function X → [−∞,+∞] but is the equivalence class

{g : X → [−∞,+∞] | f ∼ g} ,

where f ∼ g ⇐⇒ f = g μ-almost everywhere. The classical derivative of afunction (a limit of a difference quotient) is connected to the Radon–Nikodymderivative in the case of the Lebesgue measure μ = λR, but this shall not beinvestigated here.

Definition C.4.35 (Absolutely continuous measures). A signed measure ν :M→ Ris absolutely continuous with respect to a measure μ : M → [0,∞], denoted byν & μ, if

∀E ⊂M : μ(E) = 0 ⇒ ν(E) = 0.

Example. If ν(E) =∫

E

f dμ as in (C.15) then ν & μ.

The following (ε, δ)-result justifies the term absolute continuity here:

Theorem C.4.36. Let ν : M → R be a signed measure and μ : M → [0,∞] ameasure. Then the following conditions are equivalent:

(a) ν & μ.(b) ∀ε > 0 ∃δ > 0 ∀E ∈M : μ(E) < δ ⇒ |ν(E)| < ε.

Proof. The (ε, δ)-condition trivially implies ν & μ. On the other hand, let us showthat ν & μ, when we assume

∃ε > 0 ∀δ > 0 ∃Eδ ∈M : μ(Eδ) < δ and |ν(Eδ)| ≥ ε.

Then

E := lim supk→∞

E2−k =∞⋂

j=1

∞⋃k=j

E2−k ∈M,

and μ(E) = 0, because

μ(E) ≤ μ

⎛⎝ ∞⋃k=j

E2−k

⎞⎠ ≤∞∑

k=j

2−k = 21−j .

Now |ν|(E) > 0, because

|ν|(E) = limj→∞

|ν|

⎛⎝ ∞⋃k=j

E2−k

⎞⎠ ≥ ε.

Hence ν+(E) > 0 or ν−(E) > 0, so that |ν(A)| > 0 for some A ⊂ E, whereA ∈M. Here μ(A) = 0, so that ν & μ. �


Exercise C.4.37. Show that the following conditions are equivalent:

1. ν & μ.2. |ν| & μ.3. ν+ & μ and ν− & μ.

Theorem C.4.38 (Radon–Nikodym). Let μ : M→ [0,∞] be a finite measure andν & μ. Then there exists a Radon–Nikodym derivative dν/dμ, i.e.,

ν(E) =∫

E

dν

dμdμ

for every E ∈M.

Exercise C.4.39 (σ-finite Radon–Nikodym). A measure space is called σ-finite ifit is a countable union of sets of finite measure. Generalise the Radon–NikodymTheorem to σ-finite measure spaces. For example, if μ is σ-finite, we can find asequence Ej ↗ X with μ(Ej) <∞ and define

dν

dμ:= sup

j

dν|Ej

dμ|Ej

.

Exercise C.4.40. Let ν = λRn be the Lebesgue measure, and let (X,M, μ) =(Rn,M(λ∗Rn), μ), where μ is the counting measure; this measure space is not σ-finite, but ν & μ. Show that ν cannot be of the form ν(E) =

∫E

f dμ. Thus thereis no anologue to the Radon–Nikodym Theorem in this case.

Before proving the Radon–Nikodym Theorem C.4.38, let us deal with theessential special case of the result:

Lemma C.4.41. Let μ, ν : M → [0,∞] be finite measures such that ν ≤ μ. Thenthere exists a Radon–Nikodym derivative dν/dμ, i.e.,

ν(E) =∫

E

dν

dμdμ

for every E ∈M. Moreover,∫g+ dν =

∫g+ dν

dμdμ (C.16)

when g+ : X → [0,∞] is M-measurable.

Proof. An M-partition of a set X is a finite disjoint collection P ⊂M, for whichX =

⋃P. Let us define a partial order ≤ on the family theM-partitions by P ≤ Qif and only if for every Q ∈ Q there exists P ∈ P such that Q ⊂ P . The commonrefinement of M-partitions P,Q is the M-partition

↑ {P,Q} = {P ∩Q : P ∈ P, Q ∈ Q} .


For an M-partition P, let us define dP :M→ R by

dP(E) :=

{ν(P )μ(P ) , if x ∈ P ∈ P and μ(P ) > 0,

0, otherwise.

Then 0 ≤ dP ≤ 1, dP is simple and μ-integrable, and

dP =∑P∈P

ν(P )μ(P )

χP .

The idea in the following is that the Radon–Nikodym derivative dν/dμ will beapproximated by functions dP is the L2(μ)-sense. If P ≤ Q and E ∈ P then

ν(E) =∫

E

dP dμ =∫

E

dQ dμ, (C.17)

because ∫E

dQ dμ =∫

E

∑Q∈Q

ν(Q)μ(Q)

χQ dμ

=∑Q∈Q

ν(Q)μ(Q)

∫χQ∩E dμ

E∈P≤Q=∑

Q∈Q: Q⊂E

ν(Q)μ(Q)

μ(Q)

E∈P≤Q= ν(E).

Moreover, here

‖dP‖2L2(μE) ≤ ‖dQ‖2L2(μE) = ‖dP‖2L2(μE) + ‖dQ − dP‖2L2(μE), (C.18)

because

‖dP‖2L2(μE) ≤ ‖dP‖2L2(μE) + ‖dQ − dP‖2L2(μE)

=∫

E

d2P dμ +

∫E

(dQ − dP)2 dμ

=∫

E

d2Q dμ + 2

∫E

dP (dP − dQ) dμ

E∈P= ‖dQ‖2L2(μE) + 2∫

E

ν(E)μ(E)

(ν(E)μ(E)

− dQ

)dμ

(C.17)= ‖dQ‖2L2(μE) + 2

ν(E)μ(E)

(ν(E)μ(E)

− ν(E))

= ‖dQ‖2L2(μE).


Now

M := sup{‖dP‖2L2(μ) | P is an M-partition

}0≤dP≤1

≤ μ(X) < ∞.

Take a sequence of M-partitions Pk such that

‖dPk‖2L2(μ) −−−−→

k→∞M.

We obtain an increasing sequence of partitions Qk by common refinements:{Q1 := P1,

Qk+1 :=↑ {Pk+1,Qk} .

Let us show that the sequence of functions fk := dQkconverges to the Radon–

Nikodym derivative dν/dμ in L2(μ). First, fk ∈ L2(μ), because μ(X) <∞. More-over, these functions form a Cauchy sequence, because

‖fj − fk‖2L2(μ)

(C.18)=

∣∣∣‖fj‖2L2(μ) − ‖fk‖2L2(μ)

∣∣∣−−−−−→j,k→∞

0,

as M ≥ ‖fk‖2L2(μ) ≥ ‖dPk‖2L2(μ) →M . Since L2(μ) is a Banach space, there exists

f ∈ L2(μ) for which ‖f−fk‖L2(μ) → 0. Let us show that f = dν/dμ. Take E ∈M.Let dk := dRk

, whereRk :=↑ {Qk, {E,X \ E}} .

Then

ν(E)(C.17)

=∫

E

dk dμ

=∫

E

(dk − fk) dμ +∫

E

fk dμ

−−−−→k→∞

∫E

f dμ,

because∫

Efk dμ →

∫Ef dμ by the Monotone Convergence Theorem C.3.6 and

by Vitali’s Convergence Theorem C.4.14, and because∣∣∣∣∫E

(dk − fk) dμ

∣∣∣∣ ≤∫

E

|dk − fk| dμ

Holder≤

(∫E

|dk − fk|2 dμ

)1/2 (∫E

dμ

)1/2

≤ ‖dk − fk‖2L2(μ) μ(X)1/2

(C.18)

≤∣∣∣‖dk‖2L2(μ) − ‖fk‖2L2(μ)

∣∣∣ μ(X)1/2

−−−−→k→∞

0.


Thus f =dν/dμ. Finally, let g+≥0 beM-measurable. Take simpleM-measurablefunctions sk for which

0 ≤ sk(x) ≤ sk+1(x) −−−−→k→∞

g+(x).

Then ∫g+ dν =

∫lim

k→∞sk dμ

Mon. conv.= limk→∞

∫sk dμ

(∗)= lim

k→∞

∫sk

dν

dμdμ

Mon. conv.=∫

limk→∞

skdν

dμdμ

=∫

g+ dν

dμdμ,

where equality (∗) easily follows from∫

χE dν =∫

χEdνdμ dμ. �

Proof of the Radon–Nikodym Theorem C.4.38. Since ν = (ν+−ν−)& μ, we havealso ν+, ν− & μ. If the Radon–Nikodym derivatives dν+/dμ and dν−/dμ exist,then by the linearity of the integral we have

dν

dμ=

dν+

dμ− dν−

dμ.

Thus we may assume that ν, μ are finite measures, where ν & μ. Then alsoμ + ν : M → [0,∞] is a finite measure. By Lemma C.4.41, the Radon–Nikodymderivatives dμ/d(μ + ν) and dν/d(μ + ν) exist. Let

A :={

dμ

d(μ + ν)> 0

};

let us show that dν/dμ = g+, where g+ : X → [0,∞] is defined by

g+(x) :=

{dν

d(μ+ν)/dμ

d(μ+ν) , when x ∈ A,

0, when x ∈ Ac.

Here

μ(Ac)(C.16)

=∫

Ac

dμ

d(μ + ν)d(μ + ν) = 0,


and if E ∈M then

ν(E) = ν(A ∩ E) + ν(Ac ∩ E)μ(Ac∩E)=0, ν�μ

= ν(A ∩ E)(C.16)

=∫

A∩E

dν

d(μ + ν)d(μ + ν)

=∫

A∩E

g+ dμ

d(μ + ν)d(μ + ν)

(C.16)=

∫A∩E

g+ dμ

μ(Ac)=0=

∫E

g+ dμ.

Thus g+ = dν/dμ, and the Radon–Nikodym Theorem C.4.38 is proven. �

Exercise C.4.42. Let λ, μ, ν :M→ [0,∞] be σ-finite measures. Prove:

(a) If λ& μ, E ∈M and g is M-measurable, then∫E

g dλ =∫

E

gdλ

dμdμ.

(b) If λ& ν and μ& ν, then d(λ+μ)dν = dλ

dν + dμdν .

(c) If λ& μ and μ& ν, then dλdν = dλ

dμdμdν .

(d) If λ& μ and μ& λ, then dλdμ =

(dμdλ

)−1

.

Definition C.4.43 (Lebesgue decomposition). Let μ, ν :M→ [0,∞] be measures.A Lebesgue decomposition of ν with respect to μ is a pair (ν0, ν1) of measuresν0, ν1 :M→ [0,∞] satisfying

ν = ν0 + ν1,

{ν0⊥μ,

ν1 & μ.

Theorem C.4.44 (Existence of Lebesgue decomposition). Let μ, ν : M → [0,∞]be σ-finite measures. Then there exists a unique Lebesgue decomposition of ν withrespect to μ.

Proof. The Radon–Nikodym Theorem C.4.38 was formulated for a finite measure,but it can be easily generalised to σ-finite spaces: showing this was left as Exer-cise C.4.39. Let

A :={

dμ

d(μ + ν)> 0

}.


Define measures ν0, ν1 :M→ [0,∞] by{ν0(E) := ν(Ac ∩ E),ν1(E) := ν(A ∩ E).

Clearly ν = ν0 + ν1, and ν0⊥μ because{ν0(A) = ν(Ac ∩A) = ν(∅) = 0,

μ(Ac)(C.16)

=∫

Acdμ

d(μ+ν) d(μ + ν) = 0.

We will now prove that ν1 & μ. Let Ak :={

dμ

d(μ + ν)≥ 1/k

}. Take E ∈M such

that μ(E) = 0. Now ν1(E) = 0, because

ν1(E) = ν(A ∩ E)≤ (μ + ν)(A ∩ E)= lim

k→∞(μ + ν)(Ak ∩ E)

≤∫

Ak∩E

kdμ

d(μ + ν)d(μ + ν)

Radon−Nikodym= k μ(Ak ∩ E)≤ k μ(E) = 0.

Proving the uniqueness part is left as Exercise C.4.45. �

Exercise C.4.45. Show that the Lebesgue decomposition in Theorem C.4.44 isunique.

C.4.4 Integration as functional on function spaces

Assume (X,M, μ) is a measure space, possibly with topology. On function spaceslike Lp(μ) or C(X) = C(X, R) (when X is, e.g., a compact Hausdorff space),integration acts as a bounded linear functional by

f �→∫

fg dμ,

when g is a suitable weight function on X. It is natural to study necessary andsufficient conditions for g, and ask whether all the bounded linear functionals areof this form. The general functional analytic outline is as follows: Let V be a realBanach space, e.g., Lp(μ) or C(X) = C(X, R). The dual of V is the Banach space

V ′ = L(V, R) := {φ : V → R | φ bounded and linear} ,


endowed with the (operator) norm

φ �→ ‖φ‖ := supf∈V : ‖f‖V ≤1

|φ(f)|,

see Definition B.4.15 and Exercise B.4.16. Given a “concrete” space V , we wouldlike to discover an intuitive representation of the dual.

C.4.5 Integration as functional on Lp(μ)

Now we are going to find a concrete presentation for the dual of V = Lp(μ), where(X,M, μ) is a measure space. We shall assume that μ(X) <∞, though often thistechnical assumption can be removed, since everything works for σ-finite measuresjust as well.

Lemma C.4.46. Let μ be a finite measure. Let 1 ≤ p ≤ ∞, and let q = p′ be itsLebesgue conjugate, i.e., 1/p + 1/q = 1. Let g ∈ Lq(μ). Then φg ∈ Lp(μ)′, where

φg(f) :=∫

fg dμ,

and ‖φg‖ = ‖g‖Lq .

Exercise C.4.47. Prove Lemma C.4.46.

Remark C.4.48. You may generalise Lemma C.4.46 as follows: the conclusion holdsfor a general measure μ if 1 < p ≤ ∞, and for a σ-finite measure μ if 1 ≤ p ≤ ∞.

Remark C.4.49. The next Theorem C.4.50 roughly says that the dual of Lp “is”Lq, under some technical assumptions. The result holds for a general measure μ if1 < p <∞, and for a σ-finite measure if 1 ≤ p <∞.

Theorem C.4.50 (Dual of Lp(μ)). Let μ be a finite measure. Let 1 ≤ p < ∞, andlet q = p′ be its Lebesgue conjugate. Then the mapping

(g �→ φg) : Lq(μ)→ Lp(μ)′

is an isometric isomorphism, i.e., Lp(μ)′ ∼= Lq(μ).

Proof. By the previous Lemma C.4.46, it suffices to show that ψ ∈ Lp(μ)′ is ofthe form ψ = φg for some g ∈ Lq(μ). Let us define ν :M→ R such that

ν(E) := ψ(χE),

where χE ∈ Lp(μ) because μ(X) <∞.


The idea in the proof is to show that dν/dμ ∈ Lq(μ) and that

ψ(f) =∫

fdν

dμdμ. (C.19)

The first step is to show that ν is a signed measure: Let {Ej}∞j=1 ⊂M be a disjointcollection. Then ν(

⋃∞j=1 Ej) =

∑∞j=1 ν(Ej), because∣∣∣∣∣∣ν(

∞⋃j=1

Ej)−k∑

j=1

ν(Ej)

∣∣∣∣∣∣ =

∣∣∣∣∣∣ψ(χ⋃∞j=1 Ej

)−k∑

j=1

ψ(Ej)

∣∣∣∣∣∣=

∣∣∣∣∣∣ψ(∞∑

j=k+1

χEj)

∣∣∣∣∣∣≤ ‖ψ‖

∞∑j=k+1

‖χEj‖Lp(μ)

μ(X)<∞−−−−−−→k→∞

0,

where we used the linearity and boundedness of ψ, and the disjointness of {Ej}∞j=1.Thus ν is a signed measure. Moreover ν & μ, because if μ(E) = 0 then χE = 0μ-almost everywhere, implying

ν(E) = ψ(χE) = 0

as ψ∈Lp(μ)′. Thus dν/dμ∈L1(μ) exists by the Radon–Nikodym Theorem C.4.38.We have to show that dν/dμ ∈ Lq(μ) and that (C.19) holds for every f ∈ Lp(μ).At least

ψ(χE) = ν(E)Radon−Nikodym

=∫

χEdν

dμdμ

for every E ∈M; by the linearity of ψ, this extends to

ψ(s) =∫

sdν

dμdμ (C.20)

for every simple M-measurable s : X → R. Next we show that dν/dμ ∈ Lq(μ).

For a moment, let p = 1 (so that q =∞). We shall soon see that ‖dν/dμ‖Lq ≤‖ψ‖. Take any M > ‖ψ‖ and let

AM :={∣∣∣∣dν

dμ

∣∣∣∣ > M

}.


Then

M μ(AM ) =∫

AM

M dμ

≤∫

AM

∣∣∣∣dν

dμ

∣∣∣∣ dμ

= χAMsgn

(dν

dμ

)dν

dμdμ

(C.20)= ψ

(χAM

sgn(

dν

dμ

))≤ ‖ψ‖

∥∥∥∥χAMsgn

(dν

dμ

)∥∥∥∥Lp(μ)

p=1

≤ ‖ψ‖ μ(AM ).

Since M > ‖ψ‖ and μ(AM ) < ∞, we must have μ(AM ) = 0, so ‖dν/dμ‖L∞(μ) ≤‖ψ‖.

Now let 1 < p < ∞ (so that ∞ > q > 1). Take simple M-measurablefunctions hk : X → R such that

0 ≤ hk(x) ≤ hk+1(x) −−−−→k→∞

∣∣∣∣dν

dμ(x)

∣∣∣∣ .

Then ∥∥∥∥dν

dμ

∥∥∥∥q

Lq(μ)

=∫ ∣∣∣∣dν

dμ

∣∣∣∣ dμFatou≤ lim inf

k→∞

∫hq

k dμ,

so that dν/dμ ∈ Lq(μ) follows if we show that ‖hk‖Lq(μ) ≤ constant < ∞ forevery k ∈ Z+:

‖hk‖qLq(μ) =

∫hq

k dμ

≤∫

hq−1k

∣∣∣∣dν

dμ

∣∣∣∣ dμ

=∫

hq−1k sgn

(dν

dμ

)dν

dμdμ

(C.20)= ψ

(hq−1

k sgn(

dν

dμ

))≤ ‖ψ‖

∥∥∥hq−1k

∥∥∥Lp(μ)

= ‖ψ‖ ‖hk‖Lq/p(μ) ,

because p(q − 1) = q. Hence ‖hk‖Lq(μ) = ‖hk‖q(1−1/p)Lq(μ) ≤ ‖ψ‖.


Finally, we have to show that (C.19) holds for f ∈ Lp(μ). Take simple M-measurable functions fk : X → [−∞,+∞] such that fk → f in Lp(μ). Then∣∣∣∣ψ(f)−

∫f

dν

dμdμ

∣∣∣∣(C.20)

=∣∣∣∣ψ(f − fk) +

∫(fk − f)

dν

dμdμ

∣∣∣∣≤ |ψ(f − fk)|+

∫|fk − f |

∣∣∣∣dν

dμ

∣∣∣∣ dμ

Holder≤ ‖ψ‖ ‖f − fk‖Lp(μ) + ‖fk − f‖Lp(μ)

∥∥∥∥dν

dμ

∥∥∥∥Lq(μ)

−−−−→k→∞

0.

Thus the proof is complete. �Exercise C.4.51. Generalise Theorem C.4.50 to the case where μ is σ-finite (and1 ≤ p <∞).

Exercise C.4.52. Generalise Theorem C.4.50 to the case where μ is any measureand 1 < p < ∞ (so that 1 < q < ∞ also). (Hint: apply the result of Exer-cise C.4.51.)

Remark C.4.53. We have not dealt with the dual of L∞(μ). This case actuallyresembles the other Lp-cases, but is slightly different, see details, e.g., in [153].Often, however, L∞(μ)′ ∼= L1(μ).

Exercise C.4.54. Let X = [0, 1] and μ = (λR)X . Show that there exists ψ ∈ L∞(μ)′

which is not of the form f �→∫

fg dμ for any g ∈ L1(μ).(Hint: Define a suitable bounded linear functional f �→ ϕ(f) for continuous func-tions f , and extend it to ψ using the Hahn–Banach Theorem, see Theorem B.4.25.)

Exercise C.4.55. Let (X,M, μ) be a measure space, where X is uncountable,M = {E ⊂ X : E or Ec is countable} and μ is the counting measure. Show thatthere exists ψ ∈ L1(μ)′ which is not of the form f �→

∫fg dμ for any g ∈ L∞(μ).

(Hint: You may use that there exists S ∈ P(X) \M, which follows by using theHausdorff Maximal Principle or other equivalents to the Axiom of Choice.)

Theorem C.4.56 (Converse of Holder’s inequality). Let μ be a σ-finite measure,1 ≤ p ≤ ∞, and 1

p + 1q = 1. Let S be the space of all simple functions that vanish

outside a set of finite measure. Let g be M-measurable such that fg ∈ L1(μ) forall f ∈ S, and such that

Mq(g) := sup{|∫

fg dμ| : f ∈ S, ‖f‖Lp(μ) = 1}

is finite. Then g ∈ Lq(μ) and Mq(g) = ‖g‖Lq(μ).


Proof. From Holder’s inequality (Theorem C.4.4) we have the inequality Mq(g) ≤||g||Lq(μ). For the proof of ||g||Lq(μ) ≤ Mq(g) we follow [35]. Assume first thatq < ∞. Let En ⊂ X be an increasing sequence of sets such that 0 < μ(En) < ∞for all n, and such that

⋃∞n=1 En = X. Let ϕn be a sequence of simple functions

such that ϕn → g pointwise and |ϕn| ≤ g, and let gn := ϕnχEn , where χEn isthe characteristic function of the set En. Then gn → g pointwise, |gn| ≤ |g| andgn ∈ S. Define fn := ||gn||1−q

Lq(μ)|gn|q−1 g|g| when g = 0 and fn := 0 when g = 0.

The relation 1p + 1

q = 1 implies (q−1)p = q, so that ||fn||Lp(μ) = 1, and by Fatou’slemma C.3.10 we have:

||g||Lq(μ) ≤ lim inf ||gn||Lq(μ) = lim inf∫|fngn| dμ

≤ lim inf∫|fng| dμ = lim inf

∫fng dμ ≤Mq(g).

The case q =∞ is slightly different. Take ε > 0 and denote A := {x ∈ X : |g(x)| ≥M∞(g)+ ε. We need to show that μ(A) = 0. If μ(A) > 0, there exists some B ⊂ Asuch that 0 < μ(B) < ∞. Let us define f := 1

μ(B)g|g|χB when g = 0 and f := 0

when g = 0. Then ||f ||L1(μ) = 1 and∫

fg dμ ≥M∞(g) + ε, a contradiction. �

C.4.6 Integration as functional on C(X)

Measure theory and topology have fundamental connections, as exemplified in thispassage. For our purposes, it is enough to study compact Hausdorff spaces, thoughanalogies hold for locally compact Hausdorff spaces. Let (X, τ) be a compactHausdorff space and let C(X) = C(X, R) denote the Banach space of continuousfunctions f : X → R, endowed with the supremum norm:

‖f‖ = ‖f‖C(X) := supx∈X

|f(x)|.

Appealing to the “geometry” of X, we are going to characterise the dual C(X)′ =L(C(X), R).

Exercise C.4.57. Let (X, τ) be a compact Hausdorff space. Actually, C(X) containsall the information about (X, τ): a set S ⊂ X is closed if and only if S = {f = 0}for some f ∈ C(X). Prove this.

Remark C.4.58. Let (X, τ) be a topological space. Recall that the vector space ofsigned (Borel) measures

M(X) = M(Σ(τ)) := {ν : Σ(τ)→ R | ν is a signed measure}

is a Banach space with the norm

‖ν‖ := |ν|(X).


Lemma C.4.59. Let ν : M → R be a signed measure on X, where τ ⊂ M. Forf ∈ C(X), let

Tν(f) :=∫

f dν

=∫

f dν+ −∫

f dν−.

Then Tν ∈ C(X)′ and ‖Tν‖ ≤ ‖ν‖.

Proof. Each f ∈ C(X) is M-measurable, as τ ⊂M. Furthermore, ν+, ν− :M→[0,∞] are finite measures. Consequently, f ∈ C(X) is ν±-integrable, so Tν(f) ∈ Ris well defined, and

|Tν(f)| ≤∫|f | dν+ +

∫|f | dν−

≤ ‖f‖(ν+(X) + ν−(X)

)= ‖f‖ ‖ν‖.

The operator Tν : C(X)→ R is linear since integration is linear. �Theorem C.4.60 (F. Riesz’s Topological Representation Theorem). Let (X, τ) bea compact Hausdorff space. Let M(X) and Tν ∈ C(X)′ be as above. Then

(ν �→ Tν) : M(X)→ C(X)′

is an isometric isomorphism.

In other words, bounded linear functionals on C(X) are exactly integrationswith respect to signed measures, with the natural norms coinciding. We shall soonprove the Riesz Representation Theorem C.4.60 step-wise.

Definition C.4.61 (Positive functionals). Let (X, τ) be a compact Hausdorff space.A functional T : C(X)→ R is called positive if T (f) ≥ 0 whenever f ≥ 0.

Exercise C.4.62. Show that a positive linear functional T ∈ C(X)′ is bounded andthat

‖T‖ = T1,

where 1 ∈ C(X) is the constant function x �→ 1.

Lemma C.4.63. Let T ∈ C(X)′, where (X, τ) is a compact Hausdorff space. Thenthere exist positive T+, T− ∈ C(X)′ such that

T = T+ − T−,

‖T‖ = ‖T+‖+ ‖T−‖.

Proof. For f = f+ − f− ∈ C(X), let us define

T+(f) := T+(f+)− T+(f−),


whereT+(g+) := sup

{T (h+) | h+ ∈ C(X), 0 ≤ h+ ≤ g+

}.

Obviously, 0 = T (0) ≤ T+(g+) ≤ T (‖g+‖1) = ‖T‖ ‖g+‖. Thereby the functionalT+ : C(X) → R is well defined and positive. Let us show that T+ is linear. If0 < λ+ ∈ R then

T+(λ+f+) = sup{Th | h ∈ C(X), 0 ≤ h ≤ λ+f+

}= sup

{T (λ+h) | h ∈ C(X), 0 ≤ h ≤ f+

}T linear= λ+ T+(f+);

from this we easily see that

T+(λf) = λ T+(f)

for every λ ∈ R and f ∈ C(X). Next,

T+(f+ + g+) = T+(f+) + T−(g+)

whenever 0 ≤ f+, g+ ∈ C(X), because

if

⎧⎪⎨⎪⎩0 ≤ h ≤ f+ + g+,

0 ≤ h1 ≤ f+,

0 ≤ h2 ≤ g+

then

⎧⎪⎨⎪⎩0 ≤ h1 + h2 ≤ f+ + g+,

0 ≤ min(f+, h) ≤ f+,

0 ≤ h−min(f+, h) ≤ g+.

Since(f + g)+ + f− + g− = (f + g)− + f+ + g+,

we get

T+((f + g)+) + T+(f−) + T+(g−) = T+((f + g)−) + T+(f+) + T+(g+),

so that

T+(f + g) = T+((f + g)+)− T+((f + g)−)=

(T+(f+)− T+(f−)

)+

(T+(g+)− T+(g−)

)= T+(f) + T+(g).

Hence we have seen that T+ : C(X)→ R is linear and positive, and that ‖T+‖ ≤‖T‖. Next, let us define T− := T+ − T ∈ C(X)′. Then T− is positive, because

T−(f+) = sup{Th− T (f+) | h ∈ C(X) : 0 ≤ h ≤ f+

}.


Finally,

‖T‖ = ‖T+ − T−‖≤ ‖T+‖+ ‖T−‖= T+(1) + T−(1)= 2 T+(1)− T (1)= sup {T (2h− 1) | h ∈ C(X) : 0 ≤ h ≤ 1}= sup {T (g) | g ∈ C(X) : −1 ≤ g ≤ 1}= ‖T‖,

so ‖T‖ = ‖T+‖+ ‖T−‖. �Remark C.4.64. Recall that the support supp(f) ⊂ X of a function f ∈ C(X) isthe closure of the set {f = 0}. Moreover, abbreviations

K ≺ f, f ≺ U

mean that 0 ≤ f ≤ 1, K ⊂ X is compact such that χK ≤ f , and U ⊂ X is opensuch that supp(f) ⊂ U .

Theorem C.4.65. Let T+ ∈ C(X)′ be positive, where (X, τ) is a compact Hausdorffspace. Then there exists a finite Borel measure μ : Σ(τ)→ [0,∞] such that

Tf =∫

f dμ

for every f ∈ C(X).

Proof. Let us define a measurelet m : τ → [0,∞] such that

m(U) := sup {Tf | f ≺ U} .

Indeed, m(∅) = T (0) = 0. Thus m generates an outer measure m∗ : P(X)→ [0,∞]by

m∗(E) = inf

⎧⎨⎩∞∑

j=1

m(Uj) : E ⊂∞⋃

j=1

Uj ∈ τ, Uj ∈ τ

⎫⎬⎭ .

We have to show μ := m∗|Σ(τ) is the desired measure. First,

m∗(E) = inf {m(U) : E ⊂ U ∈ τ} (C.21)

follows, if we show that

m

⎛⎝ ∞⋃j=1

Uj

⎞⎠ ≤∞∑

j=1

m(Uj).


So let f ≺ ⋃∞j=1 Uj . Now supp(f) ⊂ X is compact, so supp(f) ⊂ ⋃n

j=1 Uj for somen ∈ Z+. Let {gj}n

j=1 be a partition of unity for which

K ≺n∑

j=1

g, gj ≺ Uj .

Thereby

Tf = T (fn∑

j=1

gj)

T linear=n∑

j=1

T (fgj)

fgj≺Uj

≤n∑

j=1

m(Uj)

≤∞∑

j=1

m(Uj),

proving (C.21). Next, we show that τ ⊂ M(m∗) by proving that m∗(A ∪ B) =m∗(A) + m∗(B) whenever A ⊂ U ∈ τ and B ⊂ U c; let us assume the non-trivialcase m∗(A),m∗(B) < ∞. Given ε > 0, there exists V ∈ τ such that A ∪ B ⊂ Vand m∗(A ∪B) + ε > m(V ). Moreover, let{

f ≺ U ∩ V : m(U ∩ V ) < Tf + ε,

g ≺ supp(f)c ∩ V : m(supp(f)c ∩ V ) < Tg + ε.

We notice that U ∈M(m∗), because

m∗(A ∪B) + ε > m(V )f+g≺V

≥ T (f + g)T linear= Tf + Tg

> m(U ∩ V ) + m(supp(f)c ∩ V )− 2ε

≥ m∗(A) + m∗(B)− 2ε

≥ m∗(A ∪B)− 2ε.

Thus we can define the Borel measure μ := m∗|Σ(τ). Notice that m(U) = μ(U),μ(X) = T1 < ∞ and that m∗ is Borel-regular. If χE ≤ g ≤ χF , where g ∈ C(X)and E,F ∈ Σ(τ), then ∫

χE dμ ≤∫

g dμ ≤∫

χF dμ; (C.22)


moreover,μ(E) ≤ Tg ≤ μ(F ), (C.23)

because

μ(E)E⊂{g≥1}≤ μ({g ≥ 1})

δ>0≤ μ({g > 1− δ})

{g>1−δ}∈τ= sup {Tf : f ≺ {g > 1− δ}}

T positive, 0<δ<1

≤ T (g/(1− δ))T linear−−−−−→

δ→0Tg

implies μ(E) ≤ Tg, which implies

μ(X \ F )χF c≤1−g

T (1− g) = μ(X)− Tg,

so Tg ≤ μ(F ). Let us show that T (f+) =∫

f+ dμ, when 0 ≤ f+ ∈ C(X). Takeε > 0. Let us define fε ∈ C(X) by

fε(x) := min{f+(x), ε

}.

Then

f+ =∞∑

k=0

(f(k+1)ε − fkε

), (C.24)

where the sum is actually finite, as f+ is bounded. Combining

εχ{f+≥(k+1)ε} ≤ f(k+1)ε − fkε ≤ εχ{f+≥kε}

with inequalities (C.22,C.23), we get{εμ({f+ ≥ (k + 1)ε}) ≤

∫(f(k+1)ε − fkε) dμ ≤ εμ({f+ ≥ kε}),

εμ({f+ ≥ (k + 1)ε}) ≤ T (f(k+1)ε − fkε) ≤ εμ({f+ ≥ kε}).(C.25)

We obtain∣∣∣∣∫ f+ dμ− T (f+)∣∣∣∣ (C.24)

≤∞∑

k=0

∣∣∣∣∫ (f(k+1)ε − fkε) dμ− T (f(k+1)ε − fkε)∣∣∣∣

(C.25)

≤ ε

∞∑k=0

(μ({f+ ≥ kε})− μ({f+ ≥ (k + 1)ε})

)= εμ(X)

μ(X)<∞−−−−−−→ε→0

0,

i.e., T (f+) =∫

f+ dμ. Consequently, Tf =∫

f dμ for every f ∈ C(X). �


Proof of the Riesz Representation Theorem C.4.60. Let T ∈ C(X)′. Then T =T+ − T− by Lemma C.4.63, and Theorem C.4.65 provides finite Borel measuresμ, λ : Σ(τ)→ [0,∞] such that{

T+(f) =∫

f dμ,

T−(f) =∫

f dλ

for all f ∈ C(X). Thus μ− λ : Σ(τ)→ R is a signed measure for which

Tf =∫

f d(μ− λ)

for every f ∈ C(X). Moreover,

‖T‖ = ‖Tμ−λ‖Lemma C.4.59

≤ ‖μ− λ‖≤ ‖μ‖+ ‖λ‖= μ(X) + λ(X) = T+1 + T−1= ‖T+‖+ ‖T−‖= ‖T‖,

so ‖T‖ = ‖μ− ν‖. �Remark C.4.66. The Riesz Representation Theorem C.4.60 can be generalised.For instance, let (X, τ) be a locally compact Hausdorff space, which is also secondcountable (i.e., τ has a countable base). Endow the vector spaces

C0(X) := {f ∈ C(X) | {f ≥ ε} ⊂ X is compact for every ε > 0} ,

M(X) := {ν : Σ(τ)→ R | ν is a signed measure}

with respective complete norms f �→ ‖f‖ = sup{|f(x)| : x ∈ X} and ‖ν‖ :=|ν|(X); the rough idea is that a function f ∈ C0(X) “vanishes at infinity”. Then

(ν �→ Tν) : M(X)→ C0(X)′

is an isometric isomorphism, where

Tνf :=∫

f dν =∫

f dν+ −∫

f dν−.

We shall not prove this claim here.

Exercise C.4.67. Let μ be a finite Borel measure on a compact metric space (X, d).Prove that C(X) is dense in Lp(μ) (with the natural embedding) if 1 ≤ p < ∞.What could be a problem with p =∞?

Exercise C.4.68. Let us define (τhf)(x) := f(x+h), when f : R→ [−∞,∞]. Showthat ‖τhf − f‖Lp(Rn) →h→0 0 if f ∈ Lp(Rn) and 1 ≤ p <∞. Why is p =∞?

C.5. Product measure and integral 181

C.5 Product measure and integral

Let (X,Mμ, μ) and (Y,Mν , ν) be measure spaces. We are going to study thepossibility of changing the order of iterated integrals, i.e., whether∫

X

∫Y

f dν dμ =∫

Y

∫X

f dμ dν

under some reasonable assumptions on μ, ν and f : X × Y → [−∞,+∞]. Ofcourse, there are many technical issues involved: for instance,∫

X

∫Y

f dν dμ =∫

X

(x �→

∫Y

(y �→ f(x, y)) dν

)dν,

where y �→ f(x, y) is Mν-measurable for μ-almost every x ∈ X, and so on. Westart by defining the product measure μ× ν on X × Y .

Definition C.5.1 (Product measures). Let A := {A × B : A ∈ Mμ, B ∈ Mν}.Define a measurelet m : A → [0,∞] on X × Y by

m(A×B) := μ(A) ν(B).

The outer measure m∗ : P(X × Y ) → [0,∞] generated by the measurelet m iscalled the product outer measure of the measures μ and ν. The product measureof μ and ν is the restricted measure μ× ν := m∗|Mμ×ν

, where Mμ×ν :=M(m∗).

Remark C.5.2. Recall that m generates m∗ by

m∗(S) = inf

⎧⎨⎩∞∑

j=1

m(Aj ×Bj) : S ⊂∞⋃

j=1

Aj ×Bj , {Aj ×Bj}∞j=1 ⊂ A

⎫⎬⎭ .

Actually, we can do better here: we may demand that {Aj×Bj}∞j=1 ⊂ A is disjoint.Why is that? For any two families {Aj}∞j=1 ⊂Mμ and {Bj}∞j=1 ⊂Mν , let{

F1 := A1,

Fk+1 := Ak+1 \⋃k

j=1 Aj ,

{G1 := B1,

Gk+1 := Bk+1 \⋃k

j=1 Bj .

Clearly, {Fj ×Gk | j, k ∈ Z+} ⊂ A is a disjoint family, and it is easy to check that

E =⋃ {

Fj ×Gk | j, k ∈ Z+ : Fj ×Gk ⊂ E}

,

where E =⋃∞

i=1 Ai. Moreover,

∞∑j=1

m(Aj ×Bj) ≥∑

j,k≤n: Fj×Gk⊂E

m(Fj ×Gk)

for each n ∈ Z+. Letting n→∞ yields the claim.


Proposition C.5.3. Let A ∈ Mμ and B ∈ Mν . Then A × B ∈ Mμ×ν , and hencethe A-generated σ-algebra Σ(A) ⊂Mμ×ν . Moreover,

(μ× ν)(A×B) = μ(A) ν(B).

Remark C.5.4. Of course∫

X

∫Y

χA×B dν dμ =∫

Y

∫X

χA×B dμ dν = μ(A)ν(B)here, but this certainly does not prove the claims above.

Proof. To prove that A×B ∈Mμ×ν , it suffices to show that

m∗(S ∪ T ) ≥ m∗(S) + m∗(T )

whenever S ⊂ A×B and T ⊂ (A×B)c such that m∗(S),m∗(T ) <∞. Take ε > 0.Let {Aj ×Bj}∞j=1 ⊂ A be disjoint such that

S ∪ T ⊂∞⋃

j=1

Aj ×Bj ,

ε + m∗(S ∪ T ) >∞∑

j=1

m(Aj ×Bj).

Let us define Sj , Tj , Uj ⊂ X × Y by

Sj := (Aj ×Bj) ∩ (A×B),Tj := (Aj ×Bj) ∩ (A× (Y \B)) ,

Uj := (Aj ×Bj) ∩ ((X \A)× Y ) .

Then {Sj , Tj , Uj}∞j=1 ⊂ A is disjoint, and Aj ×Bj = Sj ∪ Tj ∪ Uj . Moreover,

S ⊂∞⋃

j=1

Sj , T ⊂∞⋃

j=1

(Tj ∪ Uj),

so that

ε + m∗(S ∪ T ) >∞∑

j=1

m(Aj ×Bj)

=∞∑

j=1

m(Sj) +∞∑

j=1

(m(Tj) + m(Uj))

≥ m∗(S) + m∗(T ).


Thus we have shown that A×B ∈Mμ×ν . Finally, if {Aj ×Bj}∞j=1 ⊂ A is a coverof A×B then

m∗(A×B)trivial≤ m(A×B)= μ(A) ν(B)

=∫

X

∫Y

χA×B dν dμ

≤∫

X

∫Y

∞∑j=1

χAj×Bj dν dμ

Mon. Conv.=∞∑

j=1

m(Aj ×Bj).

Therefore m∗(A×B) = μ(A) ν(B). �

Exercise C.5.5. Show that for the Lebesgue measures, λRm × λRn = λRm+n .

Definition C.5.6. For x ∈ X, the x-slice Sx ⊂ Y of a set S ⊂ X × Y is

Sx := {y ∈ Y | (x, y) ∈ S} .

Remark C.5.7. Let

B = {R ∈ Σ(A) : Rx ∈Mν for all x ∈ X} .

Clearly X × Y ∈ A ⊂ B. If R ∈ B then also Rc = (X × Y ) \ R ∈ B, because(Rc)x = (Rx)c. Similarly, if {Rj}∞j=1 ⊂ B then

⋃∞j=1 Rj ∈ B, because (

⋃∞j=1 Rj)x =⋃∞

j=1(Rj)x. Thus Σ(A) ⊂ B.

Lemma C.5.8. The product outer measure m∗ is Σ(A)-regular: for any S ⊂ X×Ythere exists R ∈ Σ(A) such that S ⊂ R and m∗(S) = m∗(R). Moreover, if μ, ν arefinite then the x-slice Rx ∈ Mν for every x ∈ X, x �→ ν(Rx) is Mμ-measurable,and

m∗(R) =∫

X

∫Y

χR dν dμ.

Proof. For each k ∈ Z+, take a disjoint family {Akj ×Bkj}∞j=1 ⊂ A such that

S ⊂∞⋃

j=1

Akj ×Bkj ,

m∗(S) +1k

≥∞∑

j=1

m(Akj ×Bkj).


Let Rn :=n⋂

k=1

∞⋃j=1

(Akj×Bkj) and R :=⋂∞

n=1 Rn. Then S ⊂ R ∈ Σ(A). Moreover,

we have m∗(S) = m∗(R), because

m∗(S) ≤ m∗(R)

≤ m∗(∞⋃

j=1

Anj ×Bnj)

=∞∑

j=1

m(Anj ×Bnj)

≤ m∗(S) +1n

.

The set Rn is the union of a disjoint family {Cnj ×Dnj}∞j=1 ⊂ A, and

χRxn(y) =

∞∑j=1

χCnj (x) χDnj (y).

Consequently, χRxn

: Y → R is Mν-measurable for all x ∈ X,

1 ≤ χRxn(y) ≤ χRx

n+1(y) −−−−→

n→∞ χRx(y) ≥ 0.

Lebesgue’s Dominated Convergence Theorem C.3.22 yields

hn(x) :=∫

Y

χRxn

dνν(Y )<∞−−−−−→

n→∞

∫Y

χRx dν =: h(x).

Then hn : X → [0,∞) is Mμ-measurable and

hn(x) =∞∑

j=1

ν(Dnj) χCnj(x).

Moreover,∞ > ν(Y ) ≥ hn(x) ≥ hn+1(x) −−−−→

n→∞ h(x) ≥ 0,

so also h : X → [0,∞) is Mμ-measurable and∫X

∫Y

χRn dν dμ =∫

X

hn dμμ(X)<∞−−−−−−→

n→∞

∫X

h dμ =∫

X

∫Y

χR dν dμ,

where Lebesgue’s Dominated Convergence Theorem was used again. �

The following result will neatly justify calling m∗ the product outer measure.

Corollary C.5.9. Let μ, ν be finite. Then (μ× ν)∗ = m∗.


Proof. If E ⊂ X × Y then

(μ× ν)∗(E) = inf {(μ× ν)(S) | E ⊂ S ∈Mμ×ν}Lemma C.5.8= inf {m∗(R) | E ⊂ R ∈ Σ(A)}

≥ m∗(E)≥ (μ× ν)∗(E),

that is (μ× ν)∗(E) = m∗(E). �

Exercise C.5.10. Generalise Corollary C.5.9 for σ-finite measures.

Exercise C.5.11. Let μ0 = μ∗|M(μ∗) and ν0 = ν∗|M(ν∗). Is μ× ν = μ0 × ν0?

Proposition C.5.12. Let μ, ν be finite and complete, S ⊂Mμ×ν . Then the x-sliceSx ∈Mν for μ-almost every x ∈ X, and x �→ ν(Sx) isMμ-measurable. Moreover,

(μ× ν)(S) =∫

X

∫Y

χS dν dμ.

Proof. By Lemma C.5.8, there exists a set R ∈ Σ(A) ⊂ Mμ×ν such that S ⊂ Rand

(μ× ν)(S) = m∗(S)= m∗(R)

=∫

X

∫Y

χR dν dμ.

Thus the result follows, if we can show that ν(Rx) = ν(Sx) for μ-almost everyx ∈ X. Here Rx = Sx ∪ (R \ S)x. By Lemma C.5.8, there exists a set Q ∈ Σ(A)such that R \ S ⊂ Q and

m∗(R \ S) =∫

X

∫Y

χQ dν dμ.

Now m∗(R \S) = m∗(R)−m∗(S) = 0, because the measures are finite. Thereforeν(Qx) = 0 for μ-almost every x ∈ X. Because R \ S ⊂ Q and ν is complete, wesee that

(R \ S)x ∈Mν , ν((R \ S)x) = 0

for μ-almost every x ∈ X. This shows that ν(Rx) = ν(Sx) for μ-almost everyx ∈ X. �


Theorem C.5.13 (Fubini–Tonelli Theorem). Let μ, ν be finite complete measures,and let f ≥ 0 be Mμ×ν-measurable. Then

x �→ f(x, y) is Mμ-measurable ν-a.e.,

y �→∫

X

(x �→ f(x, y)) dμ is Mν-measurable, and∫X×Y

f d(μ× ν) =∫

Y

∫X

f dμ dν

=∫

X

∫Y

f dν dμ.

Proof. Take Mμ×ν-measurable simple functions fk : X × Y → [0,∞) for which

fk ≤ fk+1pointwise−−−−−−→

k→∞f.

By Proposition C.5.12, take Nk ∈Mμ such that μ(Nk) = 0 and

(y �→ fk(x, y)) : Y → [0,∞)

is Mν-measurable for all x ∈ X \Nk. Then N :=⋃∞

k=1 Nk ∈Mμ, μ(N) = 0 and

(y �→ f(x, y)) : Y → [0,∞]

is Mν-measurable for all x ∈ X \ N . Let us define gk, g : X → [0,∞] such thatgk(x) := 0 =: g(x) if x ∈ N , and otherwise

gk(x) :=∫

Y

fk(x, y) dν(y) Mon. Conv.−−−−−−−→k→∞

∫Y

f(x, y) dν(y) =: g(x).

By Proposition C.5.12, gk : X → [0,∞] is Mμ-measurable, and clearly

gk ≤ gk+1pointwise−−−−−−→

k→∞g.

Thus g : X → [0,∞] is Mμ-measurable, and∫X

∫Y

f dν dμ =∫

X

g dμ

Mon. Conv.= limk→∞

∫gk dμ

= limk→∞

∫X

∫Y

fk dν dμ

Prop. C.5.12= lim

k→∞

∫X×Y

fk d(μ× ν)

Mon. Conv.=∫

X×Y

f d(μ× ν).

This concludes the proof. �


Corollary C.5.14 (Fubini Theorem). Let f ∈ L1(μ× ν). Then∫X×Y

f d(μ× ν) =∫

X

∫Y

f dν dμ =∫

Y

∫X

f dμ dν. �

Theorem C.5.15 (σ-finite Fubini–Tonelli and Fubini Theorems). The results inProposition C.5.12, the Fubini–Tonelli Theorem C.5.13 and its Corollary C.5.14hold also for σ-finite complete measures μ, ν.

Exercise C.5.16. Prove Theorem C.5.15 by taking an increasing sequence of mea-surable sets En of finite measure such that

⋃n En = X × Y .

Exercise C.5.17. Let X = [0, 1] = Y , μ := (λR)X and ν := (E �→ #E) : P(Y ) →[0,∞]. Let f = χS , where S := {(x, y) ∈ X × Y : x = y}. Calculate

(a)∫

X

∫Y

f dν dμ,(b)

∫Y

∫X

f dμ dν,(c)

∫X×Y

f d(μ× ν).

Remark C.5.18. In Exercise C.5.17, the measure ν is not σ-finite. Thus there isno contradiction with Theorem C.5.15.

Exercise C.5.19. Let (X,M, μ) be a complete σ-finite measure space. Let f : X →[0,∞] be M-measurable. Show that∫

f dμ =∫

[0,∞)

μ({f > t}) dλR(t).

Corollary C.5.20 (Young’s inequality). Let μ, ν be σ-finite and 1 < p <∞. Assumethat K : X × Y → C is a Mμ×ν-measurable function satisfying

C1 := supy∈Y

∫X

|K(x, y)| dμ(x),

C2 := supx∈X

∫Y

|K(x, y)| dν(y),

where C1, C2 <∞. For any u ∈ Lp(ν) define Au : X → C by

Au(x) =∫

Y

K(x, y) u(y) dν(y).

Then‖Au‖Lp(μ) ≤ C

1/p1 C

1/q2 ‖u‖Lp(ν),

where q is the conjugate exponent of p.

Remark C.5.21. Notice that this defines a unique bounded linear operator A :Lp(ν)→ Lp(μ).


Remark C.5.22. It is clear that we can replace sup by the esssup in the definitionof C1, C2, where the esssup would be taken with respect to ν and μ in C1 and C2,respectively:

C1 := ν − esssupy∈Y

∫X

|K(x, y)| dμ(x),

C2 := μ− esssupx∈X

∫Y

|K(x, y)| dν(y),

with the same proof.

Proof of Corollary C.5.20. First, y �→ K(x, y) u(y) is Mν-measurable, and

|Au(x)| ≤∫

Y

(|K(x, y)|1/p|u(y)|

) (|K(x, y)|1/q

)dν(y)

Holder≤

(∫Y

|K(x, y)||u(y)|p dν(y))1/p (∫

Y

|K(x, y)| dν(y))1/q

≤(∫

Y

|K(x, y)||u(y)|p dν(y))1/p

C1/q2 .

Using this we get

‖Au‖pLp(μ) =

∫X

|Au(x)|p dμ(x)

≤ Cp/q2

∫X

∫Y

|K(x, y)||u(y)|p dν(y) dμ(x)

Fubini= Cp/q2

∫Y

|u(y)|p∫

X

|K(x, y)| dμ(x) dν(y)

≤ C1Cp/q2

∫Y

|u(y)|p dν(y)

=(C

1/p1 C

1/q2

)p

‖u‖pLp(μ),

which gives the result. �

Theorem C.5.23 (Minkowski’s inequality for integrals). Let μ, ν be σ-finite and letf : X × Y → C be a Mμ×ν-measurable function. Let 1 ≤ p <∞. Then

{∫ (∫|f(x, y)| dν(y)

)p

dμ(x)}1/p

≤∫ (∫

|f(x, y)|p dμ(x))1/p

dν(y).


Proof. If p = 1 the result follows from Theorem C.5.15 exchanging the order ofthe integration. For 1 < p <∞, taking g ∈ Lq(μ), 1

p + 1q = 1, we have∫ (∫

|f(x, y)| dν(y))|g(x)| dμ(x)

Fubini=∫ (∫

|f(x, y)||g(x)| dμ(x))

dν(y)

Holder=∫ (∫

|f(x, y)|p dμ(x))1/p

dν(y) ‖g‖Lq(μ).

Now the statement follows from the converse of Holder’s inequality (TheoremC.4.56). �

As a consequence, we obtain the second part of Minkowski’s inequality forintegrals:

Corollary C.5.24 (Monotonicity of Lp-norm). Let μ, ν be σ-finite and let f : X ×Y → C be a Mμ×ν-measurable function. Let 1 ≤ p ≤ ∞. Assume that f(·, y) ∈Lp(μ) for ν-a.e. y, and assume that the function y �→ ||f(·, y)||Lp(μ) is in L1(μ).Then f(x, ·) ∈ L1(ν) for μ-a.e. x, the function x �→

∫f(x, y) dν(y) is in Lp(μ),

and ∥∥∥∥∫f(·, y) dν(y)

∥∥∥∥Lp(μ)

≤∫‖f(·, y)‖Lp(μ) dν(y).

Proof. For p = ∞ the statement follows from Theorem C.3.14. For 1 ≤ p < ∞ itfollows from Theorem C.5.23 and Fubini’s Theorem C.5.15. �

Chapter D

Algebras

An algebra is a vector space endowed with a multiplication, satisfying some com-patibility conditions. In the sequel, we are going to deal with spectral propertiesof algebras under various additional assumptions.

D.1 Algebras

Definition D.1.1 (Algebra). A vector space A over the field C is an algebra if thereexists an element 1A ∈ A\{0} and a mapping A×A → A, (x, y) �→ xy, satisfying

x(yz) = (xy)z,

x(y + z) = xy + xz, (x + y)z = xz + yz,

λ(xy) = (λx)y = x(λy),1Ax = x = x1A,

for all x, y, z ∈ A and λ ∈ C. We briefly write xyz := x(yz). The element 1 := 1Ais called the unit of A, and an element x ∈ A is called invertible (with the uniqueinverse x−1) if there exists x−1 ∈ A such that

x−1x = 1 = xx−1.

If xy = yx for every x, y ∈ A then A is called commutative.

Remark D.1.2. Warnings: in some books the algebra axioms allow 1A to be 0,but then the resulting algebra is simply {0}; we have omitted such a triviality. Insome books the existence of a unit is omitted from the algebra axioms; what wehave called an algebra is there called a unital algebra.Example. Let us give some examples of algebras:

1. C is the most important algebra. The operations are the usual ones for com-plex numbers, and the unit element is 1C = 1 ∈ C. Clearly C is a commutativealgebra.

192 Chapter D. Algebras

2. The algebra F(X) := {f | f : X → C} of complex-valued functions on a (fi-nite or infinite) set X is endowed with the usual algebra structure (pointwiseoperations). Function algebras are commutative, because C is commutative.

3. The algebra L(V ) := {A : V → V | A is linear} of linear operators on a vectorspace V = {0} over C is endowed with the usual vector space structure andwith the multiplication (A,B) �→ AB (composition of operators); the unitelement is 1L(V ) = (v �→ v) : V → V , the identity operator on V . Thisalgebra is non-commutative if V is at least two-dimensional.

Exercise D.1.3. Let A be an algebra and x, y ∈ A. Prove the following claims:

(a) If x, xy are invertible then y is invertible.(b) If xy, yx are invertible then x, y are invertible.

Exercise D.1.4. Give an example of an algebra A and elements x, y ∈ A suchthat xy = 1A = yx. Prove that then (yx)2 = yx = 0. (Hint: Such an algebra isnecessarily infinite-dimensional.)

Exercise D.1.5 (Commutators). In an algebra A, let [A,B] = AB −BA. If λ is ascalar and A,B,C are elements of the algebra A show that

[B,A] = −[A,B],[λA, B] = λ[A,B],

[A + B,C] = [A,C] + [B,C],[AB,C] = A[B,C] + [A,C]B,

C[A,B]C−1 = [CAC−1, CBC−1].

Definition D.1.6 (Spectrum). Let A be an algebra. The spectrum σ(x) of an ele-ment x ∈ A is the set

σA(x) = σ(x) = {λ ∈ C : λ1− x is not invertible}.

Example. Let us give some examples of invertibility and spectra:

1. An element λ ∈ C is invertible if and only if λ = 0; the inverse of an invertibleλ is the usual λ−1 = 1/λ. Generally, σC(λ) = {λ}.

2. An element f ∈ F(X) is invertible if and only if f(x) = 0 for every x ∈ X.The inverse of an invertible f is g with g(x) = f(x)−1. Generally, σF(X)(f) =f(X) := {f(x) | x ∈ X}.

3. An element A ∈ L(V ) is invertible if and only if it is a bijection (if and onlyif 0 ∈ σL(V )(A)).

Exercise D.1.7. Let A be an algebra and x, y ∈ A. Prove the following claims:

(a) 1− yx is invertible if and only if 1− xy is invertible.(b) σ(yx) ⊂ σ(xy) ∪ {0}.(c) If x is invertible then σ(xy) = σ(yx).

D.1. Algebras 193

Definition D.1.8 (Ideals). Let A be an algebra. An ideal J ⊂ A (more precisely,a two-sided ideal) is a vector subspace J = A satisfying

∀x ∈ A ∀y ∈ J : xy, yx ∈ J ,

i.e., xJ ,J x ⊂ J for every x ∈ A. A maximal ideal is an ideal not contained inany other ideal.

Remark D.1.9. In some books our ideals are called proper ideals, and there idealis either a proper ideal or the whole algebra. In the case of (proper) ideals, thevector space A/J := {x + J | x ∈ A} becomes an algebra with the operation(x+J , y +J ) �→ xy +J and the unit element 1A/J := 1A+J . It is evident thatno proper ideal contains any invertible elements. We will drop the word “proper”since it is incorporated in Definition D.1.8.

Remark D.1.10. Let J ⊂ A be an ideal. Because x1 = x for every x ∈ A, wenotice that 1 ∈ J . Therefore an invertible element x ∈ A cannot belong to anideal (since x−1x = 1 ∈ J ).

Example. Let us give examples of ideals. Intuitively, an ideal of an algebra is asubspace resembling a multiplicative zero; consider equations x0 = 0 = 0x.

1. Let A be an algebra. Then {0} ⊂ A is an ideal.

2. The only ideal of C is {0} ⊂ C.

3. Let X be a set, and ∅ = S ⊂ X. Now

I(S) := {f ∈ F(X) | ∀x ∈ S : f(x) = 0}

is an ideal of the function algebra F(X). If x ∈ X then I({x}) is a maximalideal of F(X), because it is of co-dimension 1 in F(X). Notice that I(S) ⊂I({x}) for every x ∈ S; an ideal may be contained in many different maximalideals (cf. Krull’s Theorem D.1.13 in the sequel).

4. Let X be an infinite-dimensional Banach space. The set

LC(X) := {A ∈ L(X) | A is compact}

of compact linear operators X → X is an ideal of the algebra L(X) ofbounded linear operators X → X.

Definition D.1.11 (Semisimple algebras). The radical Rad(A) of an algebraA is theintersection of all the maximal ideals of A; A is called semisimple if Rad(A) = {0}.Exercise D.1.12 (Ideals spanned by sets). Show that any intersection of ideals isan ideal. Hence for any set S ⊂ A in an algebra A there exists a smallest possibleideal J ⊂ A such that S ⊂ J ; this J is called the ideal spanned by the set S.

Theorem D.1.13 (W. Krull). Every ideal is contained in a maximal ideal.


Proof. Let J be an ideal of an algebra A. Let P be the set of those ideals of A thatcontain J . The inclusion relation is the natural partial order on P ; the HausdorffMaximal Principle (Theorem A.4.9) says that there is a maximal chain C ⊂ P .Let M :=

⋃C. Clearly J ⊂ M. Let λ ∈ C, x, y ∈ M and z ∈ A. Then there

exists I ∈ C such that x, y ∈ I, so that

λx ∈ I ⊂M, x + y ∈ I ⊂M, xz, zx ∈ I ⊂M;

moreover,1 ∈

⋂I∈C

(A \ I) = A \⋃I∈C

I = A \M,

so that M = A. We have proven that M is an ideal. The maximality of the chainC implies that M is maximal. �

Lemma D.1.14. Let A be a commutative algebra and let M be an ideal. Then Mis maximal if and only if [0] is the only non-invertible element of A/M.

Proof. Of course, here [x] means x + M, where x ∈ A. Assume that M is amaximal ideal. Take [x] = [0], so that x ∈ M. Define

J := Ax +M = {ax + m | a ∈ A, m ∈M}.

Then clearly J =M⊂ J , and J is a vector subspace of A. If y ∈ A then

J y = yJ = yAx + yM⊂ Ax +M = J ,

so that either J is an ideal or J = A. But since M is a maximal ideal containedproperly in J , we must have J = A. Thus there exist a ∈ A and m ∈ M suchthat ax + m = 1A. Then

[a][x] = 1A/M = [x][a],

[x] is invertible in A/M.Conversely, assume that all the non-zero elements are invertible in A/M.

Assume that J ⊂ A is an ideal containing M. Suppose J = M, and pick x ∈J \M. Now [x] = [0], so that for some y ∈ A we have [x][y] = [1A]. Thereby

1A ∈ xy +M x∈J⊂ J +M⊂ J + J = J ,

which is a contradiction, since no ideal can contain invertible elements. Thereforewe must have J =M, meaning that M is maximal. �

Definition D.1.15 (Quotient algebra). Let A be an algebra with an ideal J . Forx ∈ A, let us denote

[x] := x + J = {x + j | j ∈ J }.

D.1. Algebras 195

Then the set A/J := {[x] | x ∈ A} can be endowed with a natural algebrastructure. Indeed, let us define

λ[x] := [λx], [x] + [y] := [x + y], [x][y] := [xy], 1A/J := [1A];

first of all, these operations are well defined, since if λ ∈ C and j, j1, j2 ∈ J then

λ(x + j) = λx + λj ∈ [λx],(x + j1) + (y + j2) = (x + y) + (j1 + j2) ∈ [x + y],

(x + j1)(y + j2) = xy + j1y + xj2 + j1j2 ∈ [xy].

Secondly, [1A] = 1A + J = J = [0], because 1A ∈ J . Moreover,

(x + j1)(1A + j2) = x + j1 + xj2 + j1j2 ∈ [x],(1A + j2)(x + j1) = x + j1 + j2x + j2j1 ∈ [x].

Now the reader may verify that A/J is really an algebra; it is called the quotientalgebra of A modulo J .

Remark D.1.16. Notice that A/J is commutative if A is commutative. Also noticethat [0] = J is the zero element in the quotient algebra.

Definition D.1.17 (Algebra homomorphism). Let A and B be algebras. A mappingφ : A → B is called an algebra homomorphism (or simply a homomorphism) if itis a linear mapping satisfying

φ(xy) = φ(x)φ(y)

for every x, y ∈ A (multiplicativity) and

φ(1A) = 1B.

The set of all homomorphisms A → B is denoted by

Hom(A,B).

A bijective homomorphism φ : A → B is called an isomorphism, denoted byφ : A ∼= B.

Example. Let us give examples of algebra homomorphisms:

1. The only homomorphism C → C is the identity mapping, i.e., Hom(C, C) ={x �→ x}.

2. Let x ∈ X. Let us define the evaluation mapping φx : F(X) → C by f �→f(x). Then φx ∈ Hom(F(X), C).

3. Let J be an ideal of an algebra A, and denote [x] = x+J . Then (x �→ [x]) ∈Hom(A,A/J ).


Exercise D.1.18. Let φ ∈ Hom(A,B). If x ∈ A is invertible then φ(x) ∈ B isinvertible. For any x ∈ A, σB(φ(x)) ⊂ σA(x).

Exercise D.1.19. Let A be the set of matrices(α β0 α

)(α, β ∈ C).

Show that A is a commutative algebra. Classify (up to an isomorphism) all thetwo-dimensional algebras. (Hint: Prove that in a two-dimensional algebra either∃x = 0 : x2 = 0 or ∃x ∈ {1,−1} : x2 = 1.)

Proposition D.1.20. Let A and B be algebras, and φ ∈ Hom(A,B). Then φ(A) ⊂ Bis a subalgebra, Ker(φ) := {x ∈ A | φ(x) = 0} is an ideal of A, and A/Ker(φ) ∼=φ(A).

Exercise D.1.21. Prove Proposition D.1.20.

Definition D.1.22 (Tensor product algebra). The tensor product algebra of a K-vector space V is the K-vector space

T :=∞⊕

m=0

⊗mV,

where ⊗0V := K, ⊗m+1V := (⊗mV ) ⊗ V ; the multiplication of this algebra isgiven by

(x, y) �→ xy := x⊗ y

with the identifications W ⊗ K ∼= W ∼= K ⊗W for a K-vector space W , so thatthe unit element 1T ∈ T is the unit element 1 ∈ K.

D.2 Topological algebras

Definition D.2.1 (Topological algebra). A topological space A with the structureof an algebra is called a topological algebra if

1. {0} ⊂ A is a closed subset, and2. the algebraic operations are continuous, i.e., the mappings

((λ, x) �→ λx) : C×A → A,

((x, y) �→ x + y) : A×A → A,

((x, y) �→ xy) : A×A → A

are continuous.

Remark D.2.2. Similarly, a topological vector space is a topological space and avector space, in which {0} is a closed subset and the vector space operations(λ, x) �→ λx and (x, y) �→ x + y are continuous.

D.2. Topological algebras 197

Remark D.2.3. Some books omit the assumption that {0} should be a closed set;then, e.g., any algebra A with a topology τ = {∅,A} would become a topologicalalgebra. However, such generalisations are seldom useful. And it will turn out soon,that actually our topological algebras are indeed Hausdorff spaces! {0} being aclosed set puts emphasis on closed ideals and continuous homomorphisms, as weshall see later.Example. Let us give examples of topological algebras:

1. The commutative algebra C endowed with its usual topology (given by theabsolute value norm x �→ |x|) is a topological algebra.

2. If (X, x �→ ‖x‖) is a normed space, X = {0}, then L(X) is a topologicalalgebra with the norm

A �→ ‖A‖ := supx∈X:‖x‖≤1

‖Ax‖.

Notice that L(C) ∼= C, and L(X) is non-commutative if dim(X) ≥ 2.3. Let X be a set. Then

Fb(X) := {f ∈ F(X) | f is bounded}

is a commutative topological algebra with the supremum norm

f �→ ‖f‖ := supx∈X

|f(x)|.

Similarly, if X is a topological space then the algebra

Cb(X) := {f ∈ C(X) | f is bounded}

of bounded continuous functions on X is a commutative topological algebrawhen endowed with the supremum norm.

4. If (X, d) is a metric space then the algebra

Lip(X) := {f : X → C | f is Lipschitz continuous and bounded}

is a commutative topological algebra with the norm

f �→ ‖f‖ := max

{supx∈X

|f(x)|, supx�=y

|f(x)− f(y)|d(x, y)

}.

5. E(R) := C∞(R) is a commutative topological algebra with the metric

(f, g) �→∞∑

m=1

2−m pm(f − g)1 + pm(f − g)

, where pm(f) := max|x|≤m,k≤m

|f (k)(x)|.

This algebra is not normable (can not be endowed with a norm).


6. The topological dual E ′(R) of E(R) is the space of compactly supported dis-tributions (see Definition 1.4.8). There the multiplication is the convolution,which is defined for nice enough f, g by

(f, g) �→ f ∗ g, (f ∗ g)(x) :=∫ ∞

−∞f(x− y) g(y) dy.

The unit element of E(R) is the Dirac delta distribution δ0 at the origin0 ∈ R. This is a commutative topological algebra with the weak∗-topology,but it is not metrisable.

7. Convolution algebras of compactly supported distributions on Lie groups arenon-metrisable topological algebras; such an algebra is commutative if andonly if the group is commutative.

Remark D.2.4. Let A be a topological algebra, U ⊂ A open, and S ⊂ A. Dueto the continuity of ((λ, x) �→ λx) : C × A → A the set λU = {λu | u ∈ U} isopen if λ = 0. Due to the continuity of ((x, y) �→ x + y) : A × A → A the setU + S = {u + s | u ∈ U, s ∈ S} is open.

Exercise D.2.5. Show that topological algebras are Hausdorff spaces.

Remark D.2.6. Notice that in the previous exercise you actually need only thecontinuities of the mappings (x, y) �→ x + y and x �→ −x, and the fact that {0} isa closed set. Indeed, the commutativity of the addition operation is not needed,so that you can actually prove a proposition “Topological groups are Hausdorffspaces”!

Exercise D.2.7. Let {Aj | j ∈ J} be a family of topological algebras. EndowA :=

∏j∈J

Aj with a structure of a topological algebra.

Exercise D.2.8. Let A be an algebra and a normed space. Prove that it is atopological algebra if and only if there exists a constant C <∞ such that

‖xy‖ ≤ C ‖x‖ ‖y‖

for every x, y ∈ A.

Exercise D.2.9. Let A be an algebra. The commutant of a subset S ⊂ A is

Γ(S) := {x ∈ A | ∀y ∈ S : xy = yx}.

Prove the following claims:

(a) Γ(S) ⊂ A is a subalgebra; Γ(S) is closed if A is a topological algebra.(b) S ⊂ Γ(Γ(S)).(c) If xy = yx for every x, y ∈ S then Γ(Γ(S)) ⊂ A is a commutative subalgebra,

where σΓ(Γ(S))(z) = σA(z) for every z ∈ Γ(Γ(S)).

D.2. Topological algebras 199

Closed ideals

In topological algebras, the good ideals are the closed ones.Example. Let A be a topological algebra; then {0} ⊂ A is a closed ideal. Let Bbe another topological algebra, and φ ∈ Hom(A,B) be continuous. Then it is easyto see that Ker(φ) = φ−1({0}) ⊂ A is a closed ideal; this is actually a canonicalexample of closed ideals.

Proposition D.2.10. Let A be a topological algebra and J its ideal. Then eitherJ = A or J ⊂ A is a closed ideal.

Proof. Let λ ∈ C, x, y ∈ J , and z ∈ A. Take V ∈ V(λx). Then there existsU ∈ V(x) such that λU ⊂ V (due to the continuity of the multiplication by ascalar). Since x ∈ J , we may pick x0 ∈ J ∩ U . Now

λx0 ∈ J ∩ (λU) ⊂ J ∩ V,

which proves that λx ∈ J . Next take W ∈ V(x+ y). Then for some U ∈ V −→(x)and V ∈ V(y) we have U +V ⊂W (due to the continuity of the mapping (x, y) �→x + y). Since x, y ∈ J , we may pick x0 ∈ J ∩ U and y0 ∈ J ∩ V . Now

x + y ∈ J ∩ (U + V ) ⊂ J ∩W,

which proves that x + y ∈ J . Finally, we should show that xz, zx ∈ J , but thisproof is so similar to the previous steps that it is left for the reader as an easytask. �Definition D.2.11 (Topology for quotient algebra). Let J be an ideal of a topo-logical algebra A. Let τ be the topology of A. For x ∈ A, define [x] = x +J , andlet [S] = {[x] | x ∈ S}. Then it is easy to check that {[U ] | U ∈ τ} is a topologyof the quotient algebra A/J ; it is called the quotient topology.

Remark D.2.12. Let A be a topological algebra and J and ideal in A. Thequotient map (x �→ [x]) ∈ Hom(A,A/J ) is continuous: namely, if x ∈ A and[V ] ∈ VA/J ([x]) for some V ∈ τ then U := V + J ∈ V(x) and [U ] = [V ].

Lemma D.2.13. Let J be an ideal of a topological algebra A. Then the algebraoperations on the quotient algebra A/J are continuous.

Proof. Let us check the continuity of the multiplication in the quotient algebra:Suppose [x][y] = [xy] ∈ [W ], where W ⊂ A is an open set (recall that every openset in the quotient algebra is of the form [W ]). Then

xy ∈W + J .

Since A is a topological algebra, there are open sets U ∈ VA(x) and V ∈ VA(y)satisfying

UV ⊂W + J .


Now [U ] ∈ VA/J ([x]) and [V ] ∈ VA/J ([y]). Furthermore, [U ][V ] ⊂ [W ] because

(U + J )(V + J ) ⊂ UV + J ⊂W + J ;

we have proven the continuity of the multiplication ([x], [y]) �→ [x][y]. As aneasy exercise, we leave it for the reader to verify the continuity of the mappings(λ, [x]) �→ λ[x] and ([x], [y]) �→ [x] + [y]. �Exercise D.2.14. Complete the previous proof by showing the continuity of themappings (λ, [x]) �→ λ[x] and ([x], [y]) �→ [x] + [y].

With Lemma D.2.13, we conclude:

Proposition D.2.15. Let J be an ideal of a topological algebra A. Then A/J is atopological algebra if and only if J is closed.

Proof. If the quotient algebra is a topological algebra then {[0]} = {J } is a closedsubset of A/J ; since the quotient homomorphism is a continuous mapping, J =Ker(x �→ [x]) ⊂ A must be a closed set.

Conversely, suppose J is a closed ideal of a topological algebra A. Then wededuce that

(A/J ) \ {[0]} = [A \ J ]

is an open subset of the quotient algebra, so that {[0]} ⊂ A/J is closed. �Remark D.2.16. Let X be a topological vector space and M be its subspace. Thereader should be able to define the quotient topology for the quotient vector spaceX/M = {[x] := x + M | x ∈ X}. Now X/M is a topological vector space if andonly if M is a closed subspace.

Let M ⊂ X be a closed subspace. If d is a metric on X then there is a naturalmetric for X/M :

([x], [y]) �→ d([x], [y]) := infz∈M

d(x− y, z),

and if X is a complete metric space then X/M is also complete. Moreover, ifx �→ ‖x‖ is a norm on X then there is a natural norm for X/M :

[x] �→ ‖[x]‖ := infz∈M

‖x− z‖.

D.3 Banach algebras

Definition D.3.1 (Banach algebra). An algebra A is called a Banach algebra if itis a Banach space satisfying

‖xy‖ ≤ ‖x‖ ‖y‖for all x, y ∈ A and

‖1‖ = 1.

D.3. Banach algebras 201

Exercise D.3.2. Let K be a compact space. Show that C(K) is a Banach algebrawith the norm f �→ ‖f‖ = maxx∈K |f(x)|.Example. Let X be a Banach space. Then the Banach space L(X) of bounded lin-ear operators X → X is a Banach algebra if the multiplication is the compositionof operators, since

‖AB‖ ≤ ‖A‖ ‖B‖for every A,B ∈ L(X); the unit is the identity operator I : X → X, x �→ x.Actually, this is not far away from characterising all the Banach algebras:

Theorem D.3.3 (Characterisation of Banach algebras). A Banach algebra A is iso-metrically isomorphic to a norm closed subalgebra of L(X) for a Banach space X.

Proof. Here X := A. For x ∈ A, let us define

m(x) : A → A by m(x)y := xy.

Obviously m(x) is a linear mapping, m(xy) = m(x)m(y), m(1A) = 1L(A), and

‖m(x)‖ = supy∈A: ‖y‖≤1

‖xy‖

≤ supy∈A: ‖y‖≤1

(‖x‖ ‖y‖) = ‖x‖ = ‖m(x)1A‖

≤ ‖m(x)‖ ‖1A‖ = ‖m(x)‖;

briefly, m = (x �→ m(x)) ∈ Hom(A,L(A)) is isometric. Thereby m(A) ⊂ L(A) isa closed subspace and hence a Banach algebra. �Proposition D.3.4. A maximal ideal in a Banach algebra is closed.

Proof. In a topological algebra, the closure of an ideal is either an ideal or the wholealgebra. Let M be a maximal ideal of a Banach algebra A. The set G(A) ⊂ Aof all invertible elements is open, and M∩ G(A) = ∅ (because no ideal containsinvertible elements). Thus M⊂M⊂ A\G(A), so that M is an ideal containinga maximal ideal M; thus M =M. �Proposition D.3.5. Let J be a closed ideal of a Banach algebra A. Then the quo-tient vector space A/J is a Banach algebra; moreover, A/J is commutative if Ais commutative.

Proof. Let us denote [x] := x + J for x ∈ A. Since J is a closed vector subspace,the quotient space A/J is a Banach space with norm

[x] �→ ‖[x]‖ = infj∈J

‖x + j‖.

Let x, y ∈ A and ε > 0. Then there exist i, j ∈ J such that

‖x + i‖ ≤ ‖[x]‖+ ε, ‖y + j‖ ≤ ‖[y]‖+ ε.


Now (x + i)(y + j) ∈ [xy], so that

‖[xy]‖ ≤ ‖(x + i)(y + j)‖≤ ‖x + i‖ ‖y + j‖≤ (‖[x]‖+ ε) (‖[y]‖+ ε)= ‖[x]‖ ‖[y]‖+ ε(‖[x]‖+ ‖[y]‖+ ε);

since ε > 0 is arbitrary, we have

‖[x][y]‖ ≤ ‖[x]‖ ‖[y]‖.

Finally, ‖[1]‖ ≤ ‖1‖ = 1 and ‖[x]‖ = ‖[x][1]‖ ≤ ‖[x]‖ ‖[1]‖, so that we have‖[1]‖ = 1. �Exercise D.3.6. Let A be a Banach algebra, and let x, y ∈ A satisfy

x2 = x, y2 = y, xy = yx.

Show that either x = y or ‖x − y‖ ≥ 1. Give an example of a Banach algebra Awith elements x, y ∈ A such that x2 = x = y = y2 and ‖x− y‖ < 1.

Proposition D.3.7. Let A be a Banach algebra. Then Hom(A, C) ⊂ A′ and ‖φ‖ = 1for every φ ∈ Hom(A, C).

Proof. Let x ∈ A, ‖x‖ < 1. Let

yn :=n∑

j=0

xj ,

where x0 := 1. If n > m then

‖yn − ym‖ = ‖xm + xm+1 + · · ·+ xn‖≤ ‖x‖m

(1 + ‖x‖+ · · ·+ ‖x‖n−m

)= ‖x‖m 1− ‖x‖n−m+1

1− ‖x‖ →n>m→∞ 0;

thus (yn)∞n=1 ⊂ A is a Cauchy sequence. There exists y = limn→∞ yn ∈ A, becauseA is complete. Since xn → 0 and

yn(1− x) = 1− xn+1 = (1− x)yn,

we deduce y = (1−x)−1. Suppose λ = φ(x), |λ| > ‖x‖; now ‖λ−1x‖ = |λ|−1 ‖x‖ <1, so that 1− λ−1x is invertible. Then

1 = φ(1) = φ((1− λ−1x)(1− λ−1x)−1

)= φ

(1− λ−1x

)φ

((1− λ−1x)−1

)= (1− λ−1φ(x)) φ

((1− λ−1x)−1

)= 0,


a contradiction; hence∀x ∈ A : |φ(x)| ≤ ‖x‖,

that is ‖φ‖ ≤ 1. Finally, φ(1) = 1, so that ‖φ‖ = 1. �Lemma D.3.8. Let A be a Banach algebra. The set G(A) ⊂ A of its invertibleelements is open. The mapping (x �→ x−1) : G(A)→ G(A) is a homeomorphism.

Proof. Take x ∈ G(A) and h ∈ A. As in the proof of the previous Proposition, wesee that

x− h = x(1− x−1h)

is invertible if ‖x−1‖ ‖h‖ < 1, that is ‖h‖ < ‖x−1‖−1; thus G(A) ⊂ A is open.The mapping x �→ x−1 is clearly its own inverse. Moreover

‖(x− h)−1 − x−1‖ = ‖(1− x−1h)−1x−1 − x−1‖

≤ ‖(1− x−1h)−1 − 1‖ ‖x−1‖ = ‖∞∑

n=1

(x−1h)n‖ ‖x−1‖

≤ ‖h‖( ∞∑

n=1

‖x−1‖n+1 ‖h‖n−1

)→h→0 0;

hence x �→ x−1 is a homeomorphism. �Exercise D.3.9 (Topological zero divisors). Let A be a Banach algebra. We saythat x ∈ A is a topological zero divisor if there exists a sequence (yn)∞n=1 ⊂ A suchthat ‖yn‖ = 1 for all n and

limn→∞xyn = 0 = lim

n→∞ ynx.

(a) Show that if (xn)∞n=1 ⊂ G(A) satisfies xn → x ∈ ∂G(A) then ‖x−1n ‖ → ∞.

(b) Using this result, show that the boundary points of G(A) are topological zerodivisors.(c) In what kind of Banach algebras 0 is the only topological zero divisor?

Theorem D.3.10 (Gelfand, 1939). Let A be a Banach algebra and x ∈ A. Thenthe spectrum σ(x) ⊂ C is a non-empty compact set.

Proof. Let x ∈ A. Then σ(x) belongs to a 0-centered disc of radius ‖x‖ in thecomplex plane: for if λ ∈ C, |λ| > ‖x‖ then 1 − λ−1x is invertible, equivalentlyλ1− x is invertible.

The mapping g : C → A, λ �→ λ1 − x, is continuous; the set G(A) ⊂ A ofinvertible elements is open, so that

C \ σ(x) = g−1(G(A))

is open. Thus σ(x) ∈ C is closed and bounded, i.e., compact by the Heine–BorelTheorem (Corollary A.13.7).


The hard part is to prove the non-emptiness of the spectrum. Let us definethe resolvent mapping R : C \ σ(x)→ G(A) by

R(λ) = (λ1− x)−1.

We know that this mapping is continuous, because it is composed of continuousmappings

(λ �→ λ1− x) : C \ σ(x)→ G(A) and (y �→ y−1) : G(A)→ G(A).

We want to show that R is weakly holomorphic, that is f ◦ R ∈ H(C \ σ(x)) forevery f ∈ A′ = L(A, C). Let z ∈ C \ σ(x), f ∈ A′. Then we calculate

(f ◦R)(z + h)− (f ◦R)(z)h

= f

(R(z + h)−R(z)

h

)= f

(R(z + h)R(z)−1 − 1

hR(z)

)= f

(R(z + h)(R(z + h)−1 − h1)− 1

hR(z)

)= f(−R(z + h)R(z))

→h→0 f(−R(z)2),

because f and R are continuous; thus R is weakly holomorphic.Suppose |λ| > ‖x‖. Then

‖R(λ)‖ = ‖(λ1− x)−1‖= |λ|−1 ‖(1− x/λ)−1‖

= |λ|−1

∥∥∥∥∥∥∞∑

j=0

(x/λ)j

∥∥∥∥∥∥≤ |λ|−1

∞∑j=0

‖x/λ‖−j

= |λ|−1 11− ‖x/λ‖

=1

|λ| − ‖x‖→|λ|→∞ 0.

Thereby(f ◦R)(λ)→|λ|→∞ 0


for every f ∈ A′. To get a contradiction, suppose σ(x) = ∅. Then f ◦R ∈ H(C) is 0by Liouville’s Theorem D.6.2 for every f ∈ A′; the Hahn–Banach Theorem B.4.25says that then R(λ) = 0 for every λ ∈ C; this is a contradiction, since 0 ∈ G(A).Thus σ(x) = ∅. �Exercise D.3.11. Let A be a Banach algebra, x ∈ A, Ω ⊂ C an open set, andσ(x) ⊂ Ω. Then

∃δ > 0 ∀y ∈ A : ‖y‖ < δ ⇒ σ(x + y) ⊂ Ω.

Exercise D.3.12. Alternatively, in the proof of Theorem D.3.10 one could use theNeumann series

R(λ) = R(λ0)∞∑

k=0

((λ0 − λ)R(λ0))k,

for all λ0 ∈ C\σ(x) and |λ − λ0|‖R(λ0)‖ < 1. Then R(λ) is analytic in C\σ(x)and satisfies R(λ) → 0 as λ → ∞. Consequently, use Liouville’s theorem (Theo-rem D.6.2) to conclude the statement.

Corollary D.3.13 (Gelfand–Mazur Theorem). Let A be a Banach algebra where0 ∈ A is the only non-invertible element. Then A is isometrically isomorphic to C.

Proof. Take x ∈ A, x = 0. Since σ(x) = ∅, pick λ(x) ∈ σ(x). Then λ(x)1 − x isnon-invertible, so that it must be 0; x = λ(x)1. By defining λ(0) = 0, we have analgebra isomorphism

λ : A → C.

Moreover, |λ(x)| = ‖λ(x)1‖ = ‖x‖. �Exercise D.3.14. Let A be a Banach algebra, and suppose that there exists C <∞such that

‖x‖ ‖y‖ ≤ C ‖xy‖for every x, y ∈ A. Show that A ∼= C isometrically.

Definition D.3.15 (Spectral radius). Let A be a Banach algebra. The spectralradius of x ∈ A is

ρ(x) := supλ∈σ(x)

|λ|;

this is well defined, because due to Gelfand’s Theorem D.3.10 the spectrum is non-empty. In other words, D(0, ρ(x)) = {λ ∈ C : |λ| ≤ ρ(x)} is the smallest 0-centeredclosed disk containing σ(x) ⊂ C. Notice that ρ(x) ≤ ‖x‖, since λ1−x = λ(1−x/λ)is invertible if |λ| > ‖x‖.Theorem D.3.16 (Spectral Radius Formula (Beurling, 1938; Gelfand, 1939)). LetA be a Banach algebra, x ∈ A. Then

ρ(x) = limn→∞ ‖x

n‖1/n.


Proof. For x = 0 the claim is trivial, so let us assume that x = 0. By Gelfand’sTheorem D.3.10, σ(x) = ∅. Let λ ∈ σ(x) and n ≥ 1. Notice that in an algebra, ifboth ab and ba are invertible then the elements a, b are invertible. Therefore

λn1− xn = (λ1− x)

(n−1∑k=0

λn−1−kxk

)=

(n−1∑k=0

λn−1−kxk

)(λ1− x)

implies that λn ∈ σ(xn). Thus |λn| ≤ ‖xn‖, so that

ρ(x) = supλ∈σ(x)

|λ| ≤ lim infn→∞ ‖xn‖1/n.

Let f ∈ A′ and λ ∈ C, |λ| > ‖x‖. Then

f(R(λ)) = f((λ1− x)−1

)= f

(λ−1(1− λ−1x)−1

)= f

(λ−1

∞∑n=0

λ−nxn

)

= λ−1∞∑

n=0

f(λ−nxn).

This formula is true also when |λ| > ρ(x), because f◦R is holomorphic in C\σ(x) ⊃C\D(0, ρ(x)). Hence if we define Tλ,x,n ∈ A′′ = L(A′, C) by Tλ,x,n(f) := f(λ−nxn),we obtain

supn∈N

|Tλ,x,n(f)| = supn∈N

|f(λ−nxn)| <∞ (when |λ| > ρ(x))

for every f ∈ A′; the Banach–Steinhaus Theorem B.4.29 applied on the family{Tλ,x,n}n∈N shows that

Mλ,x := supn∈N

‖Tλ,x,n‖ <∞,

so that we have

‖λ−nxn‖ Hahn−Banach= supf∈A′:‖f‖≤1

|f(λ−nxn)|

= supf∈A′:‖f‖≤1

|Tλ,x,n(f)|

= ‖Tλ,x,n‖≤ Mλ,x.

Hence‖xn‖1/n ≤M

1/nλ,x |λ| →n→∞ |λ|,

when |λ| > ρ(x). Thuslim sup

n→∞‖xn‖1/n ≤ ρ(x);

collecting the results, the Spectral Radius Formula is verified. �

D.4. Commutative Banach algebras 207

Remark D.3.17. The Spectral Radius Formula contains startling information: thespectral radius ρ(x) is purely an algebraic property (though related to a topo-logical algebra), but the quantity lim ‖xn‖1/n relies on both algebraic and metricproperties! Yet the results are equal!Remark D.3.18. ρ(x)−1 is the radius of convergence of the A-valued power series

λ �→∞∑

n=0

λnxn.

Remark D.3.19. Let A be a Banach algebra and B a Banach subalgebra. If x ∈ Bthen

σA(x) ⊂ σB(x)

and the inclusion can be proper, but the spectral radii for both Banach algebrasare the same, since

ρA(x) = limn→∞ ‖x

n‖1/n = ρB(x).

Exercise D.3.20. Let A be a Banach algebra, x, y ∈ A. Show that ρ(xy) = ρ(yx).Show that if x ∈ A is nilpotent (i.e., xk = 0 for some k ∈ N) then σ(x) = {0}.Exercise D.3.21. Let A be a Banach algebra and x, y ∈ A such that xy = yx.Prove that ρ(xy) ≤ ρ(x)ρ(y).

Exercise D.3.22. In the proof of Theorem D.3.16 argue as follows. For λ > ||x||note that the resolvent satisfies

R(λ) = λ−1∞∑

k=0

(λ−1x)k,

and this Laurent series converges for all |λ| > ρ(x). Consequently, its (Hadamard)radius of convergence satisfies

ρ(x) ≤ lim infn→∞ ||xn||1/n.

At the same time, the convergence for |λ| > ρ(x) implies limn→∞ λ−nxn = 0,which means that ||xn|| ≤ |λ|n for large enough n.

D.4 Commutative Banach algebras

In this section we are interested in maximal ideals of commutative Banach alge-bras. We shall learn that such algebras are closely related to algebras of continuousfunctions on compact Hausdorff spaces: there is a natural, far from trivial, homo-morphism from a commutative Banach algebra A to an algebra of functions onthe set Hom(A, C), which can be endowed with a canonical topology – relatedmathematics is called the Gelfand theory. In the sequel, one should ponder thisdilemma: which is more fundamental, a space or algebras of functions on it?


Example. Let us give examples of commutative Banach algebras:

1. Our familiar C(K), when K is a compact space.

2. L∞([0, 1]), when [0, 1] is endowed with the Lebesgue measure.

3. A(Ω) := C(Ω)∩H(Ω), when Ω ⊂ C is open, H(Ω) are holomorphic functionsin Ω, and Ω ⊂ C is compact.

4. M(Rn), the convolution algebra of complex Borel measures on Rn, with theDirac delta distribution at 0 ∈ Rn as the unit element, and endowed withthe total variation norm.

5. The algebra of matrices(

α β0 α

), where α, β ∈ C; notice that this algebra

contains nilpotent elements!

Definition D.4.1 (Spectrum and characters of an algebra). The spectrum of analgebra A is

Spec(A) := Hom(A, C),

i.e., the set of homomorphisms A → C; such a homomorphism is called a characterof A.

Remark D.4.2. The concept of spectrum is best suited for commutative algebras,as C is a commutative algebra; here a character A → C should actually be con-sidered as an algebra representation A → L(C). In order to fully capture thestructure of a non-commutative algebra, we should study representations of typeA → L(X), where the vector spaces X are multi-dimensional; for instance, if His a Hilbert space of dimension 2 or greater then Spec(L(H)) = ∅. However, thespectrum of a commutative Banach algebra is rich, as there is a bijective corre-spondence between characters and maximal ideals. Moreover, the spectrum of thealgebra is akin to the spectra of its elements:

Theorem D.4.3 (Gelfand, 1940). Let A be a commutative Banach algebra. Then:

(a) Every maximal ideal of A is of the form Ker(h) for some h ∈ Spec(A);(b) Ker(h) is a maximal ideal for every h ∈ Spec(A);(c) x ∈ A is invertible if and only if ∀h ∈ Spec(A) : h(x) = 0;(d) x ∈ A is invertible if and only if it is not in any ideal of A;(e) σ(x) = {h(x) | h ∈ Spec(A)}.

Proof. (a) Let M⊂ A be a maximal ideal; let [x] := x +M for x ∈ A. Since A iscommutative and M is maximal, every non-zero element in the quotient algebraA/M is invertible. We know that M is closed, so that A/M is a Banach algebra.Due to the Gelfand–Mazur Theorem (Corollary D.3.13), there exists an isometricisomorphism λ ∈ Hom(A/M, C). Then

h = (x �→ λ([x])) : A → C


is a character, and

Ker(h) = Ker((x �→ [x]) : A → A/M) =M.

(b) Let h : A → C be a character. Now h is a linear mapping, so that theco-dimension of Ker(h) in A equals the dimension of h(A) ⊂ C, which clearly is1. Any ideal of co-dimension 1 in an algebra must be maximal, so that Ker(h) ismaximal.

(c) If x ∈ A is invertible and h ∈ Spec(A) then h(x) ∈ C is invertible, thatis h(x) = 0. For the converse, assume that x ∈ A is non-invertible. Then

Ax = {ax | a ∈ A}

is an ideal of A (notice that 1 = ax = xa would mean that a = x−1). Hence byKrull’s Theorem D.1.13, there is a maximal ideal M ⊂ A such that Ax ⊂ M.Then (a) provides a character h ∈ Spec(A) for which Ker(h) = M. Especially,h(x) = 0.

(d) This follows from (a,b,c) directly.(e) (c) is equivalent to

“x ∈ A is non-invertible if and only if ∃h ∈ Spec(A) : h(x) = 0”,

which is equivalent to

“λ1− x is non-invertible if and only if ∃h ∈ Spec(A) : h(x) = λ”. �

Exercise D.4.4. Let A be a Banach algebra and x, y ∈ A such that xy = yx. Provethat σ(x + y) ⊂ σ(x) + σ(y) and σ(xy) ⊂ σ(x)σ(y).

Exercise D.4.5. Let A be the algebra of those functions f : R→ C for which

f(x) =∑n∈Z

fn eix·n, ‖f‖ =∑n∈Z

|fn| <∞.

Show that A is a commutative Banach algebra. Show that if f ∈ A and ∀x ∈ R :f(x) = 0 then 1/f ∈ A.

Definition D.4.6 (Gelfand transform). Let A be a commutative Banach algebra.The Gelfand transform x of an element x ∈ A is the function

x : Spec(A)→ C, x(φ) := φ(x).

Let A := {x : Spec(A)→ C | x ∈ A}. The mapping

A → A, x �→ x,

is called the Gelfand transform of A. We endow the set Spec(A) with the A-induced topology, called the Gelfand topology; this topological space is called the


maximal ideal space of A (for a good reason, in the light of the previous theorem).In other words, the Gelfand topology is the weakest topology on Spec(A) makingevery x a continuous function, i.e., the weakest topology on Spec(A) for whichA ⊂ C(Spec(A)).

Theorem D.4.7 (Gelfand, 1940). Let A be a commutative Banach algebra. ThenK = Spec(A) is a compact Hausdorff space in the Gelfand topology, the Gelfandtransform is a continuous homomorphism A → C(K), and ‖x‖ = sup

φ∈K|x(φ)| =

ρ(x) for every x ∈ A.

Proof. The Gelfand transform is a homomorphism, since

λx(φ) = φ(λx) = λφ(x) = λx(φ) = (λx)(φ),x + y(φ) = φ(x + y) = φ(x) + φ(y) = x(φ) + y(φ) = (x + y)(φ),

xy(φ) = φ(xy) = φ(x)φ(y) = x(φ)y(φ) = (xy)(φ),

1A(φ) = φ(1A) = 1 = 1C(K)(φ),

for every λ ∈ C, x, y ∈ A and φ ∈ K. Moreover,

x(K) = {x(φ) | φ ∈ K} = {φ(x) | φ ∈ Spec(A)} = σ(x),

implying‖x‖ = ρ(x) ≤ ‖x‖.

Clearly K is a Hausdorff space. What about compactness?Now K = Hom(A, C) is a subset of the closed unit ball of the dual Banach

space A′; by the Banach–Alaoglu Theorem B.4.36, this unit ball is compact inthe weak∗-topology. Recall that the weak∗-topology τA′ of A′ is the A-inducedtopology, with the interpretation A ⊂ A′′; thus the Gelfand topology τK is therelative weak∗-topology, i.e.,

τK = τA′ |K .

To prove that τK is compact, it is sufficient to show that K ⊂ A′ is closed in theweak∗-topology.

Let f ∈ A′ be in the weak∗-closure of K. We have to prove that f ∈ K, i.e.,

f(xy) = f(x)f(y) and f(1) = 1.

Let x, y ∈ A, ε > 0. Let S := {1, x, y, xy}. Using the notation of the proof of theBanach–Alaoglu Theorem B.4.36,

U(f, S, ε) = {ψ ∈ A′ : z ∈ S ⇒ |ψz − fz| < ε}

is a weak∗-neighbourhood of f . Thus choose hε ∈ K ∩ U(f, S, ε). Then

|1− f(1)| = |hε(1)− f(1)| < ε;


ε > 0 being arbitrary, we have f(1) = 1. Noticing that |hε(x)| ≤ ‖x‖, we get

|f(xy)− f(x)f(y)|≤ |f(xy)− hε(xy)|+ |hε(xy)− hε(x)f(y)|+ |hε(x)f(y)− f(x)f(y)|= |f(xy)− hε(xy)|+ |hε(x)| · |hε(y)− f(y)|+ |hε(x)− f(x)| · |f(y)|≤ ε (1 + ‖x‖+ |f(y)|).

This holds for every ε > 0, so that actually

f(xy) = f(x)f(y);

we have proven that f is a homomorphism, f ∈ K. �Exercise D.4.8 (Radicals). Let A be a commutative Banach algebra. Its radicalRad(A) is the intersection of all the maximal ideals of A. Show that

Rad(A) = Ker(x �→ x) = {x ∈ A | ρ(x) = 0},

where x �→ x is the Gelfand transform. Show that nilpotent elements of A belongto the radical.

Exercise D.4.9. Let X be a finite set. Describe the Gelfand transform of F(X).

Exercise D.4.10. Describe the Gelfand transform of the algebra of matrices(

α β0 α

),

where α, β ∈ C.

Theorem D.4.11 (When is Spec(C(X)) homeomorphic to X?). Let X be a compactHausdorff space. Then Spec(C(X)) is homeomorphic to X.

Proof. For x ∈ X, let us define the function

hx : C(X)→ C, f �→ f(x) (evaluation at x ∈ X).

This is clearly a homomorphism, and hence we may define the mapping

φ : X → Spec(C(X)), x �→ hx.

Let us prove that φ is a homeomorphism.If x, y ∈ X, x = y, then Urysohn’s Lemma (Theorem A.12.11) provides

f ∈ C(X) such that f(x) = f(y). Thereby hx(f) = hy(f), yielding φ(x) = hx =hy = φ(y); thus φ is injective. It is also surjective: namely, let us assume thath ∈ Spec(C(X)) \ φ(X). Now Ker(h) ⊂ C(X) is a maximal ideal, and for everyx ∈ X we may choose

fx ∈ Ker(h) \Ker(hx) ⊂ C(X).

Then Ux := f−1x (C \ {0}) ∈ V(x), so that

U = {Ux | x ∈ X}


is an open cover of X, which due to the compactness has a finite subcover{Uxj

}nj=1 ⊂ U . Since fxj

∈ Ker(h), the function

f :=n∑

j=1

|fxj|2 =

n∑j=1

fxjfxj

belongs to Ker(h). Clearly f(x) = 0 for every x ∈ X. Therefore g ∈ C(X) withg(x) = 1/f(x) is the inverse element of f ; this is a contradiction, since no invertibleelement belongs to an ideal. Thus φ must be surjective.

We have proven that φ : X → Spec(C(X)) is a bijection. Thereby X andSpec(C(X)) can be identified as sets. The Gelfand-topology of Spec(C(X)) isthen identified with the C(X)-induced topology σ of X, which is weaker than theoriginal topology τ of X. Hence φ : (X, τ)→ Spec(C(X)) is continuous. Actually,σ = τ , because a continuous bijection from a compact space to a Hausdorff spaceis a homeomorphism, see Proposition A.12.7. �

Corollary D.4.12. Let X and Y be compact Hausdorff spaces. Then the Banachalgebras C(X) and C(Y ) are isomorphic if and only if X is homeomorphic to Y .

Proof. By Theorem D.4.11, X ∼= Spec(C(X)) and Y ∼= Spec(C(Y )). If C(X) andC(Y ) are isomorphic Banach algebras then

X ∼= Spec(C(X))C(X)∼=C(Y )∼= Spec(C(Y )) ∼= Y.

Conversely, a homeomorphism φ : X → Y begets a Banach algebra isomor-phism

Φ : C(Y )→ C(X), (Φf)(x) := f(φ(x)),

as the reader easily verifies. �

Exercise D.4.13. Let K be a compact Hausdorff space, ∅ = S ⊂ K, and J ⊂ C(K)be an ideal. Let us define

I(S) := {f ∈ C(K) | ∀x ∈ S : f(x) = 0},V (J ) := {x ∈ K | ∀f ∈ J : f(x) = 0}.

Prove that

(a) I(S) ⊂ C(K) a closed ideal,(b) V (J ) ⊂ K is a closed non-empty subset,(c) V (I(S)) = S (hint: Urysohn), and(d) I(V (J )) = J .

Lesson to be learned: topology of K goes hand in hand with the (closed) idealstructure of C(K).

D.5. C∗-algebras 213

D.5 C∗-algebras

Now we are finally in the position to abstractly characterise algebras C(X) amongBanach algebras: according to Gelfand and Naimark, the category of compactHausdorff spaces is equivalent to the category of commutative C∗-algebras. Theclass of C∗-algebras behaves nicely, and the related functional analysis adequatelydeserves the name “non-commutative topology”.

Definition D.5.1 (Involutive algebra). An algebra A is a ∗-algebra (“star-algebra”or an involutive algebra) if there is a mapping (x �→ x∗) : A → A satisfying

(λx)∗ = λx∗, (x + y)∗ = x∗ + y∗, (xy)∗ = y∗x∗, (x∗)∗ = x,

for all x, y ∈ A and λ ∈ C; such a mapping is called an involution. In otherwords, an involution is a conjugate-linear anti-multiplicative self-invertible map-ping A → A.

A ∗-homomorphism φ : A → B between involutive algebras A and B is analgebra homomorphism satisfying

φ(x∗) = φ(x)∗

for every x ∈ A. The set of all ∗-homomorphisms between ∗-algebras A and B isdenoted by Hom∗(A,B).

Definition D.5.2 (C∗-algebra). A C∗-algebra A is an involutive Banach algebrasuch that

‖x∗x‖ = ‖x‖2

for every x ∈ A.

Example. Let us consider some involutive algebras:

1. The Banach algebra C is a C∗-algebra with the involution λ �→ λ∗ = λ, i.e.,the complex conjugation.

2. If K is a compact space then C(K) is a commutative C∗-algebra with theinvolution f �→ f∗ by complex conjugation, f∗(x) := f(x).

3. L∞([0, 1]) is a C∗-algebra, when the involution is as above.

4. A(D(0, 1)) = C(D(0, 1)

)∩H(D(0, 1)) is an involutive Banach algebra with

f∗(z) := f(z), but it is not a C∗-algebra. Here H(D(0, 1)) are functionsholomorphic in the unit disc.

5. The radical of a commutative C∗-algebra is always the trivial ideal {0}, andthus 0 is the only nilpotent element. Hence for instance the algebra of matrices(

α β0 α

)(where α, β ∈ C) cannot be a C∗-algebra.


6. If H is a Hilbert space then L(H) is a C∗-algebra when the involution is theusual adjunction A �→ A∗, and clearly any norm-closed involutive subalgebraof L(H) is also a C∗-algebra. Actually, there are no others, but in the sequelwe shall not prove the related Gelfand–Naimark Theorem D.5.3:

Theorem D.5.3 (Gelfand–Naimark Theorem (1943)). If A is a C∗-algebra thenthere exists a Hilbert space H and an isometric ∗-homomorphism onto a closedinvolutive subalgebra of L(H).

However, we shall characterise the commutative case: the Gelfand transformof a commutative C∗-algebra A will turn out to be an isometric isomorphismA → C(Spec(A)), so that A “is” the function algebra C(K) for the compactHausdorff space K = Spec(A)! Before going into this, we prove some relatedresults.

Proposition D.5.4. Let A be a ∗-algebra. Then 1∗ = 1, x ∈ A is invertible if andonly if x∗ ∈ A is invertible, and σ(x∗) = σ(x) := {λ | λ ∈ σ(x)}.

Proof. First,1∗ = 1∗1 = 1∗(1∗)∗ = (1∗1)∗ = (1∗)∗ = 1;

second,

(x−1)∗x∗ = (xx−1)∗ = 1∗ = 1 = 1∗ = (x−1x)∗ = x∗(x−1)∗;

third,λ1− x∗ = (λ1∗)∗ − x∗ = (λ1)∗ − x∗ = (λ1− x)∗,

which concludes the proof. �Proposition D.5.5. Let A be a C∗-algebra, and x = x∗ ∈ A. Then σ(x) ⊂ R.

Proof. Assume that λ ∈ σ(x) \ R, i.e., λ = λ1 + iλ2 for some λj ∈ R with λ2 = 0.Hence we may define y := (x − λ11)/λ2 ∈ A. Now y∗ = y. Moreover, i ∈ σ(y),because

i1− y =λ1− x

λ2.

Take t ∈ R. Then t + 1 ∈ σ(t1− iy), because

(t + 1)1− (t1− iy) = −i(i1− y).

Thereby

(t + 1)2 ≤ ρ(t1− iy)2

≤ ‖t1− iy‖2C∗= ‖(t1− iy)∗(t1− iy)‖

t∈R, y∗=y= ‖(t1 + iy)(t1− iy)‖ = ‖t21 + y2‖≤ t2 + ‖y‖,

so that 2t + 1 ≤ ‖y‖ for every t ∈ R; a contradiction. �

D.5. C∗-algebras 215

Corollary D.5.6. Let A be a C∗-algebra, φ : A → C a homomorphism, and x ∈ A.Then φ(x∗) = φ(x), i.e., φ is a ∗-homomorphism.

Proof. Define the “real part” and the “imaginary part” of x by

u :=x + x∗

2, v :=

x− x∗

2i.

Then x = u + iv, u∗ = u, v∗ = v, and x∗ = u − iv. Since a homomorphism mapsinvertibles to invertibles, we have φ(u) ∈ σ(u); we know that σ(u) ⊂ R, becauseu∗ = u. Similarly we obtain φ(v) ∈ R. Thereby

φ(x∗) = φ(u− iv) = φ(u)− iφ(v) = φ(u) + iφ(v) = φ(u + iv) = φ(x);

this means that Hom∗(A, C) = Hom(A, C). �Exercise D.5.7. Let A be a Banach algebra, B a closed subalgebra, and x ∈ B.Prove the following facts:

(a) G(B) is open and closed in G(A) ∩ B.(b) σA(x) ⊂ σB(x) and ∂σB(x) ⊂ ∂σA(x).(c) If C \ σA(x) is connected then σA(x) = σB(x).

Using the results of the exercise above, the reader can prove the following impor-tant fact on the invariance of the spectrum in C∗-algebras:

Exercise D.5.8. Let A be a C∗-algebra and B a C∗-subalgebra. Show that σB(x) =σA(x) for every x ∈ B.

Lemma D.5.9. Let A be a C∗-algebra. Then ‖x‖2 = ρ(x∗x) for every x ∈ A.

Proof. Now

‖(x∗x)2‖ = ‖(x∗x)(x∗x)‖ = ‖(x∗x)∗(x∗x)‖ C∗= ‖x∗x‖2,

so that by induction‖(x∗x)2

n‖ = ‖x∗x‖2n

for every n ∈ N. Therefore applying the Spectral Radius Formula, we get

ρ(x∗x) = limn→∞ ‖(x

∗x)2n‖1/2n

= limn→∞ ‖x

∗x‖2n/2n

= ‖x∗x‖,

the result we wanted. �Exercise D.5.10. Let A be a C∗-algebra. Show that there can be at most one C∗-algebra norm on an involutive Banach algebra. Moreover, prove that if A,B areC∗-algebras then φ ∈ Hom∗(A,B) is continuous and has norm ‖φ‖ = 1.

Theorem D.5.11 (Commutative Gelfand–Naimark). Let A be a commutative C∗-algebra. Then the Gelfand transform (x �→ x) : A → C(Spec(A)) is an isometric∗-isomorphism.


Proof. Let K = Spec(A). We already know that the Gelfand transform is a Banachalgebra homomorphism A → C(K). Let x ∈ A and φ ∈ K. Since φ is actually a∗-homomorphism, we get

x∗(φ) = φ(x∗) = φ(x) = x(φ) = x∗(φ);

the Gelfand transform is a ∗-homomorphism.Now we have proven that A ⊂ C(K) is an involutive subalgebra separating

the points of K. Stone–Weierstrass Theorem A.14.4 thus says that A is dense inC(K). If we can show that the Gelfand transform A → A is an isometry then wemust have A = C(K): Take x ∈ A. Then

‖x‖2 = ‖x∗x‖ = ‖x∗x‖ Gelfand= ρ(x∗x) Lemma= ‖x‖2,

i.e., ‖x‖ = ‖x‖. �Exercise D.5.12. Show that an injective ∗-homomorphism between C∗-algebras isan isometry. (Hint: Gelfand transform.)

Exercise D.5.13. A linear functional f on a C∗-algebra A is called positive iff(x∗x) ≥ 0 for every x ∈ A. Show that the positive functionals separate thepoints of A.

Exercise D.5.14. Prove that the involution of a C∗-algebra cannot be altered with-out destroying the C∗-property ‖x∗x‖ = ‖x‖2.Definition D.5.15 (Normal element). An element x of a C∗-algebra is called normalif x∗x = xx∗.

We use the commutative Gelfand–Naimark Theorem to create the so-calledcontinuous functional calculus at a normal element – a non-commutative C∗-algebra admits some commutative studies:

Theorem D.5.16 (Functional calculus at the normal element). Let A be a C∗-algebra, and x ∈ A be a normal element. Let ι = (λ �→ λ) : σ(x)→ C. Then thereexists a unique isometric ∗-homomorphism φ : C(σ(x)) → A such that φ(ι) = xand φ(C(σ(x))) is the C∗-algebra generated by x, i.e., the smallest C∗-algebracontaining {x}.

Proof. Let B be the C∗-algebra generated by x. Since x is normal, B is commu-tative. Let Gel = (y �→ y) : B → C(Spec(B)) be the Gelfand transform of B. Thereader may easily verify that

x : Spec(B)→ σ(x)

is a continuous bijection from a compact space to a Hausdorff space; hence it is ahomeomorphism. Let us define the mapping

Cx : C(σ(x))→ C(Spec(B)), (Cxf)(h) := f(x(h)) = f(h(x));

D.6. Appendix: Liouville’s Theorem 217

Cx can be thought as a “transpose” of x. Let us define

φ = Gel−1 ◦ Cx : C(σ(x))→ B ⊂ A.

Then φ : C(σ(x))→ A is obviously an isometric ∗-homomorphism. Furthermore,

φ(ι) = Gel−1(Cx(ι)) = Gel−1(x) = Gel−1(Gel(x)) = x.

Due to the Stone–Weierstrass Theorem A.14.4, the ∗-algebra generated by ι ∈C(σ(x)) is dense in C(σ(x)); since the ∗-homomorphism φ maps the generator ιto the generator x, the uniqueness of φ follows �Remark D.5.17. The ∗-homomorphism φ : C(σ(x))→ A above is called the (con-tinuous) functional calculus at the normal element φ(ι) = x ∈ A. If p = (z �→∑n

j=1 ajzj) : C→ C is a polynomial then it is natural to define p(x) :=

∑nj=1 ajx

j .Then actually

p(x) = φ(p);

hence it is natural to define f(x) := φ(f) for every f ∈ C(σ(x)). It is easy to checkthat if f ∈ C(σ(x)) and h ∈ Spec(B) then f(h(x)) = h(f(x)).

Exercise D.5.18. Let A be a C∗-algebra, x ∈ A normal, f ∈ C(σ(x)), and g ∈C(f(σ(x))). Show that σ(f(x)) = f(σ(x)) and that (g ◦ f)(x) = g(f(x)).

D.6 Appendix: Liouville’s Theorem

Here we prove Liouville’s Theorem D.6.2 from complex analysis which was usedin the proof of Gelfand’s Theorem D.3.10.

Definition D.6.1 (Holomorphic function). Let Ω ⊂ C be open. A function f : Ω→C is called holomorphic in Ω, denoted by f ∈ H(Ω), if the limit

f ′(z) := limh→0

f(z + h)− f(z)h

exists for every z ∈ Ω. Then Cauchy’s integral formula provides a power seriesrepresentation

f(z) =∞∑

n=0

cn(z − a)n

converging uniformly on the compact subsets of the disk

D(a, r) = {z ∈ C : |z − a| < r} ⊂ Ω;

here cn = f (n)(a)/n!, where f (0) = f and f (n+1) = f (n)′.

Theorem D.6.2 (Liouville’s Theorem). Let f ∈ H(C) such that |f | is bounded.Then f is constant, i.e., f(z) ≡ f(0) for every z ∈ C.


Proof. Since f ∈ H(C), we have a power series representation

f(z) =∞∑

n=0

cnzn

converging uniformly on the compact sets in the complex plane. Thereby

12π

∫ 2π

0

|f(reiφ)|2 dφ =12π

∫ 2π

0

∑n,m

cn cm rn+m ei(n−m)φ dφ

=∑n,m

cn cm rn+m 12π

∫ 2π

0

ei(n−m)φ dφ

=∞∑

n=0

|cn|2r2n

for every r > 0. Hence the fact

∞∑n=0

|cn|2r2n =12π

∫ 2π

0

|f(reiφ)|2 dφ ≤ supz∈C

|f(z)|2 <∞

implies cn = 0 for every n ≥ 1; thus f(z) ≡ c0 = f(0). �

A more general Liouville’s theorem for harmonic functions will be given inTheorem 2.6.14, with a proof relying on the Fourier analysis instead.

Part II

Commutative Symmetries

In Part II we present the theory of pseudo-differential operators on commutativegroups.

The first commutative case is the Euclidean space Rn where the theory ofpseudo-differential operators is developed most and many things may be consid-ered well-known, so here we only review basics which are useful to contrast itwith constructions on other spaces. We start by introducing elements of Fourieranalysis in Chapter 1, trying to make an independent exposition of the theory,reducing references to general measure theory to a minimum. In Chapter 2 we de-velop the most important elements of the theory of pseudo-differential operatorson Rn. There we do not aim at developing a comprehensive treatment since thereare several excellent monographs already available. Instead, we focus in Chapter4 on aspects of the theory that have analogues on the torus, and on more general(compact) Lie groups in Part IV. From this point of view Chapters 1 and 2 can beregarded as an introduction to the subject and that is why we have taken specialcare to accommodate a possibly less experienced reader there.

The second commutative case is the case of the torus Tn = Rn/Zn consideredin Chapter 4. On one hand, pseudo-differential operators on Tn can be viewedas a special case of periodic pseudo-differential operators on Rn, with all theconsequences. However, in this way one may lose many important features of theunderlying torus. On the other hand, carrying out the analysis in the intrinsiclanguage of the underlying space is usually a more natural point of view that alsohas chances of extension to other Lie groups that are not so intimately related tothe Euclidean space.

Here the literature on the general theory of periodic pseudo-differential oper-ators in the “toroidal language” is rather non-existent and only a few results seemto be available. This fact is quite surprising given that Fourier analysis on Tn isnothing else but the periodic Fourier transform on Rn and, as such, constitutes astarting point of applications of Fourier analysis to numerous problems in applied

220

mathematics and engineering. In particular, such applications (and especially reallife or computational applications) do often rely on the toroidal language of theFourier coefficients and the Fourier series rather than on the Euclidean languageof the Fourier transform.

Since every connected commutative Lie group G can be identified with theproduct G ∼= Tn ×Rm, the combination of these two settings essentially exhaustsall compact commutative Lie groups. Indeed, every compact (disconnected) com-mutative group is isomorphic to the product of a torus and a finite commutativegroup, so that being connected is not really a restriction and thus it is sufficientto study the case of the torus again.

In Chapter 5 we discuss commutator characterisations of pseudo-differentialoperators on Rn and Tn, as well as on closed manifolds which becomes usefulin the sequel. In particular, Section 5.2 contains a short introduction to pseudo-differential operators on manifolds.

Chapter 1

Fourier Analysis on Rn

In this chapter we review basic elements of Fourier analysis on Rn. Consequently,we introduce spaces of distributions, putting emphasis on the space of tempereddistributions S ′(Rn). Finally, we discuss Sobolev spaces and approximation offunctions and distributions by smooth functions. Throughout, we fix the measureon Rn to be Lebesgue measure. For convenience, we may repeat a few definitionsin the context of Rn although they may have already appeared in Chapter Con measure theory. From this point of view, the present chapter can be readessentially independently. The notation used in this chapter and also in Chapter2 is 〈ξ〉 = (1 + |ξ|2)1/2 where |ξ| = (ξ2

1 + · · ·+ ξ2n)1/2, ξ ∈ Rn.

1.1 Basic properties of the Fourier transform

Let Ω ⊂ Rn be a measurable subset of Rn. For simplicity, we may always think ofΩ being open or closed in Rn. In this section we will mostly have Ω = Rn.

Definition 1.1.1 (Lp-spaces). Let 1 ≤ p < ∞. A function f : Ω → C is said to bein Lp(Ω) if it is measurable and its norm

||f ||Lp(Ω) :=(∫

Ω

|f(x)|p dx

)1/p

is finite. In the case p = ∞, f is said to be in L∞(Ω) if it is measurable andessentially bounded, i.e., if

||f ||L∞(Ω) := esssupx∈Ω |f(x)| <∞.

Here esssupx∈Ω |f(x)| is defined as the smallest M such that |f(x)| ≤ M foralmost all x ∈ Ω.

In particular, L1(Ω) is the space of absolutely integrable functions on Ω with||f ||L1(Ω) =

∫Ω|f(x)| dx. We will often abbreviate the ||f ||Lp(Ω) norm by ||f ||Lp ,

or by ||f ||p, if the choice of Ω is clear from the context.

222 Chapter 1. Fourier Analysis on Rn

We note that it is customary to abuse the notation slightly by talking aboutfunctions in Lp(Rn) while in reality elements in Lp(Rn) are equivalence classesof functions that are equal almost everywhere. However, this is a minor technicalissue, see Definition C.4.6 for details.

Definition 1.1.2 (Fourier transform in L1(Rn)). For f ∈ L1(Rn) we define itsFourier transform by

(FRnf)(ξ) = (Ff)(ξ) = f(ξ) :=∫

Rn

e−2πix·ξf(x) dx.

Remark 1.1.3. Other similar definitions are often encountered in the literature.For example, one can use e−ix·ξ instead of e−2πix·ξ, multiply the integral by theconstant (2π)−n/2, etc. Changes in definitions may lead to changes in constantsin formulae. It may also seem that our notation for the Fourier transform is a bitexcessive. However, f is a useful shorthand notation, while FRnf is useful in thesequel when we want to explicitly distinguish it from the Fourier transform FTnffor functions on the torus Tn considered in Chapters 3 and 4. However, in thischapter as well as in Chapter 2 we may omit the subscript and write simply Fsince there should be no confusion.

It is easy to check that F : L1(Rn) → L∞(Rn) is a bounded linear operatorwith norm one:

||f ||∞ ≤ ||f ||1. (1.1)

Moreover, if f ∈ L1(Rn), its Fourier transform f is continuous, which followsfrom Lebesgue’s dominated convergence theorem. For Lebesgue’s dominated con-vergence theorem on general measure spaces we refer to Theorem C.3.22, but forcompleteness, we also state it here in a form useful to us:

Theorem 1.1.4 (Lebesgue’s dominated convergence theorem). Let (fk)∞k=1 be asequence of measurable functions on Ω such that fk → f pointwise almost every-where on Ω as k → ∞. Suppose there is an integrable function g ∈ L1(Ω) suchthat |fk| ≤ g for all k. Then f is integrable and

∫Ω

f dx = limk→∞∫Ω

fk dx.

Exercise 1.1.5. Prove that if f ∈ L1(Rn) then f is continuous everywhere.

Exercise 1.1.6. Let u, f ∈ L1(Rn) satisfy Lu = f , where L = ∂2

∂x21

+ · · · + ∂2

∂x2n

isthe Laplace operator. Prove that

∫Rn f(x) dx = 0.

Exercise 1.1.7. Let u, f ∈ L1(Rn) satisfy (1− L)u = f. Suppose that f satisfies

|f(ξ)| ≤ C

(1 + |ξ|)n−1, for all ξ ∈ Rn.

Prove that u is a bounded continuous function on Rn.

It is quite difficult to characterise the image of the space L1(Rn) under theFourier transform. But we have the following theorem:

1.1. Basic properties of the Fourier transform 223

Theorem 1.1.8 (Riemann–Lebesgue lemma). Let f ∈ L1(Rn). Then its Fouriertransform f is a continuous function on Rn vanishing at infinity, i.e., f(ξ) → 0as ξ →∞.

Proof. It is enough to make an explicit calculation for f being a characteristicfunction of a cube and then use a standard limiting argument. Thus, let f be acharacteristic function of the unit cube, i.e., f(x) = 1 for x ∈ [−1, 1]n and f(x) = 0otherwise. Then

f(ξ) =∫

[−1,1]ne−2πix·ξ dx

=n∏

j=1

∫ 1

−1

e−2πixjξj dxj

=n∏

j=1

1−2πiξj

e−2πixjξj |1−1

=(

i2π

)n 1ξ1 · · · ξn

n∏j=1

[e−2πiξj − e2πiξj

]=

n∏j=1

sin(2πξj)πξj

.

The product of exponents is bounded, so the whole expression tends to zero asξ →∞ away from the coordinate axis. In case some of the ξj ’s are zero, an obviousmodification of this argument yields the same result. �Exercise 1.1.9. Complete the proof of Theorem 1.1.8 in the case when some of theξj ’s are zero.

Definition 1.1.10 (Multi-index notation). When working in Rn, the following no-tation is extremely useful. For multi-indices α = (α1, . . . , αn) and β = (β1, . . . , βn)with integer entries αj , βj ≥ 0, we define

∂αϕ(x) =∂α1

∂xα11

· · · ∂αn

∂xαnn

ϕ(x)

and xβ = xβ11 · · ·xβn

n . For such multi-indices we will write α, β ≥ 0. For multi-indices α and β, α ≤ β means αj ≤ βj for all j ∈ {1, . . . , n}. The length of themulti-index α will be denoted by |α| = α1 + · · ·+ αn, and α! = α1! · · ·αn!.

Space L1(Rn) has its limitations for the Fourier analysis because its elementsmay be quite irregular. The following space is an excellent alternative because itselements are smooth and have strong decay properties, thus allowing us not toworry about the convergence of integrals as well as allowing the use of analytictechniques such as integration by parts, etc.


Definition 1.1.11 (Schwartz space S(Rn)). We define the Schwartz space S(Rn) ofrapidly decreasing functions as follows. We say that ϕ ∈ S(Rn) if ϕ is smooth onRn and if

supx∈Rn

∣∣xβ∂αϕ(x)∣∣ <∞

for all multi-indices α, β ≥ 0.

Exercise 1.1.12. Show that a smooth function f is in the Schwartz space if andonly if for all α ≥ 0 and N ≥ 0 there is a constant Cα,N such that |∂αϕ(x)| ≤Cα,N (1 + |x|)−N for all x ∈ Rn.

The space S(Rn) is a topological space. Let us now introduce the convergenceof functions in S(Rn).

Definition 1.1.13 (Convergence in S(Rn)). We will say that ϕj → ϕ in S(Rn) asj →∞, if ϕj , ϕ ∈ S(Rn) and if

supx∈Rn

|xβ∂α(ϕj − ϕ)(x)| → 0 as j →∞, (1.2)

for all multi-indices α, β ≥ 0.

Remark 1.1.14. The Schwartz space S(Rn) contains C∞-smooth functions on Rn

that decay rapidly at infinity, i.e.,

S(Rn) :={

f ∈ C∞(Rn) : pαβ(ϕ) := supx∈Rn

∣∣xβ∂αϕ(x)∣∣ <∞ (α, β ∈ Nn

0 )}

.

If one is familiar with functional analysis, one can take the expressions pαβ(ϕ) asseminorms on the space S(Rn), see Definition B.4.1. This collection turns S(Rn)into a locally convex linear topological space. Moreover, it is also a Frechet spacewith the natural topology induced by the seminorms pαβ (see Exercise B.3.9), andit is a nuclear Montel space (see Exercises B.3.37 and B.3.51).

Definition 1.1.15 (Useful notation Dα). Since the definition of the Fourier trans-form contains the complex exponential, it is often convenient to use the notationDj = 1

2πi∂

∂xjand Dα = Dα1

1 · · ·Dαnn . If Dj is applied to a function of ξ it will ob-

viously mean 12πi

∂∂ξj

. However, there should be no confusion with this convention.If we want to additionally emphasize the variable for differentiation, we will writeDα

x or Dαξ .

The following theorem relates multiplication with differentiation, with re-spect to the Fourier transform.

Theorem 1.1.16. Let ϕ ∈ S(Rn). Then Djϕ(ξ) = ξjϕ(ξ) and xjϕ(ξ) = −Djϕ(ξ).

Proof. From the definition of the Fourier transform we readily see that

Djϕ(ξ) =∫

Rn

e−2πix·ξ(−xj)ϕ(x) dx.


This gives the second formula. Since the integrals converge absolutely, we canintegrate by parts with respect to xj in the following expression to get

ξjϕ(ξ) =∫

Rn

(−Dj e−2πix·ξ)ϕ(x) dx =

∫Rn

e−2πix·ξDjϕ(x) dx.

This implies the first formula. Note that we do not get boundary terms whenintegrating by parts because function ϕ vanishes at infinity. �Remark 1.1.17. This theorem allows one to tackle some differential equationsalready. For example, let us look at the equation Lu = f with the Laplace operatorL = ∂2

∂x21+· · ·+ ∂2

∂x2n. Taking the Fourier transform and using the theorem we arrive

at the equation −4π2|ξ|2u = f . If we knew how to invert the Fourier transformwe could find the solution u = −F−1

(1

4π2|ξ|2 f).

Corollary 1.1.18. Let ϕ ∈ S(Rn). Then

ξβDαξ ϕ(ξ) =

∫Rn

e−2πix·ξDβx((−x)αϕ(x)) dx.

Hence alsosupξ∈Rn

∣∣ξβDαξ ϕ(ξ)

∣∣ ≤ C supx∈Rn

((1 + |x|)n+1

∣∣Dβx(xαϕ(x))

∣∣),with C =

∫Rn(1 + |x|)−n−1 dx <∞.

Here we used the following useful criterion:

Exercise 1.1.19 (Integrability criterion). Show that we have∫

Rndx

(1+|x|)ρ < ∞ ifand only if ρ > n. We also have

∫|x|≤1

dx|x|ρ <∞ if and only if ρ < n. Both of these

criteria can be easily checked by passing to polar coordinates.

Remark 1.1.20 (Fourier transform in S(Rn)). Corollary 1.1.18 implies that theFourier transform F maps S(Rn) to itself. In fact, later we will show that muchmore is true. Let us note for now that Corollary 1.1.18 together with Lebesgue’sdominated convergence theorem imply that the Fourier transform F : S(Rn) →S(Rn) is continuous, i.e., ϕj → ϕ in S(Rn) implies ϕj → ϕ in S(Rn).

Theorem 1.1.21 (Fourier inversion formula). The Fourier transform F : ϕ �→ ϕ isan isomorphism of S(Rn) into S(Rn), whose inverse is given by

ϕ(x) =∫

Rn

e2πix·ξ ϕ(ξ) dξ. (1.3)

This formula is called the Fourier inversion formula and the inverse Fourier trans-form is denoted by

(F−1Rn f)(x) ≡ (F−1f)(x) :=

∫Rn

e2πix·ξ f(ξ) dξ.

Thus, we can say that

F ◦ F−1 = F−1 ◦ F = identity on S(Rn).


The proof of this theorem will rely on several lemmas which have a signifi-cance on their own.

Lemma 1.1.22 (Multiplication formula for the Fourier transform). Let f, g ∈L1(Rn). Then

∫Rn fg dx =

∫Rn fg dx.

Proof. We will apply Fubini’s theorem. Thus,∫Rn

fg dx =∫

Rn

[∫Rn

e−2πix·yf(y) dy

]g(x) dx

=∫

Rn

[∫Rn

e−2πix·yg(x) dx

]f(y) dy

=∫

Rn

gf dy,

proving the lemma. �Lemma 1.1.23 (Fourier transform of Gaussian distributions). We have the equality∫

Rn

e−2πix·ξ e−επ2|x|2 dx = (πε)−n/2 e−|ξ|2/ε,

for every ε > 0. By the change of 2πx→ x and ε→ 2ε it is equivalent to∫Rn

e−ix·ξ e−ε|x|2/2 dx =(

2π

ε

)n/2

e−|ξ|2/(2ε). (1.4)

Proof. We will use the standard identities∫ ∞

−∞e−t2/2 dt =

√2π and

∫Rn

e−|x|2/2 dx = (2π)n/2. (1.5)

In fact, (1.4) will follow from the one-dimensional case, when we have∫ ∞

−∞e−itτ e−t2/2 dt = e−τ2/2

∫ ∞

−∞e−(t+iτ)2/2 dt

= e−τ2/2

∫ ∞

−∞e−t2/2 dt

=√

2π e−τ2/2,

where we used the Cauchy theorem about changing the contour of integration foranalytic functions and formula (1.5). Changing t→ √

εt and τ → τ/√

ε gives

√ε

∫ ∞

−∞e−itτ e−εt2/2 dt =

√2π e−τ2/(2ε).

Extending this to n dimensions yields (1.4). �


Proof of Theorem 1.1.21. For ϕ ∈ S(Rn), we want to prove (1.3), i.e., that

ϕ(x) =∫

Rn

e2πix·ξ ϕ(ξ) dξ.

By the Lebesgue dominated convergence theorem (Theorem 1.1.4) we can replacethe right-hand side of this formula by∫

Rn

e2πix·ξϕ(ξ) dξ

= limε→0

∫Rn

e2πix·ξϕ(ξ) e−2επ2|ξ|2 dξ

= limε→0

∫Rn

∫Rn

e2πi(x−y)·ξϕ(y) e−2επ2|ξ|2 dy dξ (change y → y + x)

= limε→0

∫Rn

∫Rn

e−2πiy·ξϕ(y + x) e−2επ2|ξ|2 dy dξ (Fubini’s theorem)

= limε→0

∫Rn

ϕ(y + x) dy

(∫Rn

e−2πiy·ξ e−2επ2|ξ|2 dξ

)(F.T. of Gaussian)

= limε→0

∫Rn

ϕ(y + x)(2πε)−n/2 e−|y|2/(2ε) dy (change y =

√εz)

= (2π)−n/2 limε→0

∫Rn

ϕ(√

εz + x) e−|z|2/2 dz (use (1.5))

= ϕ(x).

This finishes the proof. �Remark 1.1.24. In fact, in the proof we implicitly established the following usefulrelation between Fourier transforms and translations of functions. Let h ∈ Rn anddefine (τhf)(x) = f(x− h). Then we also see that

(τhf)(ξ) =∫

Rn

e−2πix·ξ(τhf)(x) dx

=∫

Rn

e−2πix·ξf(x− h) dx (change y = x− h)

=∫

Rn

e−2πi(y+h)·ξf(y) dy

= e−2πih·ξ f(ξ).

Exercise 1.1.25 (Fourier transform and linear transformations). Let f ∈ L1(Rn).Show that if A ∈ Rn×n satisfies det A = 0, and B = (AT )−1, then

f ◦A = |det A|−1f ◦B.

In particular, conclude that the Fourier transform commutes with rotations: if Ais orthogonal (i.e., AT = A∗ = A−1, so that A defines a rotation of Rn), then


f ◦A = f ◦A. Consequently, conclude also that the Fourier transform of a radialfunction is radial: if f(x) = h1(|x|) for some h1, then f(ξ) = h2(|ξ|) for some h2.

Definition 1.1.26 (Convolutions). For functions f, g ∈ L1(Rn), we define theirconvolution by

(f ∗ g)(x) :=∫

Rn

f(x− y) g(y) dy. (1.6)

It is easy to see that f ∗ g ∈ L1(Rn) with norm inequality

||f ∗ g||L1(Rn) ≤ ||f ||L1(Rn)||g||L1(Rn) (1.7)

and that f ∗ g = g ∗ f. Also, in particular for f, g ∈ S(Rn), integrals are absolutelyconvergent and we can differentiate under the integral sign to get

∂α(f ∗ g) = ∂αf ∗ g = f ∗ ∂αg. (1.8)

Remark 1.1.27. We can note that a more rigorous way of defining the convolutionwould be first defining (1.6) for f, g ∈ S(Rn) and then extending it to a mapping∗ : L1(Rn)× L1(Rn)→ L1(Rn) by (1.7) avoiding the convergence question of theintegral in (1.6) for functions from L1(Rn).

Exercise 1.1.28. Prove the commutativity of convolution: if f, g ∈ L1(Rn), thenf ∗ g = g ∗ f . If f, g,∈ S(Rn), prove formula (1.8).

Exercise 1.1.29. Prove the associativity of convolution: if f, g, h ∈ L1(Rn), provethat

(f ∗ g) ∗ h = f ∗ (g ∗ h).

The following properties relate convolutions with Fourier transforms.

Theorem 1.1.30. Let ϕ, ψ ∈ S(Rn). Then we have

(i)∫

Rn ϕ ψ dx =∫

Rn ϕ ψ dξ;

(ii) ϕ ∗ ψ(ξ) = ϕ(ξ)ψ(ξ);

(iii) ϕ ψ(ξ) = (ϕ ∗ ψ)(ξ).

Proof. (i) Let us denote

χ(ξ) = ψ(ξ) =∫

Rn

e2πix·ξψ(x) dx = F−1(ψ)(ξ),

so that χ = ψ. It follows now that∫Rn

ϕψ =∫

Rn

ϕχ =∫

Rn

ϕχ =∫

Rn

ϕψ,

where we used the multiplication formula for the Fourier transform in Lemma1.1.22.

1.2. Useful inequalities 229

(ii) We can easily calculate

ϕ ∗ ψ(ξ) =∫

Rn

e−2πix·ξ(ϕ ∗ ψ)(x) dx =∫

Rn

∫Rn

e−2πix·ξϕ(x− y)ψ(y) dy dx

=∫

Rn

∫Rn

e−2πi(x−y)·ξϕ(x− y) e−2πiy·ξψ(y) dy dx

=∫

Rn

∫Rn

e−2πiz·ξϕ(z) e−2πiy·ξψ(y) dy dz = ϕ(ξ)ψ(ξ),

where we used the substitution z = x− y. We leave (iii) as Exercise 1.1.31. �Exercise 1.1.31. Prove part (iii) of Theorem 1.1.30.

1.2 Useful inequalities

This section will be devoted to several important inequalities which are very usefulin Fourier analysis and in many types of analysis involving spaces of functions.

Proposition 1.2.1 (Cauchy’s inequality). For all a, b ∈ R we have ab ≤ a2

2 + b2

2 .

Moreover, for any ε > 0, we also have ab ≤ εa2 + b2

4ε . As a consequence, weimmediately obtain Cauchy’s inequality for functions:∫

Ω

|f(x)g(x)| dx ≤ 12

∫Ω

(|f(x)|2 + |g(x)|2) dx,

which is||fg||L1(Ω) ≤

12

(||f ||2L2(Ω) + ||g||2L2(Ω)

).

Proof. The first inequality follows from 0 ≤ (a− b)2 = a2 − 2ab + b2. The secondone follows if we apply the first one to ab = (

√2εa)(b/

√2ε). �

Proposition 1.2.2 (Cauchy–Schwarz inequality). Let x, y ∈ Rn. Then we have|x · y| ≤ |x||y|.

Proof. For ε > 0, we have 0 ≤ |x ± εy|2 = |x|2 ± 2εx · y + ε2|y|2. This implies±x ·y ≤ 1

2ε |x|2 + ε2 |y|2. Setting ε = |x|

|y| , we obtain the required inequality, providedy = 0 (if x = 0 or y = 0 it is trivial).

An alternative proof may be given as follows. We can observe that the in-equality 0 ≤ |x + εy|2 = |x|2 + 2εx · y + ε2|y|2 implies that the discriminant of thequadratic (in ε) polynomial on the right-hand side must be non-positive, whichmeans |x · y|2 − |x|2|y|2 ≤ 0. �Proposition 1.2.3 (Young’s inequality). Let 1 < p, q <∞ be such that 1

p + 1q = 1.

Thenab ≤ ap

p+

bq

qfor all a, b > 0.


Moreover, if ε > 0, we have ab ≤ εap + C(ε)bq for all a, b > 0, where C(ε) =(εp)−q/pq−1.

As a consequence, we immediately obtain that if f ∈ Lp(Ω) and g ∈ Lq(Ω),then fg ∈ L1(Ω) with

||fg||L1 ≤ 1p||f ||pLp +

1q||g||qLq .

Proof. To prove the first inequality, we will use the fact that the exponentialfunction x �→ ex is convex (a function f : R → R is called convex if f(τx + (1 −τ)y) ≤ τf(x) + (1− τ)f(y), for all x, y ∈ R and all 0 ≤ τ ≤ 1). This implies

ab = eln a+ln b = e1p ln ap+ 1

q ln bq ≤ 1p

eln ap

+1q

eln bq

=ap

p+

bq

q.

The second inequality with ε follows if we apply the first one to the productab =

((εp)1/pa

)(b/(εp)1/p

). �

Proposition 1.2.4 (Holder’s inequality). Let 1 ≤ p, q ≤ ∞ with 1p + 1

q = 1. Letf ∈ Lp(Ω) and g ∈ Lq(Ω). Then fg ∈ L1(Ω) and

||fg||L1(Ω) ≤ ||f ||Lp(Ω)||g||Lq(Ω).

In the formulation we use the standard convention of setting 1/∞ = 0. In the caseof p = q = 2 Holder’s inequality is often called the Cauchy–Schwarz inequality.

Holder’s inequality in the setting of general measures was given in TheoremC.4.4, but here we give a short proof also in Rn for transparency.

Proof. In the case p = 1 or p = ∞ the inequality is obvious, so let us assume1 < p < ∞. Let us first consider the case when ||f ||Lp = ||g||Lq = 1. Then byYoung’s inequality with 1 < p, q <∞, we have

||fg||L1 ≤ 1p||f ||pLp +

1q||g||qLq =

1p

+1q

= 1 = ||f ||Lp ||g||Lq ,

which is the desired inequality. Now, let us consider general f, g. We observe thatwe may assume that ||f ||Lp = 0 and ||g||Lq = 0, since otherwise one of the functionsis zero almost everywhere in Ω and Holder’s inequality becomes trivial. It followsfrom the considered case that∫

Ω

∣∣∣∣ f

||f ||pg

||g||q

∣∣∣∣ dx ≤ 1,

which implies the general case by the linearity of the integral. �

1.2. Useful inequalities 231

Proposition 1.2.5 (General Holder’s inequality). Let 1 ≤ p1, . . . , pm ≤ ∞ be suchthat 1

p1+ · · · + 1

pm= 1. Let fk ∈ Lpk(Ω) for all k = 1, . . . ,m. Then the product

f1 · · · fm ∈ L1(Ω) and

||f1 · · · fm||L1(Ω) ≤m∏

k=1

||fk||Lpk (Ω).

This inequality readily follows from Holder’s inequality by induction on thenumber of functions.

Exercise 1.2.6. Prove Proposition 1.2.5. Formulate and prove the correspondinggeneral version of Theorem C.4.4.

Proposition 1.2.7 (Minkowski’s inequality). Let 1 ≤ p ≤ ∞. Let f, g ∈ Lp(Ω).Then

||f + g||Lp(Ω) ≤ ||f ||Lp(Ω) + ||g||Lp(Ω).

In particular, this means that || · ||Lp satisfies the triangle inequality and is a norm,so Lp(Ω) is a normed space.

Minkowski’s inequality in the setting of general measures was given in The-orem C.4.5.

Proof. The cases of p = 1 or p =∞ follow from the triangle inequality for complexnumbers and are, therefore, trivial. So we may assume 1 < p <∞. Then we have

||f + g||pLp(Ω) =∫

Ω

|f + g|p dx ≤∫

Ω

|f + g|p−1(|f |+ |g|) dx

=∫

Ω

|f + g|p−1|f | dx +∫

Ω

|f + g|p−1|g| dx

(use Holder’s inequality with p = p, q =p

p− 1)

≤(∫

Ω

|f + g|p dx

) p−1p

[(∫Ω

|f |p dx

) 1p

+(∫

Ω

|g|p dx

) 1p

]= ||f + g||p−1

Lp(Ω)

(||f ||Lp(Ω) + ||g||Lp(Ω)

),

which implies the desired inequality. �

Proposition 1.2.8 (Young’s inequality for convolutions). Let 1 ≤ p ≤ ∞, f ∈L1(Rn) and g ∈ Lp(Rn). Then f ∗ g ∈ Lp(Rn) and

||f ∗ g||Lp ≤ ||f ||L1 ||g||Lp .

Proof. We will not prove it from the beginning because the proof is much shorter ifwe use Minkowski’s inequality for integrals in Theorem C.5.23 or the monotonicity


of the Lp-norm in Corollary C.5.24. Indeed, we can write

||f ∗ g||Lp = ||∫

Rn

f(y) g(· − y) dy||Lp

≤∫|f(y)| ||g(· − y)||Lp dy = ||f ||L1 ||g||Lp . �

Exercise 1.2.9. Let f ∈ L1(Rn) and g ∈ Ck(Rn) be such that ∂αg ∈ L∞(Rn) forall |α| ≤ k. Prove that f ∗ g ∈ Ck. Consequently, show that ∂α(f ∗ g) = f ∗ ∂αg atall points.

Proposition 1.2.10 (General Young’s inequality for convolutions). Let 1 ≤ p, q, r ≤∞ be such that 1

p + 1q = 1+ 1

r . Let f ∈ Lp(Rn) and g ∈ Lq(Rn). Then f∗g ∈ Lr(Rn)and

||f ∗ g||Lr ≤ ||f ||Lp ||g||Lq .

Proof. The proof follows by the Riesz–Thorin interpolation theorem C.4.18 fromProposition 1.2.8 and the estimate ||f ∗ g||L∞ ≤ ||f ||Lp ||g||Lq in the case of1p + 1

q = 1. �

Exercise 1.2.11. If 1p + 1

q = 1, f ∈ Lp(Rn) and g ∈ Lq(Rn), prove the estimate

||f ∗ g||L∞ ≤ ||f ||Lp ||g||Lq .

(Hint: Holder’s inequality.)

Remark 1.2.12. If 1p + 1

q = 1, f ∈ Lp(Rn) and g ∈ Lq(Rn), one can actually showthat f ∗ g is not only bounded, but also uniformly continuous. Consequently, if1 < p, q <∞, then f ∗ g(x)→ 0 as x→∞.

Exercise 1.2.13. Prove this remark. (Hint: for the uniform continuity use Holder’sinequality. For the second part check that the statement is obviously true forcompactly supported f and g, and then pass to the limit as supports of f and ggrow; this is possible in view of the uniform continuity.)

Proposition 1.2.14 (Interpolation for Lp-norms). Let 1 ≤ s ≤ r ≤ t ≤ ∞ be suchthat 1

r = θs + 1−θ

t for some 0 ≤ θ ≤ 1. Let f ∈ Ls(Ω)⋂

Lt(Ω). Then f ∈ Lr(Ω)and

||f ||Lr(Ω) ≤ ||f ||θLs(Ω)||f ||1−θLt(Ω).

Proof. To prove this, we use that θrs + (1−θ)r

t = 1 and so we can apply Holder’sinequality in the following way:∫

Ω

|f |r dx =∫

Ω

|f |θr|f |(1−θ)r dx

≤(∫

Ω

|f |θr· sθr dx

) θrs

(∫Ω

|f |(1−θ)r· t(1−θ)r dx

) (1−θ)rt

,

which is the desired inequality. �

1.3. Tempered distributions 233

1.3 Tempered distributions

In this section we will introduce several spaces of distributions and will extendthe Fourier transform to more general spaces of functions than S(Rn) or L1(Rn)considered in Section 1.1. The main problem with the immediate extension isthat the integral in the definition of the Fourier transform in Definition 1.1.2 mayno longer converge if we go beyond the space L1(Rn) of integrable functions. Wegive preference to tempered distributions over general distributions since our mainfocus in this chapter is Fourier analysis.

Definition 1.3.1 (Tempered distributions S ′(Rn)). We define the space of tempereddistributions S ′(Rn) as the space of all continuous linear functionals on S(Rn). Thismeans that u ∈ S ′(Rn) if it is a functional u : S(Rn)→ C such that:

1. u is linear, i.e., u(αϕ + βψ) = αu(ϕ) + βu(ψ) for all α, β ∈ C and all ϕ, ψ ∈S(Rn);

2. u is continuous, i.e., u(ϕj)→ u(ϕ) in C whenever ϕj → ϕ in S(Rn).

We can also define the convergence in the space S ′(Rn) of tempered distributions.1

Let uj , u ∈ S ′(Rn). We will say that uj → u in S ′(Rn) as j →∞ if uj(ϕ)→ u(ϕ)in C as j →∞, for all ϕ ∈ S(Rn).

Functions in S(Rn) are called the test functions for tempered distributionsin S ′(Rn). Another notation for u(ϕ) will be 〈u, ϕ〉.

Here one can also recall the definition of the convergence ϕj → ϕ in S(Rn)from (1.2), which said that ϕj → ϕ in S(Rn) as j → ∞, if ϕj , ϕ ∈ S(Rn) and ifsupx∈Rn |xβ∂α(ϕj − ϕ)(x)| → 0 as j →∞, for all multi-indices α, β ≥ 0.

1.3.1 Fourier transform of tempered distributions

Here we show that the Fourier transform can be extended from S(Rn) to S ′(Rn)by duality. We also establish Plancherel’s and Parseval’s equalities on the spaceL2(Rn).

Definition 1.3.2 (Fourier transform of tempered distributions). If u ∈ S ′(Rn), wecan define its (generalised) Fourier transform by setting

u(ϕ) := u(ϕ),

for all ϕ ∈ S(Rn).

Proposition 1.3.3 (Fourier transform on S ′(Rn)). The Fourier transform fromDefinition 1.3.2 is well defined and continuous from S ′(Rn) to S ′(Rn).

1We will not discuss here topological properties of spaces of distributions. See Remark A.19.3for some properties as well as Section B.3 and Section 10.12.


Proof. First, we can readily see that if u ∈ S ′(Rn) then also u ∈ S ′(Rn). Indeed,since ϕ ∈ S(Rn), it follows that ϕ ∈ S(Rn) and so u(ϕ) is a well-defined complexnumber. Moreover, u is linear since both u and the Fourier transform F are linear.Finally, u is continuous because ϕj → ϕ in S(Rn) implies ϕj → ϕ in S(Rn) byRemark 1.1.20, and hence

u(ϕj) = u(ϕj)→ u(ϕ) = u(ϕ)

by the continuity of both u from S(Rn) to C and of the Fourier transform F as amapping from S(Rn) to S(Rn) (see Corollary 1.1.18).

Now, it follows that it is also continuous as a mapping from S ′(Rn) to S ′(Rn),i.e., uj → u in S ′(Rn) implies that uj → u in S ′(Rn). Indeed, if uj → u in S ′(Rn),we have uj(ϕ) = uj(ϕ) → u(ϕ) = u(ϕ) for all ϕ ∈ S(Rn), which means thatuj → u in S ′(Rn). �

Now we give two immediate but important principles for distributions.

Proposition 1.3.4 (Convergence principle). Let X be a topological subspace inS ′(Rn) (i.e., convergence in X implies convergence in S ′(Rn)). Suppose that uj →u in S ′(Rn) and that uj → v in X. Then u ∈ X and u = v.

This statement is simply a consequence of the fact that the space S ′(Rn) isHausdorff, hence it has the uniqueness of limits property (recall that a topologicalspace is called Hausdorff if any two points have open disjoint neighbourhoods, i.e.,open disjoint sets containing them). The convergence principle is also related toanother principle which we call

Proposition 1.3.5 (Uniqueness principle for distributions). Let u, v ∈ S ′(Rn) andsuppose that u(ϕ) = v(ϕ) for all ϕ ∈ S(Rn). Then u = v.

This can be reformulated by saying that if an element o ∈ S ′(Rn) satisfieso(ϕ) = 0 for all ϕ ∈ S(Rn), then o is the zero element in S ′(Rn).

Exercise 1.3.6. Let f ∈ Lp(Rn), 1 ≤ p ≤ ∞, and assume that we have∫Rn

f(x) ϕ(x) dx = 0

for all ϕ ∈ C∞(Rn) for which the integral makes sense. Prove that f = 0 almosteverywhere. Do also a local version of this statement in Exercise 1.4.20.

Remark 1.3.7 (Functions as distributions). We can interpret functions in Lp(Rn),1 ≤ p ≤ ∞, as tempered distributions. If f ∈ Lp(Rn), we define the functional uf

by

uf (ϕ) :=∫

Rn

f(x) ϕ(x) dx, (1.9)

for all ϕ ∈ S(Rn). By Holder’s inequality, we observe that |uf (ϕ)| ≤ ||f ||Lp ||ϕ||Lq ,for 1

p + 1q = 1, and hence uf (ϕ) is well defined in view of the simple inclusion


S(Rn) ⊂ Lq(Rn), for all 1 ≤ q ≤ ∞. It needs to be verified that uf is a linearcontinuous functional on S(Rn). It is clearly linear in ϕ, while its continuity followsby Holder’s inequality (Proposition 1.2.4) from

|uf (ϕj)− uf (ϕ)| ≤ ||f ||Lp ||ϕj − ϕ||Lq

and the following lemma:

Lemma 1.3.8. We have S(Rn) ⊂ Lq(Rn) with continuous embedding, i.e., ϕj → ϕin S(Rn) implies that ϕj → ϕ in Lq(Rn).

Exercise 1.3.9. Prove this lemma.

To summarise, any function f ∈ Lp(Rn) leads to a tempered distributionuf ∈ S ′(Rn) in the canonical way given by (1.9). In this way we will view functionsin Lp(Rn) as tempered distributions and continue to simply write f instead of uf .There should be no confusion with this notation since writing f(x) suggests thatf is a function while f(ϕ) suggests that it is applied to test functions and so it isviewed as a distribution uf .Remark 1.3.10 (Consistency of all definitions). With this identification, Definition1.1.2 of the Fourier transform for functions in L1(Rn) agrees with Definition 1.3.2of the Fourier transforms of tempered distributions. Indeed, let f ∈ L1(Rn). Thenwe have two ways of looking at its Fourier transforms:

1. We can use the first definition f(ξ) =∫

Rn e−2πix·ξf(x) dx, and then we knowthat f ∈ L∞(Rn). In this way we get uf ∈ S ′(Rn).

2. We can immediately think of f ∈ L1(Rn) as of tempered distribution uf ∈S ′(Rn), and the second definition then produces its Fourier transform uf ∈S ′(Rn).

Fortunately, these two approaches are consistent and produce the same tempereddistribution uf = uf ∈ S ′(Rn). Indeed, we have

uf (ϕ) =∫

Rn

f ϕ dx =∫

Rn

f ϕ dx = uf (ϕ).

Here we used the multiplication formula for the Fourier transform in Lemma 1.1.22and the fact that both u ∈ L1(Rn) and u ∈ L∞(Rn) can be viewed as tempereddistributions in the canonical way (see Remark 1.3.7). It follows that we havef(ϕ) = f(ϕ) which justifies Definition 1.3.2.Remark 1.3.11. We note that if u ∈ L1(Rn) and also u ∈ L1(Rn), then the Fourierinversion formula in Theorem 1.1.21 holds for almost all x ∈ Rn. A more generalFourier inversion formula for tempered distributions will be given in Theorem1.3.25.

Exercise 1.3.12. Let 1 ≤ p ≤ ∞. Show that if fk → f in Lp(Rn) then fk → f inS ′(Rn).


It turns out that the Fourier transform acts especially nicely on one of thespaces Lp(Rn), namely on the space L2(Rn), which is also a Hilbert space. Thesetwo facts lead to a very rich Fourier analysis on L2(Rn) which we will deal withonly briefly.

Theorem 1.3.13 (Plancherel’s and Parseval’s formulae). Let u ∈ L2(Rn). Thenu ∈ L2(Rn) and

||u||L2(Rn) = ||u||L2(Rn) (Plancherel’s identity).

Moreover, for all u, v ∈ L2(Rn) we have∫Rn

u v dx =∫

Rn

u v dξ (Parseval’s identity).

Proof. We will use the fact (a special case of this fact follows from Theorem 1.3.31to be proved later) that S(Rn) is sequentially dense in L2(Rn), i.e., that for everyu ∈ L2(Rn) there exists a sequence uj ∈ S(Rn) such that uj → u in L2(Rn). ThenTheorem 1.1.30, (i), with ϕ = ψ = uj − uk, implies that

||uj − uk||2L2 = ||uj − uk||2L2 → 0,

since uj is a convergent sequence in L2(Rn). Thus, uj is a Cauchy sequence inthe complete (Banach, see Theorem C.4.9) space L2(Rn). It follows that it mustconverge to some v ∈ L2(Rn). By the continuity of the Fourier transform in S ′(Rn)(see Proposition 1.3.3) we must also have uj → u in S ′(Rn). By the convergenceprinciple for distributions in Proposition 1.3.4, we get that u = v ∈ L2(Rn).Applying Theorem 1.1.30, (i), again, to ϕ = ψ = uj , we get ||uj ||2L2 = ||uj ||2L2 .Passing to the limit, we get ||u||2L2 = ||u||2L2 , which is Plancherel’s formula.

Finally, for u, v ∈ L2(Rn), let uj , vj ∈ S(Rn) be such that uj → u and vj → vin L2(Rn). Applying Theorem 1.1.30, (i), to ϕ = uj , ψ = vj , and passing to thelimit, we obtain Parseval’s identity. �

Corollary 1.3.14 (Hausdorff–Young inequality). Let 1 ≤ p ≤ 2 and 1p + 1

q = 1. Ifu ∈ Lp(Rn) then u ∈ Lq(Rn) and

||u||Lq(Rn) ≤ ||u||Lp(Rn).

Proof. The statement follows by the Riesz–Thorin Interpolation Theorem C.4.18from estimates ||u||L∞(Rn) ≤ ||u||L1(Rn) in (1.1) and Plancherel’s identity||u||L2(Rn) = ||u||L2(Rn) in Theorem 1.3.13. �

1.3.2 Operations with distributions

Besides the Fourier transform, there are several other operations that can be ex-tended from functions in S(Rn) to tempered distributions in S ′(Rn).


For example, partial differentiation ∂∂xj

can be extended to a continuousoperator ∂

∂xj: S ′(Rn) → S ′(Rn). Indeed, for u ∈ S ′(Rn) and ϕ ∈ S(Rn), let us

define (∂

∂xju

)(ϕ) := −u

(∂ϕ

∂xj

).

It is necessary to include the negative sign in this definition. Indeed, if u ∈ S(Rn),then the integration by parts formula and the identification of functions withdistributions in Remark 1.3.7 yield(

∂

∂xju

)(ϕ) =

∫Rn

(∂u

∂xj

)(x)ϕ(x) dx

= −∫

Rn

u(x)(

∂ϕ

∂xj

)(x) dx = −u

(∂ϕ

∂xj

),

which explains the sign. This also shows the consistency of this definition of thederivative with the usual definition for differentiable functions.

Definition 1.3.15 (Distributional derivatives). More generally, for any multi-indexα, one can define

(∂αu)(ϕ) = (−1)|α|u(∂αϕ),

for ϕ ∈ S(Rn).

Proposition 1.3.16. If u ∈ S ′(Rn), then ∂αu ∈ S ′(Rn) and operator ∂α : S ′(Rn)→S ′(Rn) is continuous.

Proof. Indeed, if ϕk → ϕ in S(Rn), then clearly also ∂αϕk → ∂αϕ in S(Rn), and,therefore,

(∂αu)(ϕk) = (−1)|α|u(∂αϕk)→ (−1)|α|u(∂αϕ) = (∂αu)(ϕ),

which means that ∂αu ∈ S ′(Rn). Moreover, let uk → u ∈ S ′(Rn). Then ∂αuk(ϕ) =(−1)|α|uk(∂αϕ) → (−1)|α|u(∂αϕ) = ∂αu(ϕ), for all ϕ ∈ S(Rn), i.e., ∂α is contin-uous on S ′(Rn). �

Exercise 1.3.17. Show that if u ∈ S ′(Rn), then ∂α∂βu = ∂β∂αu = ∂α+βu.

Exercise 1.3.18. Let χ : R → R be the characteristic function of the interval[−1, 1], i.e., χ(y) = 1 for −1 ≤ y ≤ 1 and χ(y) = 0 for |y| > 1. Calculate thedistributional derivative χ′. Define operator T by Tf(x) = d

dx (χ ∗ f)(x), x ∈ R,f ∈ S(Rn). Prove that Tf(x) = f(x + 1)− f(x− 1).

Remark 1.3.19 (Multiplication by functions). If a smooth function f ∈ C∞(Rn)and all of its derivatives are bounded by some polynomial functions, we can definethe multiplication of a tempered distribution u by f by setting (fu)(ϕ) := u(fϕ).This is well defined since ϕ ∈ S(Rn) implies fϕ ∈ S(Rn).


Exercise 1.3.20 (Hadamard’s principal value). Show that log |x| is a tempereddistribution on R. Let u = d

dx log |x|. Show that

u(ϕ) = limε↘0

∫R\[−ε,ε]

1x

ϕ(x) dx

for all ϕ ∈ C1(R) vanishing outside a bounded set. The distribution u is called theprincipal value of 1

x and is denoted by p.v. 1x .

Remark 1.3.21 (Schwartz’ impossibility result). One has to be careful when mul-tiplying distributions as the following example shows:

0 =1x· (x · δ) =

(1x· x

)· δ = δ,

where 1x may be any inverse of x, for example p.v. 1

x . In general, distributionscan not be multiplied, as was noted by Laurent Schwartz in [104], and which iscalled the Schwartz’ impossibility result. Still, some multiplication is possible, asis demonstrated by Remark 1.3.19.

Exercise 1.3.22. Define the distribution 1x±i0 by(

1x± i0

)(ϕ) := lim

ε→0±

∫R

1x + iε

ϕ(x) dx,

for ϕ ∈ S(Rn). Prove that

1x± i0

= p.v.1x∓ iπδ.

However, as we have seen, statements on S(Rn) can usually be extended tocorresponding statements on S ′(Rn). This applies to the Fourier inversion formulaas well.

Definition 1.3.23 (Inverse Fourier transform). Define F−1 on S ′(Rn) by

(F−1u)(ϕ) := u(F−1ϕ),

for u ∈ S ′(Rn) and ϕ ∈ S(Rn).

Exercise 1.3.24. Show that F−1 : S ′(Rn)→ S ′(Rn) is well defined and continuous.

Theorem 1.3.25 (Fourier inversion formula for tempered distributions). OperatorsF and F−1 are inverse to each other on S ′(Rn), i.e.,

FF−1 = F−1F = identity on S ′(Rn).

Proof. To prove this, let u ∈ S ′(Rn) and ϕ ∈ S(Rn). Then by Theorem 1.1.21 andDefinitions 1.3.2 and 1.3.23, we get

(FF−1u)(ϕ) = (F−1u)(Fϕ) = u(F−1Fϕ) = u(ϕ),

so FF−1u = u by the uniqueness principle for distributions in Proposition 1.3.5.A similar argument applies to show that F−1F = id. �


Remark 1.3.26. To give an example of these operations, let us define the Heavisidefunction H on R by setting

H(x) ={

0, if x < 0,1, if x ≥ 0.

Clearly H ∈ L∞(R), so in particular, it is a tempered distribution in S ′(Rn). Letus also define the Dirac δ–distribution by setting δ(ϕ) = ϕ(0) for all ϕ ∈ S(R). Itis easy to see that δ ∈ S ′(Rn).

We claim first that H ′ = δ. Indeed, we have

H ′(ϕ) = −H(ϕ′) = −∫ ∞

0

ϕ′(x) dx = ϕ(0) = δ(ϕ),

hence H ′ = δ by the uniqueness principle for distributions.Let us now calculate the Fourier transform of δ. According to the definitions,

we have

δ(ϕ) = δ(ϕ) = ϕ(0) =∫

R

ϕ(x) dx = 1(ϕ),

hence δ = 1. Here we used the fact that the constant 1 is in L∞(Rn), hence alsoa tempered distribution.

Exercise 1.3.27. Check that we also have 1 = δ.

1.3.3 Approximating by smooth functions

It turns out that although elements of S ′(Rn) can be very irregular and the space isquite large, tempered distributions can still be approximated by smooth compactlysupported functions.

Definition 1.3.28 (Space C∞0 (Ω)). For an open set Ω ⊂ Rn, the space C∞0 (Ω) ofsmooth compactly supported functions is defined as the space of smooth functionsϕ : Ω→ C with compact support. Here the support of ϕ is defined as the closureof the set where ϕ is non-zero, i.e., by

suppϕ = {x ∈ Ω : ϕ(x) = 0}.

Remark 1.3.29 (How large is C∞0 (Ω)?). We can see that this space is non-empty.For example, if we define function χ(t) by χ(t) = e−1/t2 for t > 0 and by χ(t) = 0for t ≤ 0, then f(t) = χ(t)χ(1 − t) is a smooth compactly supported function onR. Consequently, ϕ(x) = f(x1) · · · f(xn) is a function in C∞0 (Rn), with suppϕ =[0, 1]n.

Another example is the function ψ defined by ψ(x) = e1/(|x|2−1) for |x| < 1and by ψ(x) = 0 for |x| ≥ 1. We have ψ ∈ C∞0 (Rn) with supp ψ = {|x| ≤ 1}.


Remark 1.3.30. For the functional analytic description of the topology of the spaceC∞0 (Ω) we refer to Exercise B.3.12. It is also a nuclear Montel space, see ExercisesB.3.35 and B.3.51.

Although these examples are quite special, products of these functions withany other smooth function as well as their derivatives are all in C∞0 (Rn). On theother hand, C∞0 (Rn) can not contain analytic functions, thus making it relativelysmall. Still, it is dense in very large spaces of functions/distributions in theirrespective topologies.

Theorem 1.3.31 (Sequential density of C∞0 (Ω) in S ′(Rn)). The space C∞0 (Rn) issequentially dense in S ′(Rn), i.e., for every u ∈ S ′(Rn) there exists a sequenceuk ∈ C∞0 (Rn) such that uk → u in S ′(Rn) as k →∞.

Lemma 1.3.32. The space C∞0 (Rn) is sequentially dense in S(Rn), i.e., for everyϕ ∈ S(Rn) there exists a sequence ϕk ∈ C∞0 (Rn) such that ϕk → ϕ in S(Rn) ask →∞.

Proof. Let ϕ ∈ S(Rn). Let us fix some ψ ∈ C∞0 (Rn) such that ψ = 1 in aneighbourhood of the origin and let us define ψk(x) = ψ(x/k). Then it can beeasily checked that ϕk = ψkϕ→ ϕ in S(Rn), as k →∞. �

Proof of Theorem 1.3.31. Let u ∈ S ′(Rn) and let ψ and ψk be as in the proof ofLemma 1.3.32. Then ψu ∈ S ′(Rn) is well defined by (ψu)(ϕ) = u(ψϕ), for allϕ ∈ S(Rn). We have that ψku → u in S ′(Rn). Indeed, we have that (ψku)(ϕ) =u(ψkϕ) → u(ϕ) by Lemma 1.3.32. Similarly, we have that ψku → u in S ′(Rn),and hence also F−1(ψku) → u in S ′(Rn) because of the continuity of the Fouriertransform in S ′(Rn), see Proposition 1.3.3. Consequently, we have

ukj = ψj(F−1(ψku))→ u in S ′(Rn) as k, j →∞.

It remains to show that ukj ∈ C∞0 (Rn). In general, let χ ∈ C∞0 (Rn) and letw = χu. We claim that F−1w ∈ C∞(Rn). Indeed, we have

(F−1w)(ϕ) = w(F−1ϕ) = wξ

(∫Rn

e2πix·ξϕ(x) dx

)=

∫Rn

wξ( e2πix·ξ)ϕ(x) dx,

where we write wξ to emphasize that w acts on the test function as the function ofξ-variable, and where we used the continuity of w and the fact that wξ( e2πix·ξ) =u(χ e2πix·ξ) is well defined. Now, it follows that F−1w can be identified with thefunction (F−1w)(x) = uξ(χ(ξ) e2πix·ξ), which is smooth with respect to x. Indeed,we can note first that the right-hand side depends continuously on x because ofthe continuity of u on S(Rn). Here we also use that everything is well defined sinceχ ∈ C∞0 (Rn). Moreover, since the function χ(ξ) e2πix·ξ is compactly supported in ξ,so are its derivatives with respect to x, and hence all the derivatives of (F−1w)(x)are also continuous in x, proving the claim and the theorem. �

1.4. Distributions 241

Exercise 1.3.33. Prove that S(Rn) is sequentially dense in L2(Rn), i.e., that forevery u ∈ L2(Rn) there exists a sequence uj ∈ S(Rn) such that uj → u in L2(Rn).Prove that this is also true for Lp(Rn), for all 1 ≤ p <∞.

Exercise 1.3.34 (Uncertainty principle). Prove that C∞0 (Rn) ∩ FC∞0 (Rn) = {0}.(Hint: it is enough to know that polynomials are dense in L2(K) for any compactK.)

Exercise 1.3.35 (Scaling operators). For λ ∈ R, λ = 0, define the mapping mλ :Rn → Rn by mλ(x) = λx.

(i) Let ϕ ∈ S(Rn). Prove that ϕ ◦mλ(ξ) = λ−n (ϕ ◦mλ−1)(ξ) for all ξ ∈ Rn.(ii) Let u ∈ S ′(Rn). Define the distribution u ◦mλ by

(u ◦mλ)(ϕ) := λ−n u(ϕ ◦mλ−1),

for all ϕ ∈ S(Rn). Prove that this definition is consistent with S(Rn), i.e.,show that if u ∈ S(Rn), (u ◦mλ)(x) = u(λx), and if we identify u with itscanonical distribution, then we have

(u ◦mλ)(ϕ) = λ−n u(ϕ ◦mλ−1)

for all ϕ ∈ S(Rn).(iii) Let u ∈ S ′(Rn). Prove that u ◦mλ = λ−n u ◦mλ−1 .

1.4 Distributions

Since our main interest is in Fourier analysis, we started with the space S ′(Rn) oftempered distributions which allows the definition and use of the Fourier trans-form. However, there is a bigger space of distributions which we will sketch here. Itwill contain some important classes of functions that S ′(Rn) does not contain. Formuch more comprehensive treatments of spaces of distributions and their proper-ties we refer the reader to monographs [8, 10, 39, 106, 105].

1.4.1 Localisation of Lp-spaces and distributions

Definition 1.4.1 (Localisations of Lp-spaces). We define local versions of the spacesLp(Ω) as follows. We will say that f ∈ Lp

loc(Ω) if ϕf ∈ Lp(Ω) for all ϕ ∈ C∞0 (Ω).We note that the spaces Lp

loc(Rn) are not subspaces of S ′(Rn) since they do not

encode any information on the global behaviour of functions. For example, e|x|2

is smooth, and hence belongs to all Lploc(R

n), 1 ≤ p ≤ ∞, but it is not in S ′(Rn).There is a natural notion of convergence in the localised spaces Lp

loc(Ω). Thus, wewill write fm → f in Lp

loc(Ω) as m → ∞, if f and fm belong to Lp(Ω)loc for allm, and if ϕfm → ϕf in Lp(Ω) as m→∞, for all ϕ ∈ C∞0 (Ω).

The difference between the space of distributions D′(Rn) that we are goingto introduce now, and the space of tempered distributions S ′(Rn) is the choice of


the set C∞0 (Rn) rather than S(Rn) as the space of test functions. At the sametime, choosing C∞0 (Ω) as test functions allows one to obtain the space D′(Ω) ofdistributions in Ω, rather than on the whole space Rn.

The definition and facts below are sketched only as they are similar to Defi-nition 1.3.1.

Definition 1.4.2 (Distributions D′(Ω)). We say that ϕk → ϕ in C∞0 (Ω) if ϕk, ϕ ∈C∞0 (Ω), if there is a compact set K ⊂ Ω such that suppϕk ⊂ K for all k, andif supx∈Ω |∂α(ϕk − ϕ)(x)| → 0 for all multi-indices α. Then D′(Ω) is defined asthe set of all linear continuous functionals u : C∞0 (Ω) → C, i.e., all functionalsu : C∞0 (Ω)→ C such that:

1. u is linear, i.e., u(αϕ + βψ) = αu(ϕ) + βu(ψ) for all α, β ∈ C and all ϕ, ψ ∈C∞0 (Ω);

2. u is continuous, i.e., u(ϕj)→ u(ϕ) in C whenever ϕj → ϕ in C∞0 (Ω).

Exercise 1.4.3 (Order of a distribution). Show that a linear operator u : C∞0 (Ω)→C belongs to D′(Ω) if and only if for every compact set K ⊂ Ω there exist constantsC and m such that

|u(ϕ)| ≤ C max|α|≤m

supx∈Ω

|∂αϕ(x)|, (1.10)

for all ϕ ∈ C∞0 (Ω) with suppϕ ⊂ K. The smallest m for which (1.10) holds iscalled the order of u in K. The smallest m which works for all compact sets ifcalled the order of the distribution u. Show that δ-distribution has order 1. Findexamples of distributions of infinite order.

Remark 1.4.4 (Distributions of zero order as measures). If u ∈ D′(Ω) is a distri-bution of order zero, by (1.10) it defines a continuous functional on C(Ω). Thenit follows from Theorem C.4.60 that u is a measure (at least when Ω is compact,or when u acts on continuous compactly supported functions).Remark 1.4.5 (Continuous inclusion S ′(Rn) ⊂ D′(Rn)). It is easy to see thatC∞0 (Rn) ⊂ S(Rn) and that if ϕk → ϕ in C∞0 (Rn), then ϕk → ϕ in S(Rn). Thus, ifu ∈ S ′(Rn) and if ϕk → ϕ in C∞0 (Rn), we have u(ϕk)→ u(ϕ), which means thatu ∈ D′(Rn). Thus, we showed that S ′(Rn) ⊂ D′(Rn). We say that uk → u ∈ D′(Ω)if uk, u ∈ D′(Ω) and if uk(ϕ)→ u(ϕ) for all ϕ ∈ C∞0 (Ω).

Exercise 1.4.6. Show that uk → u in S ′(Rn) implies uk → u in D′(Rn), i.e., theinclusion S ′(Rn) ⊂ D′(Rn) is continuous.

Exercise 1.4.7. Prove that the canonical identification in Remark 1.3.7 yields theinclusions Lp

loc(Ω) ⊂ D′(Ω) for all 1 ≤ p ≤ ∞.

Definition 1.4.8 (Compactly supported distributions E ′(Ω)). We say that ϕk → ϕin C∞(Ω) if ϕk, ϕ ∈ C∞(Ω) and if supx∈K |∂α(ϕk−ϕ)(x)| → 0 for all multi-indicesα and all compact subsets K of Ω. Then E ′(Ω) is defined as the set of all linearcontinuous functionals u : C∞(Ω) → C, i.e., all functionals u : C∞(Ω) → C suchthat:


1. u is linear, i.e., u(αϕ + βψ) = αu(ϕ) + βu(ψ) for all α, β ∈ C and all ϕ, ψ ∈C∞(Ω);

2. u is continuous, i.e., u(ϕj)→ u(ϕ) in C whenever ϕj → ϕ in C∞(Ω).

Exercise 1.4.9. Show that the restriction of u ∈ E ′(Ω) to C∞0 (Ω) is an injectivelinear mapping from E ′(Ω) to D′(Ω).

Exercise 1.4.10. Show that E ′(Ω) ⊂ D′(Ω) and that E ′(Rn) ⊂ S ′(Rn) ⊂ D′(Rn).Show also that all these inclusions are continuous.

Definition 1.4.11 (Support of a distribution). We say that u ∈ D′(Ω) is supported inthe set K ⊂ Ω if u(ϕ) = 0 for all ϕ ∈ C∞(Ω) such that ϕ = 0 on K. The smallestclosed set in which u is supported is called the support of u and is denoted bysupp u.

Exercise 1.4.12. Formulate and prove the analogue of the criterion in Exercise1.4.3 for compactly supported distributions in E ′(Ω).

Exercise 1.4.13. Show that distributions in E ′(Ω) have compact support (justifyingthe name of “compactly supported” distributions in Definition 1.4.8).

Exercise 1.4.14 (Distributions with compact support). Prove that if the support ofu ∈ D′(Rn) is compact then u is of finite order. Prove that all compactly supporteddistributions belong to E ′(Ω).

Exercise 1.4.15 (Distributions with point support). Prove that if a distributionu ∈ D′(Rn) of order m has support supp u = {0}, then there exist constantsaα ∈ C such that u =

∑|α|≤m aα∂αδ.

Definition 1.4.16 (Singular support). The singular support of u ∈ D′(Ω) is definedas the complement of the set where u is smooth. Namely, x ∈ sing supp u if thereis an open neighbourhood U of x and a smooth function f ∈ C∞(U) such thatu(ϕ) = f(ϕ) for all ϕ ∈ C∞0 (U).

Exercise 1.4.17. Show that if u ∈ D′(Ω) then its singular support is closed.

Exercise 1.4.18. Show that sing supp |x| = {0} and that sing supp δ = {0}.Exercise 1.4.19. Show that if u ∈ E ′(Rn), then u is a smooth function of the so-called slow growth (i.e., u(ξ) and all of its derivatives are of at most polynomialgrowth). Hint: the slow growth follows from testing u on the exponential functionseξ(x) = e2πix·ξ. Indeed, show first that u(ξ) = 〈u, eξ〉 and thus

∂αu(ξ) =⟨u, ∂α

ξ eξ

⟩= (−2πi)|α|〈u, xαeξ〉.

Consequently, by an analogue of (1.10) in Exercise 1.4.3 we conclude that

|∂αu(ξ)| ≤ C∑|β|≤m

sup|x|≤R

|∂βx (xαeξ(x))| ≤ C(1 + R)|α|(1 + |ξ|)m.

Exercise 1.4.20. Prove the following stronger version of Exercise 1.3.6. Let f ∈L1

loc(Rn) and assume that

∫Rn f(x) ϕ(x) dx = 0 for all ϕ ∈ C∞0 (Rn). Prove that

f = 0 almost everywhere.


1.4.2 Convolution of distributions

We can write the convolution of two functions f, g ∈ S(Rn) in the following way:

(f ∗ g)(x) =∫

Rn

f(z)g(x− z) dz =∫

Rn

f(z)(τxRg)(z) dz,

where (Rg)(x) = g(−x) and (τhg)(x) = g(x− h), so that

(τxRg)(z) = (Rg)(z − x) = g(x− z).

Recalling our identification of functions with distributions in Remark 1.3.7, wecan write (f ∗ g)(x) = f(τxRg). This can now be extended to distributions.

Definition 1.4.21 (Convolution with a distribution). For u ∈ S ′(Rn) and ϕ ∈S(Rn), define

(u ∗ ϕ)(x) := u(τxRϕ).

The definition makes sense since τxRϕ ∈ S(Rn) and since τx, R : S(Rn)→ S(Rn)are continuous.

Corollary 1.4.22. For example, for ψ ∈ S(Rn) we have δ ∗ ψ = ψ since for everyx ∈ Rn we have

(δ ∗ ψ)(x) = δ(τxRψ) = ψ(x− z)|z=0 = ψ(x).

Lemma 1.4.23. Let u ∈ S ′(Rn) and ϕ ∈ S(Rn). Then u ∗ ϕ ∈ C∞(Rn).

Proof. We can observe that (u ∗ ϕ)(x) = u(τxRϕ) is continuous in x since τx :S(Rn) → S(Rn) and u : S(Rn) → C are continuous. The same applies when welook at derivatives in x, implying that u ∗ ϕ is smooth. Here we note that we areallowed to pass the limit through u since it is a continuous functional. �Exercise 1.4.24. Prove that if u, v, ϕ ∈ S(Rn) then (u ∗ v)(ϕ) = u(Rv ∗ ϕ).

Exercise 1.4.25 (Reflection of a distribution). For v ∈ S ′(Rn), define its reflectionRv by

(Rv)(ϕ) := v(Rϕ),

for ϕ ∈ S(Rn). Prove that Rv ∈ S ′(Rn). Prove also that this definition is consistentwith the definition of (Rg)(x) = g(−x) for g ∈ C∞(Rn).

Exercise 1.4.26. Show that if v ∈ S ′(Rn), then the mapping ϕ �→ Rv ∗ ϕ iscontinuous from C∞(Rn) to C∞(Rn).

Consequently, if v ∈ S ′(Rn) and ϕ ∈ S(Rn), we have Rv ∗ ϕ ∈ C∞(Rn) byLemma 1.4.23. This motivates the following:

Definition 1.4.27 (Convolution of distributions). Let u ∈ E ′(Rn) and v ∈ S ′(Rn).Define the convolution u ∗ v of u and v by

(u ∗ v)(ϕ) := u(Rv ∗ ϕ),



Exercise 1.4.28. We see from Exercise 1.4.24 that this definition is consistent withS(Rn). Prove that if u ∈ E ′(Rn) and v ∈ S ′(Rn) then u ∗ v ∈ S ′(Rn).

Exercise 1.4.29. Prove that if u ∈ E ′(Rn) and v ∈ S ′(Rn) then

sing supp (u ∗ v) ⊂ sing supp u + sing supp v.

Exercise 1.4.30. Extend the notion of convolution to two distributions u, v ∈D′(Rn) when at least one of them has compact support.

Exercise 1.4.31 (Diagonal property). Show that the convolution u∗v of two distri-butions exists if for every compact set K the intersection (suppu×supp v)∩{(x, y) :x + y ∈ K} is compact. This allows, for example, to take a convolution of twoHeaviside functions H, yielding H ∗H = xH.

Remark 1.4.32. Let us show an example of the calculation with distributions. Letv ∈ S ′(Rn). We will show that v ∗ δ = δ ∗ v = v. Indeed, on one hand we have

(v ∗ δ)(ϕ) 1.4.27= v(Rδ ∗ ϕ) = v(δ ∗ ϕ) 1.4.22= v(ϕ)

in view of Rδ = δ:

〈Rδ, ψ〉 1.4.25= 〈δ, Rψ〉 = Rψ(0) = ψ(0) = 〈δ, ψ〉.

On the other hand, we have

(δ ∗ v)(ϕ) 1.4.27= δ(Rv ∗ ϕ) = (Rv ∗ ϕ)(0) 1.4.21= Rv(τ0Rϕ) = Rv(Rϕ) 1.4.25= v(ϕ).

Note that in view of Exercise 1.4.30 we could have taken v ∈ D′(Rn) here.

Exercise 1.4.33. Let u ∈ S ′(Rn) and v ∈ E ′(Rn). Define u ∗ v as v ∗ u, i.e.,u ∗ v := v ∗ u. Prove that this coincides with Definition 1.4.27 when u, v ∈ E ′(Rn).

Exercise 1.4.34. Prove that the extension of (1.8) holds, i.e., that

∂α(f ∗ g) = ∂αf ∗ g = f ∗ ∂αg.

Remark 1.4.35 (Non-associativity of convolution). In Exercise 1.1.29 we formu-lated the associativity of a convolution. However, for distributions one has to becareful. Indeed, recalling the relation H ′ = δ from Remark 1.3.26 and assumingthe associativity one could prove

1 1.4.32= δ ∗ 11.4.32= (δ ∗ δ) ∗ 11.4.34= (H ∗ δ′) ∗ 1“ = ” H ∗ (δ′ ∗ 1)1.4.34= H ∗ (δ ∗ 1′)= H ∗ 0= 0.


Exercise 1.4.36. Why does the associativity in Remark 1.4.35 fail? How could werestrict the spaces of distributions for the convolution to be still associative?

Exercise 1.4.37. Show that if u ∈ E ′(Rn) and v ∈ S ′(Rn), then u ∗ v = uv, wherethe product on the right-hand side makes sense in view of Remark 1.3.19 andExercise 1.4.19.

We now formulate a couple of useful properties of translations:

Exercise 1.4.38 (Translation is continuous in Lp(Rn)). Prove that translation iscontinuous in Lp(Rn), namely that translations τx : Lp(Rn)→ Lp(Rn), (τxf)(y) =f(y − x), satisfy ||τxf − f ||Lp → 0 as x→ 0, for every f ∈ Lp(Rn).

Exercise 1.4.39 (Translations of convolutions). For f, g ∈ L1(Rn), show that theconvolution of f and g satisfies

τx(f ∗ g) = (τxf) ∗ g = f ∗ (τxg).

Can you extend this to some classes of distributions?

1.5 Sobolev spaces

In this section we discuss Sobolev spaces Lpk with integer orders k ∈ N. After

introducing the necessary elements of the theory of pseudo-differential operatorswe will come back to this topic in Section 2.6.3 to also discuss Sobolev spaces Lp

s

for all real s ∈ R.

1.5.1 Weak derivatives and Sobolev spaces

There is a notion of a weak derivative which is a special case of the distributionalderivative from Definition 1.3.15. However, it allows a realisation in an integralform and we mention it here briefly.

Definition 1.5.1 (Weak derivative). Let Ω be an open subset of Rn and let u, v ∈L1

loc(Ω). We say that v is the αth-weak partial derivative of u if∫Ω

u ∂αϕ dx = (−1)|α|∫

Ω

v ϕ dx, for all ϕ ∈ C∞0 (Ω).

In this case we also write v = ∂αu.

The constant (−1)|α| stands for the consistency with the corresponding def-inition for smooth functions when using integration by parts in Ω. It is the samereason as to include the constant (−1)|α| in Definition 1.3.15. The weak derivativedefined in this way is uniquely determined:

Lemma 1.5.2. Let u ∈ L1loc(Ω). If a weak αth derivative of u exists, it is uniquely

defined up to a set of measure zero.

1.5. Sobolev spaces 247

Proof. Indeed, assume that there are two functions v, w ∈ L1loc(Ω) such that∫

Ω

u ∂αϕ dx = (−1)|α|∫

Ω

v ϕ dx = (−1)|α|∫

Ω

w ϕ dx,

for all ϕ ∈ C∞0 (Ω). Then∫Ω(v − w)ϕ dx = 0 for all ϕ ∈ C∞0 (Ω). A standard

result from measure theory (e.g., Theorem C.4.60) now implies that v = w almosteverywhere in Ω. �Exercise 1.5.3. Let us define u, v : R→ R by

u(x) ={

x, if x ≤ 1,1, if x > 1,

v(x) ={

1, if x ≤ 1,0, if x > 1,

Prove that u′ = v weakly.

Exercise 1.5.4. Define u : R→ R by

u(x) ={

x, if x ≤ 1,2, if x > 1.

Prove that u has no weak derivative. Calculate the distributional derivative of u.

Exercise 1.5.5. Prove that the Dirac δ-distribution is not an element of L1loc(R

n).

There are different ways to define Sobolev spaces2. Here we choose the oneusing weak or distributional derivatives.

Definition 1.5.6 (Sobolev spaces). Let 1 ≤ p ≤ ∞ and let k ∈ N∪{0}. The Sobolevspace Lp

k(Ω) (or W p,k(Ω)) consists of all u ∈ L1loc(Ω) such that for all multi-indices

α with |α| ≤ k, ∂αu exists weakly (or distributionally) and ∂αu ∈ Lp(Ω). Foru ∈ Lp

k(Ω), we define

||u||Lpk(Ω) :=

⎛⎝ ∑|α|≤k

||∂αu||pLp

⎞⎠1/p

=

⎛⎝ ∑|α|≤k

∫Ω

|∂αu|p dx

⎞⎠1/p

,

for 1 ≤ p <∞, and for p =∞ we define

||u||L∞k (Ω) := max

|α|≤kesssupΩ|∂αu|.

Since p ≥ 1, we know that Lploc(Ω) ⊂ L1

loc(Ω), e.g., by Holder’s inequality(Proposition 1.2.4), so we note that it does not matter whether we take a weak ora distributional derivative.

In the case p = 2, one often uses the notation Hk(Ω) for L2k(Ω), and in the

case p = 2 and k = 0, we get H0(Ω) = L2(Ω). As usual, we identify functions inLp

k(Ω) which are equal almost everywhere (see Definition C.4.6).

2We come back to this subject in Section 2.6.3.


Proposition 1.5.7. The functions ||·||Lpk(Ω) in Definition 1.5.6 are norms on Lp

k(Ω).

Proof. Indeed, we clearly have ||λu||Lpk

= |λ|||u||Lpk

and ||u||Lpk

= 0 if and only ifu = 0 almost everywhere. For the triangle inequality, the case p = ∞ is straight-forward. For 1 ≤ p <∞ and for u, v ∈ Lp

k(Ω), Minkowski’s inequality (Proposition1.2.7) implies

||u + v||Lpk

=

⎛⎝ ∑|α|≤k

||∂αu + ∂αv||pLp

⎞⎠1/p

≤

⎛⎝ ∑|α|≤k

(||∂αu||Lp + ||∂αv||Lp)p

⎞⎠1/p

≤

⎛⎝ ∑|α|≤k

||∂αu||pLp

⎞⎠1/p

+

⎛⎝ ∑|α|≤k

||∂αv||pLp

⎞⎠1/p

= ||u||Lpk

+ ||v||Lpk,


We define local versions of spaces Lpk(Ω) similarly to local versions of Lp-

spaces.

Definition 1.5.8 (Localisations of Sobolev spaces). We will say that f ∈ Lpk(Ω)loc

if ϕf ∈ Lpk(Ω) for all ϕ ∈ C∞0 (Ω). We will write fm → f in Lp

k(Ω)loc as m → ∞,if f and fm belong to Lp

k(Ω)loc for all m, and if ϕfm → ϕf in Lpk(Ω) as m →∞,

for all ϕ ∈ C∞0 (Ω).

Example (Example of a point singularity). An often encountered example of afunction with a point singularity is u(x) = |x|−a defined for x ∈ Ω = B(0, 1) ⊂ Rn,x = 0. We may ask a question: for which a > 0 do we have u ∈ Lp

1(Ω)?First we observe that away from the origin, u is a smooth function and can

be differentiated pointwise with ∂xj u = −axj |x|−a−2 and hence also |∇u(x)| =|a||x|−a−1, x = 0. In particular, |∇u| ∈ L1(Ω) for a+1 < n (here Exercise 1.1.19 isof use). We also have |∇u| ∈ Lp(Ω) for (a+1)p < n. So we must assume a+1 < nand (a + 1)p < n. Let us now calculate the weak (distributional) derivative of uin Ω. Let ϕ ∈ C∞0 (Ω). Let ε > 0. On Ω\B(0, ε) we can integrate by parts to get∫

Ω\B(0,ε)

u∂xj ϕ dx = −∫

Ω\B(0,ε)

∂xj uϕ dx +∫

∂B(0,ε)

uϕνj dσ, (1.11)

where dσ is the surface measure on the sphere ∂B(0, ε) and ν = (ν1, . . . , νn) isthe inward pointing normal on ∂B(0, ε). Now, since u = |ε|−a on ∂B(0, ε), we canestimate ∣∣∣∣∣

∫∂B(0,ε)

uϕνj dσ

∣∣∣∣∣ ≤ ||ϕ||L∞

∫∂B(0,ε)

ε−a dσ ≤ Cεn−1−a → 0


as ε→ 0, since a + 1 < n. Passing to the limit in the integration by parts formula(1.11), we get

∫Ω

u∂xjϕ dx = −

∫Ω

∂xjuϕ dx, which means that ∂xj

u is also theweak derivative of u. So, u ∈ Lp

1(Ω) if u, |∇u| ∈ Lp(Ω), which holds for (a+1)p < n,i.e., for a < (n− p)/p.

Exercise 1.5.9. Find conditions on a in the above example for which u ∈ Lpk(Ω).

1.5.2 Some properties of Sobolev spaces

Since Lp(Ω) ⊂ D′(Ω), we can work with u ∈ Lp(Ω) as with functions or as withdistributions. In particular, we can differentiate them distributionally, etc. More-over, as we have already seen, the equality of objects (be it functions, functionals,distributions, etc.) depends on the spaces in which the equality is considered. InSobolev spaces we can use tools from measure theory so we work with functionsdefined almost everywhere. Thus, an equality in Sobolev spaces (as in the followingtheorem) means pointwise equality almost everywhere.

Theorem 1.5.10 (Properties of Sobolev spaces). Let u, v ∈ Lpk(Ω), and let α be a

multi-index with |α| ≤ k. Then:

(i) ∂αu ∈ Lpk−|α|(Ω), and ∂α(∂βu) = ∂β(∂αu) = ∂α+βu, for all multi-indices

α, β such that |α|+ |β| ≤ k.

(ii) For all λ, μ ∈ C we have λu + μv ∈ Lpk(Ω) and ∂α(λu + μv) = λ∂αu + μ∂αv.

(iii) If Ω is an open subset of Ω, then u ∈ Lpk(Ω).

(iv) If χ ∈ C∞0 (Ω), then χu ∈ Lpk(Ω) and we have the Leibniz formula

∂α(χu) =∑β≤α

(α

β

)(∂βχ)(∂α−βu),

where(αβ

)= α!

β!(α−β)! is the binomial coefficient.

(v) Lpk(Ω) is a Banach space.

Proof. Statements (i), (ii), and (iii) are easy. For example, if ϕ ∈ C∞0 (Ω) then also∂βϕ ∈ C∞0 (Ω), and (i) follows from∫

Ω

∂αu ∂βϕ dx = (−1)|α|∫

Ω

u ∂α+βϕ dx = (−1)|α|+|α+β|∫

Ω

∂α+βuϕ dx,

since (−1)|α|+|α+β| = (−1)|β|.Let us now show (iv). The proof will be carried out by induction on |α|. For

|α| = 1, writing 〈u, ϕ〉 for u(ϕ) =∫Ω

uϕ dx, we get

〈∂α(χu), ϕ〉 = (−1)|α|〈u, χ∂αϕ〉= −〈u, ∂α(χϕ)− (∂αχ)ϕ〉 = 〈χ∂αu, ϕ〉+ 〈(∂αχ)u, ϕ〉,

which is what was required. Now, suppose that the Leibniz formula is valid for all|β| ≤ l, and let us take α with |α| = l+1. Then we can write α = β +γ with some


|β| = l and |γ| = 1. We get

〈χu, ∂αϕ〉 =⟨χu, ∂β(∂γϕ)

⟩= (−1)|β|

⟨∂β(χu), ∂γϕ

⟩(by induction hypothesis)

= (−1)|β|⟨∑

σ≤β

(β

σ

)∂σχ ∂β−σu, ∂γϕ

⟩(by definition)

= (−1)|β|+|γ|⟨∑

σ≤β

(β

σ

)∂γ(∂σχ ∂β−σu), ϕ

⟩(set ρ = σ + γ)

= (−1)|α|⟨∑

σ≤β

(β

σ

)(∂ρχ ∂α−ρu + ∂σχ ∂α−σu

), ϕ

⟩

= (−1)|α|⟨∑

ρ≤α

(α

ρ

)∂ρχ ∂α−ρu, ϕ

⟩,

where we used that(βσ

)+

(βρ

)=

(α−γρ−γ

)+

(α−γ

ρ

)=

(αρ

).

Now let us prove (v). We have already shown in Proposition 1.5.7 thatLp

k(Ω) is a normed space. Let us show now that the completeness of Lp(Ω) (The-orem C.4.9) implies the completeness of Lp

k(Ω). Let um be a Cauchy sequence inLp

k(Ω). Then ∂αum is a Cauchy sequence in Lp(Ω) for any |α| ≤ k. Since Lp(Ω)is complete, there exists some uα ∈ Lp(Ω) such that ∂αum → uα in Lp(Ω). Letu = u(0,...,0), so in particular, we have um → u in Lp(Ω). Let us now show that infact u ∈ Lp

k(Ω) and ∂αu = uα for all |α| ≤ k. Let ϕ ∈ C∞0 (Ω). Then

〈∂αu, ϕ〉 = (− 1)|α|〈u, ∂αϕ〉= (− 1)|α| lim

m→∞ 〈um, ∂αϕ〉= lim

m→∞ 〈∂αum, ϕ〉

= 〈uα, ϕ〉,

which implies u ∈ Lpk(Ω) and ∂αu = uα. Moreover, we have ∂αum → ∂αu in

Lp(Ω) for all |α| ≤ k, which means that um → u in Lpk(Ω) and hence Lp

k(Ω) iscomplete. �Exercise 1.5.11 (An embedding theorem). Prove that if s > k + n/2 and s ∈ Nthen Hs(Rn) ⊂ Ck(Rn) and the inclusion is continuous. Do also Exercise 2.6.17for a sharper version of this embedding.

1.5.3 Mollifiers

In Theorem 1.3.31 we saw that we can approximate quite irregular functions or(tempered) distributions by much more regular functions. The argument relied on


the use of Fourier analysis and worked well on Rn. Such a technique is very pow-erful, as could have been seen from the proof of Plancherel’s formula in Theorem1.3.13. On the other hand, when working in subsets of Rn we may be unable touse the Fourier transform (since for its definition we used the whole space Rn).Thus, we want to be able to approximate functions (or distributions) by smoothfunctions without using Fourier techniques. This turns out to be possible usingthe so-called mollification of functions.

Assume for a moment that we are in Rn again and let us first argue veryinformally. Let us first look at the Fourier transform of the convolution with aδ-distribution. Thus, for a function f we must have δ ∗ f = δf = f , if we use thatδ = 1. Taking the inverse Fourier transform we obtain the important identity

δ ∗ f = f,

which will be justified formally later. Now, if we take a sequence of smooth func-tions ηε approximating the δ-distribution, i.e., if ηε → δ in some sense as ε → 0,and if this convergence is preserved by the convolution, we should get

ηε ∗ f → δ ∗ f = f as ε→ 0.

Now, the convolution ηε ∗ f may be defined locally in Rn, and functions ηε ∗ f willbe smooth if ηε are, thus giving us a way to approximate f . We will now make thisargument precise. For this, we will deal in a straightforward manner by looking atthe limit of ηε ∗ f for a suitably chosen sequence of functions ηε, referring neitherto δ-distribution nor to the Fourier transform.

Definition 1.5.12 (Mollifiers). For an open set Ω ⊂ Rn and ε > 0 we defineΩε = {x ∈ Ω : dist(x, ∂Ω) > ε}. Let us define η ∈ C∞0 (Rn) by

η(x) =

{C e

1|x|2−1 , if |x| < 1,0, if |x| ≥ 1,

where the constant C is chosen so that∫

Rn η dx = 1. Such a function η is calleda (Friedrichs) mollifier. For ε > 0, we define

ηε(x) =1εn

η(x

ε

),

so that supp ηε ⊂ B(0, ε) and∫

Rn ηε dx = 1.

Let f ∈ L1loc(Ω). A mollification of f corresponding to η is a family f ε = ηε∗f

in Ωε, i.e.,

f ε(x) =∫

Ω

ηε(x− y)f(y) dy =∫

B(0,ε)

ηε(y)f(x− y) dy, for x ∈ Ωε.


Theorem 1.5.13 (Properties of mollifications). Let f ∈ L1loc(Ω). Then we have the

following properties.

(i) f ε ∈ C∞(Ωε).(ii) f ε → f almost everywhere as ε→ 0.

(iii) If f ∈ C(Ω), then f ε → f uniformly on compact subsets of Ω.

(iv) f ε → f in Lploc(Ω) for all 1 ≤ p <∞.

Proof. To show (i), we can differentiate f ε(x) =∫Ω

ηε(x − y)f(y) dy under theintegral sign and use the fact that f ∈ L1

loc(Ω). The proof of (ii) will rely on thefollowing

Theorem 1.5.14 (Lebesgue’s differentiation theorem). Let f ∈ L1loc(Ω). Then

limr→0

1|B(x, r)|

∫B(x,r)

|f(y)− f(x)| dy = 0 for a.e. x ∈ Ω.

Now, for all x for which the statement of Lebesgue’s differentiation theoremis true, we can estimate

|f ε(x)− f(x)| =∣∣∣∣∣∫

B(x,ε)

ηε(x− y)(f(y)− f(x)) dy

∣∣∣∣∣≤ ε−n

∫B(x,ε)

η

(x− y

ε

)|f(y)− f(x)| dy

≤ C1

|B(x, ε)|

∫B(x,ε)

|f(y)− f(x)| dy,

where the last expression goes to zero as ε → 0, by the choice of x. For (iii), letK be a compact subset of Ω. Let K0 ⊂ Ω be another compact set such that Kis contained in the interior of K0. Then f is uniformly continuous on K0 and thelimit in the Lebesgue differentiation theorem holds uniformly for x ∈ K. The sameargument as in (ii) then shows that f ε → f uniformly on K.

Finally, to show (iv), let us choose open sets U ⊂ V ⊂ Ω such that U ⊂ Vδ

and V ⊂ Ωδ for some small δ > 0. Let us show first that ||f ε||Lp(U) ≤ ||f ||Lp(V )

for all sufficiently small ε > 0. Indeed, for all x ∈ U , we can estimate

|f ε(x)| =∣∣∣∣∣∫

B(x,ε)

ηε(x− y)f(y) dy

∣∣∣∣∣≤

∫B(x,ε)

η1−1/pε (x− y)η1/p

ε (x− y)|f(y)| dy (Holder’s inequality)

≤(∫

B(x,ε)

ηε(x− y) dy

)1−1/p(∫B(x,ε)

ηε(x− y)|f(y)|p dy

)1/p

.


Since∫

B(x,ε)ηε(x− y) dy = 1, we get∫

U

|f ε(x)|p dx ≤∫

U

(∫B(x,ε)

ηε(x− y)|f(y)|p dy

)dx

≤∫

V

(∫B(y,ε)

ηε(x− y) dx

)|f(y)|p dy

=∫

V

|f(y)|p dy.

Now, let δ > 0 and let us choose g ∈ C(V ) such that ||f − g||Lp(V ) < δ (here weuse the fact that C(V ) is (sequentially) dense in Lp(V )). Then

||f ε − f ||Lp(U) ≤ ||f ε − gε||Lp(U) + ||gε − g||Lp(U) + ||g − f ||Lp(U)

≤ 2||f − g||Lp(V ) + ||gε − g||Lp(U)

< 2δ + ||gε − g||Lp(U).

Since gε → g uniformly on the closure of V by (iii), it follows that ||f ε−f ||Lp(U) ≤3δ for small enough ε > 0, completing the proof of (iv). �

As a consequence of Theorem 1.5.13 we obtain

Corollary 1.5.15. The space C∞(Ω) is sequentially dense in the space C0(Ω) ofall continuous functions with compact support in Ω. Also, C∞(Ω) is sequentiallydense in Lp

loc(Ω) for all 1 ≤ p <∞.

Exercise 1.5.16. Prove a simple but useful corollary of the Lebesgue differentiationtheorem, partly explaining its name:

Corollary 1.5.17 (Corollary of the Lebesgue differentiation theorem). Let f ∈L1

loc(Ω). Then

limr→0

1|B(x, r)|

∫B(x,r)

f(y) dy = f(x) for a.e. x ∈ Ω.

1.5.4 Approximation of Sobolev space functions

With the use of mollifications we can approximate functions in Sobolev spacesby smooth functions. We have a local approximation in localised Sobolev spacesLp

k(Ω)loc, a global approximation in Lpk(Ω), and further approximations dependent

on the regularity of the boundary of Ω. Although the set Ω is bounded, we stillsay that an approximation in Lp

k(Ω) is global if it works up to the boundary.

Proposition 1.5.18 (Local approximation by smooth functions). Assume that Ω ⊂Rn is open. Let f ∈ Lp

k(Ω) for 1 ≤ p < ∞ and k ∈ N ∪ {0}. Let f ε = ηε ∗ f inΩε be the mollification of f , ε > 0. Then f ε ∈ C∞(Ωε) and f ε → f in Lp

k(Ω)loc asε→ 0, i.e., f ε → f in Lp

k(K) as ε→ 0 for all compact K ⊂ Ω.


Proof. It was already proved in Theorem 1.5.3, (i), that f ε ∈ C∞(Ωε). Since fis locally integrable, we can differentiate the convolution under the integral signto get ∂αf ε = ηε ∗ ∂αu in Ωε. Now, let U be an open and bounded subset ofΩ containing K. Then by Theorem 1.5.3, (iv), we get ∂αf ε → ∂αf in Lp(U) asε→ 0, for all |α| ≤ k. Hence

||f ε − f ||pLp

k(U)=

∑|α|≤k

||∂αf ε − ∂αf ||pLp(U) → 0

as ε→ 0, proving the statement. �

Proposition 1.5.19 (Global approximation by smooth functions). Assume that Ω ⊂Rn is open and bounded. Let f ∈ Lp

k(Ω) for 1 ≤ p < ∞ and k ∈ N ∪ {0}. Thenthere is a sequence fm ∈ C∞(Ω)

⋂Lp

k(Ω) such that fm → f in Lpk(Ω).

Proof. Let us write Ω =⋃∞

j=1 Ωj , where Ωj = {x ∈ Ω : dist(x, ∂Ω) > 1/j}. LetVj = Ωj+3\Ωj+1 (this definition will be very important). Take also any open V0

with V0 ⊂ Ω so that Ω =⋃∞

j=0 Vj . Let χj be a partition of unity subordinate toVj , i.e., a family χj ∈ C∞0 (Vj) such that 0 ≤ χj ≤ 1 and

∑∞j=0 χj = 1 in Ω. Then

χjf ∈ Lpk(Ω) and supp(χjf) ⊂ Vj . Let us fix some δ > 0 and choose εj > 0 so small

that the function f j = ηεj∗ (χjf) is supported in Wj = Ωj+4\Ωj and satisfies

||f j −χjf ||Lpk(Ω) ≤ δ2−j−1 for all j. Let now g =

∑∞j=0 f j . Then g ∈ C∞(Ω) since

in any open set U in Ω there are only finitely many non-zero terms in the sum.Moreover, since f =

∑∞j=0 χjf , for each such U we have

||g − f ||Lpk(U) ≤

∞∑j=0

||f j − χjf ||Lpk(Ω) ≤ δ

∞∑j=0

12j+1

= δ.

Taking the supremum over all open subsets U of Ω, we obtain ||g − f ||Lpk(Ω) ≤ δ,


In general, there are many versions of these results depending on the set Ω,in particular on the regularity of its boundary. For example, we give here withoutproof the following

Further result. Let Ω be a bounded subset of Rn with C1 boundary. Let f ∈ Lpk(Ω)

for 1 ≤ p <∞ and k ∈ N ∪ {0}. Then there is a sequence fm ∈ C∞(Ω) such thatfm → f in Lp

k(Ω).

Finally, we use mollifiers to establish a smooth version of Urysohn’s lemmain Theorem A.12.11.

Theorem 1.5.20 (Smooth Urysohn’s lemma). Let K ⊂ Rn be compact and U ⊂ Rn

be open such that K ⊂ U . Then there exists f ∈ C∞0 (U) such that 0 ≤ f ≤ 1,f = 1 on K and supp f ⊂ U .

1.6. Interpolation 255

Proof. First we observe that the distance δ := dist(K, Rn\U) > 0 because K iscompact and Rn\U is closed. Let V := {x ∈ Rn : dist(x, K) < δ/3}. If η is theFriedrichs mollifier from Definition 1.5.12, then χ := ηδ/3 satisfies supp χ ⊂ {x ∈Rn : |x| ≤ δ/3} and

∫Rn χ(x) dx = 1. The desired function f can then be obtained

as f := IV ∗ χ, where IV is the characteristic function of the set V . We havethat f ∈ C∞ by Theorem 1.5.13 and supp f ⊂ U by Exercise3 1.4.29. We have0 ≤ f ≤ 1 from its definition, and f = 1 on K follows by a direct verification. �

1.6 Interpolation

The Riesz–Thorin interpolation theorem C.4.18 was already useful in establishingvarious inequalities in Lp (for example, it was used to prove the general Young’s in-equality for convolutions in Proposition 1.2.10, or the Hausdorff–Young inequalityin Corollary 1.3.14).

The aim of this section is to prove another very useful interpolation result: theMarcinkiewicz interpolation theorem. Here μ will stand for the Lebesgue measureon Rn.

Definition 1.6.1 (Distribution functions). For a function f : Rn → C we define itsdistribution function μf (λ) by

μf (λ) = μ{x ∈ Rn : |f(x)| ≥ λ}.

We have the following useful relation between the Lp-norm and the distribu-tion of a function.

Theorem 1.6.2. Let f ∈ Lp(Rn). Then we have the identity∫Rn

|f(x)|p dx = p

∫ ∞

0

μf (λ)λp−1 dλ.

Proof. Let us define a measure on R by setting

ν((a, b]) := μf (b)− μf (a)= −μ{x ∈ Rn : a < |f(x)| ≤ b}= −μ(|f |−1((a, b])).

By the standard extension property of measures we can then extend ν to all Borelsets E ⊂ (0,∞) by setting ν(E) = −μ(|f |−1(E)). We note that this definitionis well defined since |f | is measurable if f is measurable (Theorem C.2.9). Thenwe claim that we have the following property for, say, integrable functions φ :[0,∞)→ R: ∫

Rn

ϕ ◦ |f | dμ = −∫ ∞

0

ϕ(α) dν(α). (1.12)

3In fact, in this case the property supp f ⊂ supp IV + supp χ ⊂ V + Bδ/3(0) = {x ∈ Rn :d(x, K) ≤ 2δ/3} ⊂ U can be easily checked directly.


Indeed, if ϕ = χ[a,b] is a characteristic function of a set [a, b], i.e., equal to one on[a, b] and zero on its complement, then the definition of ν implies∫

Rn

χ[a,b] ◦ |f | dμ =∫

a<|f(x)|≤b

dμ = −∫ b

a

dν = −∫ ∞

0

χ[a,b] dν,

which verifies (1.12) for characteristic functions. By the linearity of integrals, wethen have (1.12) for finite linear combinations of characteristic functions and,consequently, for all integrable functions by the monotone convergence theorem(Theorem C.3.6). Now, taking ϕ(α) = αp in (1.12), we get∫

Rn

|f |p dμ = −∫ ∞

0

αp dν(α) = p

∫ ∞

0

αp−1μf (α) dα,

where we integrated by parts in the last equality. The proof is complete. �

Definition 1.6.3 (Weak type (p, p)). We say that operator T is of weak type (p, p)if there is a constant C > 0 such that for every λ > 0 we have

μ{x ∈ Rn : |Tu(x)| > λ} ≤ C||u||pLp

λp.

Proposition 1.6.4. If T is bounded from Lp(Rn) to Lp(Rn) then T is also of weaktype (p, p).

Proof. If v ∈ L1(Rn) then for all ρ > 0 we have a simple estimate

ρμ{x ∈ Rn : |v(x)| > ρ} ≤∫|v(x)|>ρ

|v(x)| dμ(x) ≤ ||v||L1 .

Now, if we take v(x) = |Tu(x)|p and ρ = λp, this readily implies that T is of weaktype (p, p). �

The following theorem is extremely valuable in proving Lp-continuity of op-erators since it reduces the analysis to a weaker type continuity only for two valuesof indices.

Theorem 1.6.5 (Marcinkiewicz’ interpolation theorem). Let r < q and assumethat operator T is of weak types (r, r) and (q, q). Then T is bounded from Lp(Rn)to Lp(Rn) for all r < p < q.

Proof. Let u ∈ Lp(Rn). For each λ > 0 we can define functions u1 and u2 byu1(x) = u(x) for |u(x)| > λ and by u2(x) = u(x) for |u(x)| ≤ λ, and to be zerootherwise. Then we have the identity u = u1 + u2 and estimates |u1|, |u2| ≤ |u|. Itfollows that

μTu(2λ) ≤ μTu1(λ) + μTu2(λ) ≤ C1||u1||rLr

λr+ C2

||u2||qLq

λq,

1.6. Interpolation 257

since T is of weak types (r, r) and (q, q). Therefore, we can estimate∫Rn

|Tu(x)|p dx = p

∫ ∞

0

λp−1μTu(λ) dλ

≤ C1p

∫ ∞

0

λp−1−r

(∫|u|>λ

|u(x)|r dx

)dλ

+ C2p

∫ ∞

0

λp−1−q

(∫|u|≤λ

|u(x)|q dx

)dλ.

Using Fubini’s theorem, the first term on the right-hand side can be rewritten as

∫ ∞

0

λp−1−r

(∫|u|>λ

|u(x)|r dx

)dλ

=∫ ∞

0

λp−1−r

(∫Rn

χ|u|>λ|u(x)|r dx

)dλ

=∫

Rn

|u(x)|r(∫ ∞

0

λp−1−rχ|u|>λ dλ

)dx

=∫

Rn

|u(x)|r(∫ |u(x)|

0

λp−1−r dλ

)dx

=1

p− r

∫Rn

|u(x)|r|u(x)|p−r dx

=1

p− r

∫Rn

|u(x)|p dx,

where χ|u|>λ is the characteristic function of the set {x ∈ Rn : |u(x)| > λ}.Similarly, we have

∫ ∞

0

λp−1−q

(∫|u|≤λ

|u(x)|q dx

)dλ =

1q − p

∫Rn

|u(x)|p dx,


As an important tool (which will not be used here so it is given just for theinformation) for proving various results of boundedness in L1(Rn) or of weak type(1, 1), we have the following fundamental decomposition of integrable functions.

Theorem 1.6.6 (Calderon–Zygmund covering lemma). Let u ∈ L1(Rn) and λ > 0.Then there exist v, wk ∈ L1(Rn) and there exists a collection of disjoint cubes Qk,


k ∈ N, centred at some points xk, such that the following properties are satisfied:

u = v +∞∑

k=1

wk, ||v||L1 +∞∑

k=1

||wk||L1 ≤ 3||u||L1 ,

suppwk ⊂ Qk,

∫Qk

wk(x) dx = 0,

∞∑k=1

μ(Qk) ≤ λ−1||u||L1 , |v(x)| ≤ 2nλ.

This theorem is one of the starting points of the harmonic analysis of opera-tors on Lp(Rn), but we will not pursue this topic here, and can refer to, e.g., [118]or [132] for many further aspects.

Chapter 2

Pseudo-differential Operators on Rn

The subject of pseudo-differential operators on Rn is well studied and there aremany excellent monographs on the subject, see, e.g., [27, 33, 55, 71, 101, 112,130, 135, 152], as well as on the more general subject of Fourier integral operators,microlocal analysis, and related topics in, e.g., [30, 56, 45, 81, 113]. Therefore, herewe only sketch main elements of the theory. In this chapter, we use the notation〈ξ〉 = (1 + |ξ|2)1/2.

2.1 Motivation and definition

We will start with an informal observation that if T is a translation invariant linearoperator on some space of functions on Rn, then we can write

T ( e2πix·ξ) = a(ξ) e2πix·ξ for all ξ ∈ Rn. (2.1)

Indeed, more explicitly, if T acts on functions of the variable y, we can writef(x, ξ) = T ( e2πiy·ξ)(x) = (Teξ)(x), where eξ(x) = e2πix·ξ. Let (τhf)(x) = f(x−h)be the translation operator by h ∈ Rn. We say that T is translation invariant ifTτh = τhT for all h. By our assumptions on T we get

f(x + h, ξ) = T ( e2πi(y+h)·ξ)(x) = e2πih·ξT ( e2πiy·ξ)(x) = e2πih·ξf(x, ξ).

Now, setting x = 0, we get f(h, ξ) = e2πih·ξf(0, ξ), so we obtain formula (2.1)with a(ξ) = f(0, ξ). In turn, this a(ξ) can be found from formula (2.1), yielding

a(ξ) = e−2πix·ξT ( e2πiy·ξ)(x).

If we now formally apply T to the Fourier inversion formula

f(x) =∫

Rn

e2πix·ξ f(ξ) dξ

260 Chapter 2. Pseudo-differential Operators on Rn

and use the linearity of T , we obtain

Tf(x) =∫

Rn

T ( e2πix·ξ)f(ξ) dξ =∫

Rn

e2πix·ξa(ξ)f(ξ) dξ.

This formula allows one to reduce certain properties of the operator T to prop-erties of the multiplication by the corresponding function a(ξ), called the symbolof T . For example, continuity of T on L2 would reduce to the boundedness ofa(ξ), composition of two operators T1 ◦ T2 would reduce to the multiplication oftheir symbols a1(ξ)a2(ξ), etc. Pseudo-differential operators extend this construc-tion to functions which are not necessarily translation invariant. In fact, as wesaw above we can always write a(x, ξ) := e−2πix·ξ(Teξ)(x), so that we would haveT ( e2πix·ξ) = e2πix·ξa(x, ξ). Consequently, reasoning as above, we could analo-gously arrive at the formula

Tf(x) =∫

Rn

e2πix·ξ a(x, ξ) f(ξ) dξ. (2.2)

Now, in order to avoid several rather informal conclusions in the arguments above,one usually takes the opposite route and adopts formula (2.2) as the definition ofthe pseudo-differential operator with symbol a(x, ξ). Such operators are then oftendenoted by Op(a), by a(X, D), or by Ta.

The simplest and perhaps most useful class of symbols allowing this ap-proach to work well is the following class denoted by Sm

1,0(Rn × Rn), or simply by

Sm(Rn × Rn).

Definition 2.1.1 (Symbol classes Sm(Rn × Rn)). We will say that a∈Sm(Rn×Rn)if a = a(x, ξ) is smooth on Rn × Rn and if the estimates

|∂βx∂α

ξ a(x, ξ)| ≤ Aαβ(1 + |ξ|)m−|α| (2.3)

hold for all α, β and all x, ξ ∈ Rn. Constants Aαβ may depend on a, α, β but not onx, ξ. The operator T defined by (2.2) is called the pseudo-differential operator withsymbol a. The class of operators of the form (2.2) with symbols from Sm(Rn × Rn)is denoted by Ψm(Rn × Rn) or by OpSm(Rn × Rn).

Remark 2.1.2. We will insist on writing Sm(Rn × Rn) and not abbreviating it toSm or even to Sm(Rn). The reason is that in Chapter 4 we will want to distin-guish between symbol class Sm(Tn × Rn) which will be 1-periodic symbols fromSm(Rn × Rn) and symbol class Sm(Tn × Zn) which will be the class of toroidalsymbols.Remark 2.1.3 (Symbols of differential operators). Note that for partial differentialoperators symbols are just the characteristic polynomials. One can readily seethat the symbol of the differential operator L =

∑|α|≤m aα(x)∂α

x is a(x, ξ) =∑|α|≤m aα(x)(2πiξ)α and a ∈ Sm(Rn × Rn) if the coefficients aα and all of their

derivatives are smooth and bounded on Rn.

2.1. Motivation and definition 261

Remark 2.1.4 (Powers of the Laplacian). For example, the symbol of the LaplacianL = ∂2

∂x21+· · ·+ ∂2

∂x2n

is −4π2|ξ|2 and it is an element of S2(Rn × Rn). Consequently,for any μ ∈ R, we can define the operators (1−L)μ as pseudo-differential operatorswith symbol (1 + 4π2|ξ|2)μ/2 ∈ Sμ(Rn × Rn).

Exercise 2.1.5. Let u ∈ C(Rn) satisfy |u(x)| ≤ C〈x〉N for some constants C,N ,where 〈x〉 = (1 + |x|2)1/2. Let k > N + n. Let us define

vk(φ) =∫

Rn

∫Rn

e−2πix·ξu(x)〈x〉−k(1− L)k/2φ(ξ) dx dξ,

where φ ∈ S(Rn). Prove that vk ∈ S ′(Rn). Prove that there is v ∈ S ′(Rn) suchthat v = vk for all k > N + n. Show that v = u.

We now proceed in establishing basic properties of pseudo-differential oper-ators.

Theorem 2.1.6 (Pseudo-differential operators on S(Rn)). Let a ∈ Sm(Rn × Rn)and f ∈ S(Rn). We define the pseudo-differential operator with symbol a by

a(X, D)f(x) :=∫

Rn

e2πix·ξa(x, ξ)f(ξ) dξ. (2.4)

Then a(X, D)f ∈ S(Rn).

Proof. First we observe that the integral in (2.4) converges absolutely. The same istrue for all of its derivatives with respect to x by Lebesgue’s dominated convergencetheorem (Theorem 1.1.4), which implies that a(X, D)f ∈ C∞(Rn). Let us shownow that in fact a(X, D)f ∈ S(Rn). Introducing the operator

Lξ = (1 + 4π2|x|2)−1(I − Lξ)

(where Lξ is the Laplace operator with respect to ξ-variables) with the propertyLξ e2πix·ξ = e2πix·ξ, integrating (2.2) by parts N times yields

a(X, D)f(x) =∫

Rn

e2πix·ξ(Lξ)N [a(x, ξ)f(ξ)] dξ.

From this we get |a(X, D)f(x)| ≤ CN (1+ |x|)−2N for all N , so a(X, D)f is rapidlydecreasing. The same argument applies to derivatives of a(X, D)f to show thata(X, D)f ∈ S(Rn). �

The following generalisation of symbol class Sm(Rn × Rn) is often useful:

Definition 2.1.7 (Symbol classes Smρ,δ(R

n × Rn)). Let 0 ≤ ρ, δ ≤ 1. We will saythat a ∈ Sm

ρ,δ(Rn × Rn) if a = a(x, ξ) is smooth on Rn × Rn and if

|∂βx∂α

ξ a(x, ξ)| ≤ Aαβ(1 + |ξ|)m−ρ|α|+δ|β| (2.5)


for all α, β and all x, ξ ∈ Rn. Constants Aαβ may depend on a, α, β but not onx, ξ. The operator T defined by (2.2) is called the pseudo-differential operator withsymbol a of order m and type (ρ, δ). The class of operators of the form (2.2) withsymbols from Sm

ρ,δ(Rn × Rn) is denoted by Ψm

ρ,δ(Rn × Rn) or by OpSm

ρ,δ(Rn × Rn).

Definition 2.1.8 (Symbol σA of operator A). If A ∈ Ψmρ,δ(R

n × Rn) we denote itssymbol by σA = σA(x, ξ). It is well defined in view of Theorem 2.5.6 later on,which also gives a formula for σA ∈ Sm

ρ,δ(Rn × Rn).

Exercise 2.1.9. Extend the statement of Theorem 2.1.6 to operators of type (ρ, δ).Namely, let 0 ≤ ρ, δ ≤ 1, and let a ∈ Sm

ρ,δ(Rn × Rn) and f ∈ S(Rn). Show that

a(X, D)f ∈ S(Rn).

The following convergence criterion will be useful in the sequel. It followsdirectly from the Lebesgue dominated convergence Theorem 1.1.4.

Proposition 2.1.10 (Convergence criterion for pseudo-differential operators). Sup-pose we have a sequence of symbols ak ∈ Sm(Rn × Rn) which satisfies the uniformsymbolic estimates

|∂βx∂α

ξ ak(x, ξ)| ≤ Aαβ(1 + |ξ|)m−|α|,

for all α, β, all x, ξ ∈ Rn, and all k, with constants Aαβ independent of x, ξ andk. Suppose that a ∈ Sm(Rn × Rn) is such that ak(x, ξ) and all of its derivativesconverge to a(x, ξ) and its derivatives, respectively, pointwise as k → ∞. Thenak(X, D)f → a(X, D)f in S(Rn) for any f ∈ S(Rn).

Exercise 2.1.11. Verify the details of the proof of Proposition 2.1.10.

Remark 2.1.12. More general families of pseudo-differential operators are intro-duced in, e.g., [55] and [130]. Yet Sm(Rn) = Sm

1,0(Rn) contained in the Hormander

classes Smρ,δ(R

n) is definitely the most important case, and [135], [130], [55], and[118] concentrate on it. Compressed information about pseudo-differential opera-tors and nonlinear partial differential equations can be found in [131]. The spectralproperties of pseudo-differential operators are considered in [112], and we have alsoleft out the matrix-valued pseudo-differential operators.Remark 2.1.13. The relation between operators and symbols can be also viewedas follows. Let u ∈ S(Rn) and fix s < −n/2. The function ψ : Rn → Hs(Rn),ψ(ξ) = eξ, where eξ(x) = e2πix·ξ, is Bochner-integrable (see [53]) with respect tou(ξ) dξ, and therefore

(Au)(x) =∫

Rn

e2πix·ξ σA(x, ξ) u(ξ) dξ

for symbols of order zero. The distribution Au can be viewed as a σA-weightedinverse Fourier transform of u. Unfortunately, the algebra of the finite order op-erators on the Sobolev scale is too large to admit fruitful symbol analysis, whilethe non-trivial restrictions by the symbol inequalities (2.3) yield a well-behavingsubalgebra.

2.2. Amplitude representation of pseudo-differential operators 263

2.2 Amplitude representation of

pseudo-differential operators

If we write out the Fourier transform in (2.4) as an integral, we obtain

a(X, D)f(x) =∫

Rn

∫Rn

e2πi(x−y)·ξ a(x, ξ) f(y) dy dξ. (2.6)

However, a problem with this formula is that the ξ-integral does not convergeabsolutely even for f ∈ S(Rn). To overcome this difficulty, one uses the idea toapproximate a(x, ξ) by symbols with compact support. To this end, let us fixsome γ ∈ C∞0 (Rn × Rn) such that γ = 1 near the origin. Let us now defineaε(x, ξ) = a(x, ξ)γ(εx, εξ). Then one can readily check that aε ∈ C∞0 (Rn × Rn)and that the following holds:

• if a ∈ Sm(Rn × Rn), then aε ∈ Sm(Rn × Rn) uniformly in 0 < ε ≤ 1 (thismeans that constants Aαβ in symbolic inequalities in Definition 2.1.1 may bechosen independent of 0 < ε ≤ 1);

• aε → a pointwise as ε → 0, uniformly in 0 < ε ≤ 1. The same is true forderivatives of aε and a.

It follows now from the convergence criterion Proposition 2.1.10 that

aε(X, D)f → a(X, D)f in S(Rn) as ε→ 0,

for all f ∈ S(Rn). Here a(X, D)f is defined as in (2.4). Now, formula (2.6) doesmake sense for aε ∈ C∞0 , so we may define the double integral in (2.6) as the limitin S(Rn) of aε(X, D)f , i.e., take

a(X, D)f(x) := limε→0

∫Rn

∫Rn

e2πi(x−y)·ξaε(x, ξ)f(y) dy dξ, f ∈ S(Rn).

Pseudo-differential operators on S ′(Rn). Recall that we can define the L2-adjointa(X, D)∗ of an operator a(X, D) by the formula

(a(X, D)f, g)L2 = (f, a(X, D)∗g)L2 ,

f, g ∈ S(Rn), where

(u, v)L2 =∫

Rn

u(x)v(x) dx

is the usual L2-inner product. From (2.6) and this formula we can readily calculatethat

a(X, D)∗g(y) = limε→0

∫Rn

∫Rn

e2πi(y−x)·ξ aε(x, ξ) g(x) dx dξ, g ∈ S(Rn).

With the same understanding of non-convergent integrals as in (2.6) and replacingx by z to eliminate any confusion, we can write

a(X, D)∗g(y) =∫

Rn

∫Rn

e2πi(y−z)·ξ a(z, ξ) g(z) dz dξ, g ∈ S(Rn). (2.7)


Exercise 2.2.1. As before, by integration by parts, check that a(X, D)∗ : S(Rn)→S(Rn) is continuous.

Definition 2.2.2 (Pseudo-differential operators on S ′(Rn)). Let u ∈ S ′(Rn). Wedefine a(X, D)u by the formula

(a(X, D)u)(ϕ) := u(a(X, D)∗ϕ) for all ϕ ∈ S(Rn).

Remark 2.2.3 (Consistency). We clearly have

a(X, D)∗ϕ(y) =∫

Rn

∫Rn

e2πi(z−y)·ξ a(z, ξ) ϕ(z) dz dξ,

so if u, ϕ ∈ S(Rn), we have the consistency in

(a(X, D)u)(ϕ) =∫

Rn

a(X, D)u(x)ϕ(x) dx = (a(X, D)u, ϕ)L2

= (u, a(X, D)∗ϕ)L2 =∫

Rn

u(x)a(X, D)∗ϕ(x) dx = u(a(X, D)∗ϕ).

Proposition 2.2.4. If a ∈ Sm(Rn × Rn) and u ∈ S ′(Rn) then a(X, D)u ∈ S ′(Rn).Moreover, operator a(X, D) : S ′(Rn)→ S ′(Rn) is continuous.

Proof. Indeed, let uk → u in S ′(Rn). Then we have

(a(X, D)uk)(ϕ) = uk(a(X, D)∗ϕ)→ u(a(X, D)∗ϕ) = (a(X, D)u)(ϕ),

so a(X, D)uk → a(X, D)u in S ′(Rn) and, therefore, a(X, D) : S ′(Rn) → S ′(Rn)is continuous. �Exercise 2.2.5. Let 0 ≤ ρ ≤ 1 and 0 ≤ δ < 1. Show that if a ∈ Sm

ρ,δ(Rn × Rn) and

u ∈ S ′(Rn) then a(X, D)u ∈ S ′(Rn), and that the operator a(X, D) : S ′(Rn) →S ′(Rn) is continuous.

2.3 Kernel representation of

pseudo-differential operators

Summarising Sections 2.1 and 2.2, we can write pseudo-differential operators indifferent ways:

a(X, D)f(x) =∫

Rn

e2πix·ξ a(x, ξ) f(ξ) dξ =∫

Rn

∫Rn

e2πi(x−y)·ξ a(x, ξ) f(y) dy dξ

=∫

Rn

∫Rn

e2πiz·ξ a(x, ξ) f(x− z) dz dξ =∫

Rn

k(x, z) f(x− z) dz

=∫

Rn

K(x, y) f(y) dy,

2.3. Kernel representation of pseudo-differential operators 265

with kernels

K(x, y) = k(x, x− y), k(x, z) =∫

Rn

e2πiz·ξ a(x, ξ) dξ.

Theorem 2.3.1 (Kernel of a pseudo-differential operator). Let a ∈ Sm(Rn × Rn).Then the kernel K(x, y) of pseudo-differential operator a(X, D) satisfies

|∂βx,yK(x, y)| ≤ CNβ |x− y|−N

for N > m + n + |β| and x = y. Thus, for x = y, the kernel K(x, y) is a smoothfunction, rapidly decreasing as |x− y| → ∞.

Proof. We notice that k(x, ·) is the inverse Fourier transform of a(x, ·). It followsthen that (−2πiz)α∂β

z k(x, z) is the inverse Fourier transform with respect to ξ ofthe derivative ∂α

ξ

[(2πiξ)βa(x, ξ)

], i.e.,

(−2πiz)α∂βz k(x, z) = F−1

ξ

(∂α

ξ

[(2πiξ)βa(x, ξ)

])(z).

Since (2πiξ)βa(x, ξ) ∈ Sm+|β|(Rn × Rn) is a symbol of order m+ |β|, we have that∣∣∂αξ

[(2πiξ)βa(x, ξ)

]∣∣ ≤ Cαβ〈ξ〉m+|β|−|α|.

Therefore, ∂αξ

[(2πiξ)βa(x, ξ)

]is in L1(Rn

ξ ) with respect to ξ, if |α| > m + n + |β|.Consequently, its inverse Fourier transform is bounded:

(−2πiz)α∂βz k(x, z) ∈ L∞(Rn

z )

for |α| > m + n + |β|. Since taking derivatives of k(x, z) with respect to x doesnot change the argument, this implies the statement of the theorem. �

As an immediate consequence of Theorem 2.3.1 we obtain the information onhow the singular support is mapped by a pseudo-differential operator (see Remark4.10.8 for more details):

Corollary 2.3.2 (Singular supports). Let T ∈ Ψm(Rn × Rn). Then for every u ∈S ′(Rn) we have

sing supp Au ⊂ sing supp u. (2.8)

Definition 2.3.3 (Local and pseudolocal operators). An operator A is called pseu-dolocal if the property (2.8) holds for all u. This is in analogy to the term “local”where an operator A is called local if

suppAu ⊂ supp u

for all u. By Corollary 2.3.2 every operator in Ψm(Rn × Rn) is pseudolocal. Theconverse is not true, as stated in Exercise 2.3.6.


Exercise 2.3.4 (Partial differential operators are local). Let A be a linear differ-ential operator

Af(x) =∑|α|≤m

aα(x)∂αx f(x)

with coefficients aα ∈ C∞(Rn), |α| ≤ m. Prove that supp Af ⊂ supp f , for allf ∈ C∞(Rn).

Exercise 2.3.5 (Peetre’s theorem). Prove the converse to Exercise 2.3.4 which isknown as Peetre’s theorem: if A : C∞(Ω)→ C∞(Ω) is a continuous linear operatorwhich is local, then A is a partial differential operator with smooth coefficients.

Exercise 2.3.6 (Pseudo–Peetre’s theorem?). Prove that we can not add the word“pseudo” to Peetre’s theorem. Namely, a pseudolocal linear continuous operatoron C∞(Rn) does not have to be a pseudo-differential operator.

We refer to Section 4.10 for a further exploration of these properties.

We will now discuss an important class of operators which are usually takento be negligible when one works with pseudo-differential operators. One of thereasons is that whenever they are applied to distributions they produce smoothfunctions, and so such operators can be neglected from the point of view of theanalysis of singularities. However, it is important to understand these operatorsin order to know exactly what we are allowed to neglect.

Definition 2.3.7 (Smoothing operators). We can define symbols of order −∞ bysetting S−∞(Rn × Rn) :=

⋂m∈R Sm(Rn × Rn), so that a ∈ S−∞(Rn × Rn) if

a ∈ C∞ and if|∂β

x∂αξ a(x, ξ)| ≤ AαβN (1 + |ξ|)−N

holds for all N , and all x, ξ ∈ Rn. The constants AαβN may depend on a, α, β, Nbut not on x, ξ. Pseudo-differential operators with symbols in S−∞ are calledsmoothing pseudo-differential operators.

Exercise 2.3.8. Show that the class S−∞(Rn × Rn) is independent of ρ and δ inthe sense that S−∞(Rn × Rn) =

⋂m∈R Sm

ρ,δ(Rn × Rn) for all ρ and δ.

Proposition 2.3.9. Let a ∈ S−∞(Rn × Rn). Then the integral kernel K of a(X, D)is smooth on Rn × Rn.

Proof. Since a(x, ·) ∈ L1(Rn), we immediately get k ∈ L∞(Rn). Moreover,

∂βx∂α

z k(x, z) =∫

Rn

e2πiz·ξ(2πiξ)α∂βxa(x, ξ) dξ.

Since (2πiξ)α∂βxa(x, ξ) is absolutely integrable, it follows that from the Lebesgue

dominated convergence theorem (Theorem 1.1.4) that ∂βx∂α

z k is continuous. Thisis true for all α, β, hence k, and then also K, are smooth. �

Let us write kx(·) = k(x, ·).

2.4. Boundedness on L2(Rn) 267

Corollary 2.3.10. Let a ∈ S−∞(Rn × Rn). Then kx ∈ S(Rn). We have

a(X, D)f(x) = (kx ∗ f)(x)

and, consequently, a(X, D)f ∈ C∞(Rn) for all f ∈ S ′(Rn).

We note that the convolution in the corollary is understood in the sense ofdistributions, see Section 1.4.2.

Proof of Corollary 2.3.10. Now Corollary 2.3.10 follows from the fact that for a ∈S−∞ we can write

a(X, D)f(x) = (f ∗ kx)(x)

with kx(·) = k(x, ·) ∈ S(Rn). So

(a(X, D)f)(x) = f(τxRkx).

If now f ∈ S ′(Rn), it follows that a(X, D)f ∈ C∞ because of the continuity off(τxRkx) and all of its derivatives with respect to x. �Exercise 2.3.11 (Non-locality). Let T be an operator defined by

Tf(x) =∫

Rn

K(x, y) f(y) dy,

with K ∈ C∞0 (Rn × Rn). Prove that T defines a continuous operator from S(Rn)to S(Rn) and from S ′(Rn) to S ′(Rn). For operators T as above with K ≡ 0, showthat we can never have the property supp Tf ⊂ supp f for all f ∈ C∞(Rn).

2.4 Boundedness on L2(Rn)

In this section we prove that pseudo-differential operators with symbols inS0(Rn × Rn) are bounded on L2(Rn). The corresponding result in Sobolev spaceswill be given in Theorem 2.6.11. First we prepare the following general result thatshows that in many similar situations we only have to verify the estimate for theoperator on a smaller space:

Proposition 2.4.1. Let A : S ′(Rn) → S ′(Rn) be a continuous linear operator suchthat A(S(Rn)) ⊂ L2(Rn) and such that there exists C for which the estimate

||Af ||L2(Rn) ≤ C||f ||L2(Rn) (2.9)

holds for all f ∈ S(Rn). Then A extends to a bounded linear operator from L2(Rn)to L2(Rn), and estimate (2.9) holds for all f ∈ L2(Rn), with the same constant C.

Proof. Indeed, let f ∈ L2(Rn) and let fk ∈ S(Rn) be a sequence of rapidly de-creasing functions such that fk → f in L2(Rn). Such a sequence exists becauseS(Rn) is dense in L2(Rn) (Exercise 1.3.33). Then by (2.9) applied to fk − fm wehave

||A(fk − fm)||L2(Rn) ≤ C||fk − fm||L2(Rn),


so Afk is a Cauchy sequence in L2(Rn). By the completeness of L2(Rn) (TheoremC.4.9) there is some g ∈ L2(Rn) such that Afk → g in L2(Rn). On the otherhand Afk → Af in S ′(Rn) because fk → f in L2(Rn) implies that fk → f inS(Rn) (Exercise 1.3.12). By the uniqueness principle in Proposition 1.3.5 we haveAf = g ∈ L2(Rn). Passing to the limit in (2.9) applied to fk, we get ||Af ||L2(Rn) ≤C||f ||L2(Rn), with the same constant C, completing the proof. �

There are different proofs of the L2-result. For the proof of Theorem 2.4.2below we follow [118] but an alternative proof based on the calculus will be alsogiven later in Section 2.5.4.

Theorem 2.4.2 (L2-boundedness of pseudo-differential operators). Let a ∈S0(Rn × Rn). Then a(X, D) extends to a bounded linear operator from L2(Rn)to L2(Rn).

Proof. First of all, we note that by a standard functional analytic argument inProposition 2.4.1 it is sufficient to show the boundedness inequality (2.9) for A =a(X, D) only for f ∈ S(Rn), with constant C independent of the choice of f .

The proof of (2.9) will consist of two parts. In the first part we establish itfor compactly supported (with respect to x) symbols and in the second part wewill extend it to the general case of a ∈ S0(Rn × Rn).

So, let us first assume that a(x, ξ) has compact support with respect to x.This will allow us to use the Fourier transform with respect to x, in particular theformulae

a(x, ξ) =∫

Rn

e2πix·λ a(λ, ξ) dλ, a(λ, ξ) =∫

Rn

e−2πix·λ a(x, ξ) dx,

with absolutely convergent integrals. We will use the fact that a(·, ξ) ∈ C∞0 (Rn) ⊂S(Rn), so that a(·, ξ) is in the Schwartz space in the first variable. Consequently,we have a(·, ξ) ∈ S(Rn) uniformly in ξ. To see the uniformity, we can notice that

(2πiλ)α a(λ, ξ) =∫

Rn

e−2πix·λ ∂αx a(x, ξ) dx,

and hence |(2πiλ)αa(λ, ξ)| ≤ Cα for all ξ ∈ Rn. It follows that

supξ∈Rn

|a(λ, ξ)| ≤ CN (1 + |λ|)−N

for all N . Now we can write

a(X, D)f(x) =∫

Rn

e2πix·ξ a(x, ξ) f(ξ) dξ

=∫

Rn

∫Rn

e2πix·ξ e2πix·λ a(λ, ξ) f(ξ) dλ dξ

=∫

Rn

(Sf)(λ, x) dλ,

2.4. Boundedness on L2(Rn) 269

where(Sf)(λ, x) = e2πix·λ(a(λ, D)f)(x).

Here a(λ, D)f is a Fourier multiplier with symbol a(λ, ξ) independent of x, so byPlancherel’s identity Theorem 1.3.13 we get

||a(λ, D)f ||L2 = ||F(a(λ, D)f)||L2 = ||a(λ, ·)f ||L2

≤ supξ∈Rn

|a(λ, ξ)| ||f ||L2 ≤ CN (1 + |λ|)−N ||f ||L2 ,

for all N ≥ 0. Hence we get

||a(X, D)f ||L2 ≤∫

Rn

||Sf(λ, ·)||L2 dλ

≤ CN

∫Rn

(1 + |λ|)−N ||f ||L2 dλ ≤ C||f ||L2 ,

if we take N > n.Now, to pass to symbols which are not necessarily compactly supported with

respect to x, we will use the inequality∫|x−x0|≤1

|a(X, D)f(x)|2 dx ≤ CN

∫Rn

|f(x)|2 dx

(1 + |x− x0|)N, (2.10)

which holds for every x0 ∈ Rn and for every N ≥ 0, with CN independent of x0

and dependent only on constants in the symbolic inequalities for a. Let us showfirst that (2.10) implies (2.9). Writing χ|x−x0|≤1 for the characteristic function ofthe set |x− x0| ≤ 1 and integrating (2.10) with respect to x0 yields∫

Rn

(∫Rn

χ|x−x0|≤1|a(X, D)f(x)|2 dx

)dx0

≤ CN

∫Rn

(∫Rn

|f(x)|2 dx

(1 + |x− x0|)N

)dx0.

Changing the order of integration, we arrive at

vol(B(1))∫

Rn

|a(X, D)f(x)|2 dx ≤ CN

∫Rn

|f(x)|2 dx,

which is (2.9). Let us now prove (2.10).Let us prove it for x0 = 0 first. We can write f = f1 + f2, where f1 and f2

are smooth functions such that |f1| ≤ |f |, |f2| ≤ |f |, and supp f1 ⊂ {|x| ≤ 3},supp f2 ⊂ {|x| ≥ 2}. We will do the estimate for f1 first. Let us fix η ∈ C∞0 (Rn)such that η(x) = 1 for |x| ≤ 1. Then η(a(X, D)) = (ηa)(X, D) is a pseudo-differential operator with a compactly supported in x symbol η(x)a(x, ξ), thus by


the first part we have∫{|x|≤1}

|a(X, D)f1(x)|2 dx =∫

Rn

|(ηa)(X, D)f1(x)|2 dx

≤C

∫Rn

|f1(x)|2 dx

≤C

∫{|x|≤3}

|f(x)|2 dx,

which is the required estimate for f1. Let us now do the estimate for f2. If |x| ≤ 1,then x ∈ supp f2, so we can write

a(X, D)f2(x) =∫{|x|≥2}

k(x, x− y)f2(y) dy,

where k is the kernel of a(X, D). Since |x| ≤ 1 and |y| ≥ 2, we have |x − y| ≥ 1and hence by Theorem 2.3.1 we can estimate

|k(x, x− z)| ≤ C1|x− y|−N ≤ C2|y|−N

for all N ≥ 0. Thus we can estimate

|a(X, D)f2(x)| ≤C1

∫{|y|≥2}

|f2(y)||y|N dy

≤C2

∫Rn

|f(y)|(1 + |y|)N

dy

≤C3

(∫Rn

|f(y)|2(1 + |y|)N

dy

)1/2

,

where we used the Cauchy-Schwarz inequality (Proposition 1.2.4) and that

C3 = C2

(∫Rn

1(1 + |y|)N

dy

)1/2

<∞

for N > n (Exercise 1.1.19). This in turn implies∫{|x|≤1}

|a(X, D)f2(x)|2 dx ≤ C

(∫Rn

|f(y)|2(1 + |y|)N

dy

)1/2

,

which is the required estimate for f2. These estimates for f1 and f2 imply (2.10)with x0 = 0. We note that constant C0 depends only on the dimension and on theconstants in symbolic inequalities for a.

Let us now show (2.10) with an arbitrary x0 ∈ Rn. Let us define ax0(x, ξ) =a(x − x0, ξ). Then we immediately see that estimate (2.10) for a(X, D) in theball {|x − x0| ≤ 1} is equivalent to the same estimate for ax0(X, D) in the ball{|x| ≤ 1}. Finally we note that since constants in symbolic inequalities for a andax0 are the same, we obtain (2.10) with constant CN independent of x0. Thiscompletes the proof of Theorem 2.4.2. �

2.5. Calculus of pseudo-differential operators 271

2.5 Calculus of pseudo-differential operators

In this section we establish formulae for the composition of pseudo-differentialoperators, adjoint operators and and discuss the transformation of symbols underchanges of variables.

2.5.1 Composition formulae

First we analyse compositions of pseudo-differential operators.

Theorem 2.5.1 (Composition of pseudo-differential operators). Let

a ∈ Sm1(Rn × Rn) and b ∈ Sm2(Rn × Rn).

Then there exists some symbol c ∈ Sm1+m2(Rn × Rn) such that

c(X, D) = a(X, D) ◦ b(X, D).

Moreover, we have the asymptotic formula

c ∼∑α

(2πi)−|α|

α!(∂α

ξ a)(∂αx b), (2.11)

which means that for all N > 0 we have

c−∑|α|<N

(2πi)−|α|

α!(∂α

ξ a)(∂αx b) ∈ Sm1+m2−N (Rn × Rn).

Proof. Let us assume first that all symbols are compactly supported, so we canchange the order of integration freely. Indeed, we can think of, say, symbol a(x, ξ)as of aε(x, ξ) = a(x, ξ)γ(εx, εξ), make all the calculations uniformly in 0 < ε ≤ 1,use that aε ∈ Sm1(Rn × Rn) uniformly in ε, and then take a limit as ε → 0. Letus now plug

b(X, D)f(y) =∫

Rn

∫Rn

e2πi(y−z)·ξ b(y, ξ) f(z) dz dξ

intoa(X, D)g(x) =

∫Rn

∫Rn

e2πi(x−y)·η a(x, ξ) g(y) dy dη,

yielding

a(X, D)(b(X, D)f)(x)

=∫

Rn

∫Rn

∫Rn

∫Rn

e2πi(x−y)·η a(x, η) e2πi(y−z)·ξ b(y, ξ) f(z) dz dξ dy dη

=∫

Rn

∫Rn

e2πi(x−z)·ξ c(x, ξ) f(z) dz dξ,


with

c(x, ξ) =∫

Rn

∫Rn

e2πi(x−y)·(η−ξ) a(x, η) b(y, ξ) dy dη

=∫

Rn

e2πix·(η−ξ) a(x, η) b(η − ξ, ξ) dη

=∫

Rn

e2πix·η a(x, ξ + η) b(η, ξ) dη. (2.12)

Here b is the Fourier transform with respect to the first variable, and we used that

(x− y) · η + (y − z) · ξ − (x− z) · ξ = (x− y) · (η − ξ).

Asymptotic formula. Let us assume first that b(x, ξ) has compact support in x(although if we think of b as bε, it will have a compact support, but the size of thesupport is not uniform in ε, so we need to treat operators with compact support ina uniform way first). Since b is compactly supported in x and b ∈ Sm2(Rn × Rn),its Fourier transform with respect to x is rapidly decaying uniformly in ξ, so wecan estimate

|b(η, ξ)| ≤ CM (1 + |η|)−M (1 + |ξ|)m2 ,

for all M ≥ 0. The Taylor’s expansion formula for a(x, ξ+η) in the second variablegives

a(x, ξ + η) =∑|α|<N

1α!

∂αξ a(x, ξ)ηα + RN (x, ξ, η),

with a remainder RN that we will analyse later. Plugging this formula into (2.12)and looking at terms in the sum, we get∫

Rn

e2πix·η[∂α

ξ a(x, ξ)ηα]b(η, ξ) dη = (2πi)−|α|∂α

ξ a(x, ξ)∂αx b(x, ξ),

so that we have

c(x, ξ) =(2πi)−|α|

α!

∑|α|<N

(∂α

ξ a(x, ξ))∂α

x b(x, ξ) +∫

Rn

e2πix·ηRN (x, ξ, η)b(η, ξ) dη.

Remainder. For the remainder in the Taylor series we have the estimate

|RN (x, ξ, η)|≤ CN |η|N ×max

{|∂α

ξ a(x, ζ)| : |α| = N, ζ on the line between ξ and ξ + η}.

Now, if |η| ≤ |ξ|/2, then points on the line between ξ and ξ + η are proportionalto ξ, so we can estimate

|RN (x, ξ, η)| ≤ CN |η|N (1 + |ξ|)m1−N (|η| ≤ |ξ|/2).

On the other hand, if N ≥ m1, we get the following estimate for all ξ and η:

|RN (x, ξ, η)| ≤ CN |η|N (N ≥ m1).


Using the estimate for b(η, ξ) and these two estimates, we get∣∣∣∣∫Rn

e2πix·η RN (x, ξ, η) b(η, ξ) dη

∣∣∣∣≤ CN,M (1 + |ξ|)m1+m2−N

∫Rn

(1 + |η|)−M |η|N dη

+ (1 + |ξ|)m2

∫|η|≥|ξ|/2

(1 + |η|)−M |η|N dη.

Taking M large enough, we can estimate both terms on the right-hand side byC(1 + |ξ|)m1+m2−N . Making an estimate for ∂α

x ∂βξ RN in a similar way, we get∣∣∣∣∫

Rn

e2πix·η[∂α

x ∂βξ RN (x, ξ, η)

]b(η, ξ) dη

∣∣∣∣ ≤ C(1 + |ξ|)m1+m2−N−|β|,

implying the statement of the theorem for b compactly supported with respect to x.

General symbols. We note that it is sufficient to have the asymptotic formula andan estimate for the remainder for x near some fixed point x0, uniformly in x0. Letχ ∈ C∞0 (Rn) be such that supp η ⊂ {x : |x− x0| ≤ 2} and such that η(x) = 1 for|x− x0| ≤ 1. Let us decompose

b = χb + (1− η)b =: b1 + b2.

Since the symbol b1 = χb is compactly supported with respect to x, the com-position formula for a(X, D) ◦ b1(X, D) is given by the theorem, and it is equalto the claimed series for a(X, D) ◦ b(X, D) near x0. We now have to show thatthe operator a(X, D) ◦ b2(X, D) is smoothing, i.e., its symbol c2(x, ξ) is of order−∞, and does not change the asymptotic formula for the composition. Indeed, wealready know that the symbol of operator a(X, D) ◦ b2(X, D) is given by

c2(x, ξ) =∫

Rn

∫Rn

e2πi(x−y)·(η−ξ) a(x, η) b2(y, ξ) dy dη,

and we claim that

|c2(x, η)| ≤ CN (1 + |ξ|)m1+m2−N for all N ≥ 0, |x− x0| ≤12.

In the above integral for c2 we can integrate by parts to derive various properties ofits decay. For example, we can integrate by parts with respect to the η-LaplacianΔη using the identity

ΔN1η e2πi(x−y)·(η−ξ) = (−4π2)N1 |x− y|2N1 e2πi(x−y)·(η−ξ),

to see that in the integral we can replace a(x, η) by ΔN1η a(x,η)

(−4π2)N1 |x−y|2N1 , i.e.,

c2(x, ξ) =∫

Rn

∫Rn

e2πi(x−y)·(η−ξ)ΔN1

η a(x, η)(−4π2)N1 |x− y|2N1

b2(y, ξ) dy dη.


Here |x− x0| ≤ 1/2 and y ∈ supp(1− χ) implies |x− y| ≥ 1/2, so the integrationby parts is well defined. We can also integrate by parts with respect to y-LaplacianΔy using the identity

(1−Δy)N2 e2πi(x−y)·(η−ξ) = (1 + 4π2|ξ − η|2)N2 e2πi(x−y)·(η−ξ).

Moreover, we can use that

1 + |ξ| ≤ 1 + |ξ − η|+ |η| ≤ (1 + |ξ − η|)(1 + |η|),and hence

1(1 + |ξ − η|)2N2

≤ (1 + |η|)2N2

(1 + |ξ|)2N2.

Thus, integrating by parts with respect to y and using this estimate together withsymbolic estimates for a and b2, we get

|c2(x, ξ)| ≤ C

∫Rn

∫Rn

(1 + |η|)m1−2N1

(1 + |x− y|)2N1

(1 + |ξ|)m2

(1 + |ξ − η|)2N2dη

≤ C

∫Rn

(1 + |η|)m1−2N1+2N2(1 + |ξ|)m2−2N2 dy dη

≤ C(1 + |ξ|)m1+m2−N

∫Rn

(1 + |η|)N−2N1 dη,

if we take N = m1 − 2N2. Now, taking large N2 and 2N1 > N + n, we obtainthe desired estimate for c2(x, ξ). Similar estimates can be done for ∂α

x ∂βξ c2(x, ξ)

simply using symbolic inequalities for a and b2, so we obtain that c2 ∈ S−∞ for|x− x0| ≤ 1/2.

Finally, we notice that (in the general case) we have never used any infor-mation on the size of the support of symbols, so all constants depend only onthe constants in symbolic inequalities. Thus, they remain uniformly bounded in0 < ε ≤ 1 for the composition cε(X, D) = aε(X, D) ◦ bε(X, D), and we havecε ∈ Sm1+m2 with symbolic constants uniform in ε. Moreover, the asymptotic for-mula is satisfied uniformly in ε. Since cε → c pointwise as ε→ 0, we can concludethat c ∈ Sm1+m2 , that c(X, D) = a(X, D) ◦ b(X, D), and that the asymptoticformula of the theorem holds. �Exercise 2.5.2. Adopt the proof to get the following (ρ, δ) version of the composi-tion formula in Theorem 2.5.1. Namely, let 0 ≤ δ < ρ ≤ 1 and let

a ∈ Sm1ρ,δ (Rn × Rn) and b ∈ Sm2

ρ,δ (Rn × Rn).

Then there exists a symbol c ∈ Sm1+m2ρ,δ (Rn × Rn) such that c(X, D) = a(X, D) ◦

b(X, D). Moreover, we have asymptotic formula (2.11) which now means that forall N > 0 we have

c−∑|α|<N

(2πi)−|α|

α!(∂α

ξ a)(∂αx b) ∈ S

m1+m2−(ρ−δ)Nρ,δ (Rn × Rn).


Exercise 2.5.3. Let p(x, ξ) =∑|α|≤m aα(x)ξα, aα ∈ C∞(Rn), and let p(X, D) be

the differential operator with symbol p(x, ξ).

(i) Prove that

p(X, D)(f(x)g(x)) =∑α

1α!

[p(α)(X, D)f(x)][Dαg(x)]

for all f, g ∈ S, where p(α)(X, D) is a differential operator with symbolp(α)(x, ξ) = ∂α

ξ p(x, ξ).

(ii) Explain how to deduce the composition formula in Theorem 2.5.1 for differ-ential operators from (i).

(iii) Let p(x, D) ∈ Ψm(Rn × Rn) be now a general pseudo-differential operator oforder m. Let f ∈ L2(Rn) and g ∈ S(Rn). Prove that

p(X, D)(f(x)g(x))−∑|α|≤m

1α!

[p(α)(X, D)f(x)][Dαg(x)] ∈ H1(Rn).

We have already seen that when taking the adjoint of a pseudo-differentialoperator, we get the symbol which depends on the “wrong” set of variables: (y, ξ)instead of (x, ξ). Nevertheless, we want the adjoint to be a pseudo-differentialoperator as well. For this we need to study operators with symbols depending onall combinations of variables. This leads to the definition of an amplitude.

Definition 2.5.4 (Amplitudes). We will write c = c(x, y, ξ) ∈ Am(Rn) if c ∈C∞(Rn × Rn × Rn) and if∣∣∂γ

y ∂βx∂α

ξ c(x, y, ξ)∣∣ ≤ Cα,β,γ(1 + |ξ|)m−|α|

holds for all x, y, ξ ∈ Rn and all multi-indices α, β, γ. Constants Cα,β,γ may dependon c, α, β, γ but not on x, y, ξ. The corresponding operator c(X, Y,D) is definedby

c(X, Y, D)f(x) :=∫

Rn

∫Rn

e2πi(x−y)·ξ c(x, y, ξ) f(y) dy dξ. (2.13)

The operator c(X, Y,D) is called an amplitude operator with amplitude c =c(x, y, ξ). Analogously, for 0 ≤ ρ ≤ 1 and 0 ≤ δ < 1 we will write c = c(x, y, ξ) ∈Am

ρ,δ(Rn) if c ∈ C∞(Rn × Rn × Rn) and if∣∣∂γ

y ∂βx∂α

ξ c(x, y, ξ)∣∣ ≤ Cα,β,γ(1 + |ξ|)m−ρ|α|+δ(|β|+|γ|)

holds for all x, y, ξ ∈ Rn and all multi-indices α, β, γ. The class of operatorsc(X, Y,D) with c ∈ Am

ρ,δ(Rn) will be denoted by Op(Am

ρ,δ(Rn)).


Remark 2.5.5. As in (2.6), we can justify formula (2.13) by considering

cε(x, y, ξ) = c(x, y, ξ)γ(εy, εξ),

with the same γ as was considered there. Then cε → c pointwise (also with thepointwise convergence of derivatives), uniformly in Am(Rn) for 0 < ε ≤ 1, socε(X, Y,D)f → c(X, Y,D)f in S(Rn) for all f ∈ S(Rn), by a suitable extension ofProposition 2.1.10. Thus, c(X, Y,D) is well defined and continuous as an operatorfrom S(Rn) to S(Rn) if c ∈ Am(Rn) or if c ∈ Am

ρ,δ(Rn) for 0 ≤ ρ ≤ 1 and

0 ≤ δ < 1.

We are now in position to justify our starting point formula (2.1).

Theorem 2.5.6 (Quantization of operators). A continuous linear operator T :S ′(Rn)→ S ′(Rn) is a pseudo-differential operator with symbol a(x, ξ) if and only if

a(x, ξ) = e−2πix·ξT ( e2πix·ξ) ∈ S∞(Rn × Rn). (2.14)

In particular, a pseudo-differential operator T ∈ Ψm(Rn × Rn) defines its symbola ∈ Sm(Rn × Rn) uniquely, so that T = a(X, D). The symbol a(x, ξ) is defined bythe formula (2.14).

The notation used in the statement is a useful abbreviation for

e−2πix·ξT ( e2πix·ξ) = e−2πix·ξ (T ( e2πiy·ξ)

)(x)

= e−2πix·ξ(T ( e2πi〈·,ξ〉)

)(x)

= e−ξ(x)(Teξ)(x),

where eξ(x) = e2πix·ξ.Formula (2.14) for the symbol can be justified either in S ′(Rn) or as the limit

in the expressione−2πix·ξT ( e2πix·ξϕε)→ a(x, ξ),

as ϕε → 1 for a family of ϕε ∈ C∞0 (Rn). In Theorem 4.1.4 we will prove this onthe torus and in Theorem 10.4.4 we will give a proof of this statement on generalLie groups so now we leave it as

Exercise 2.5.7. Prove Theorem 2.5.6 and formula (2.14) for the symbol.

Theorem 2.5.8 (Symbols of amplitude operators). Let c ∈ Am(Rn) be an ampli-tude. Then there exists a symbol a ∈ Sm(Rn × Rn) such that a(X, D) = c(X, Y,D).Moreover, the asymptotic expansion for a is given by

a(x, ξ)−∑|α|<N

(2πi)−|α|

α!∂α

ξ ∂αy c(x, y, ξ)|y=x ∈ Sm−N (Rn × Rn), (2.15)

for all N ≥ 0.


Proof. The proof is similar to the proof of the composition formula in Theorem2.5.1. As in that proof, we first formally conclude that we must have

a(x, ξ) = e−2πix·ξ T ( e2πix·ξ)

=∫

Rn

∫Rn

e−2πix·ξ e2πi(x−y)·η c(x, y, η) e2πiy·ξ dy dη

=∫

Rn

c(x, y, η) e2πi(x−y)·(η−ξ) dy dη

=∫

Rn

c(x, η, ξ + η) e2πix·η dη,

where c = Fyc is the Fourier transform of c with respect to y and where we usedthe change of variables η �→ η + ξ. By the same justification as in Theorem 2.5.1,we may work with amplitudes compactly supported in y, and make sure that allour arguments do not depend on the size of the support of c in y. Taking theTaylor’s expansion of c(x, η, ξ + η) at ξ, we obtain

c(x, η, ξ + η) =∑|α|<N

1α!

∂αξ c(x, η, ξ)ηα + RN (x, η, ξ),

where, as before, we can show that the remainder RN (x, η, ξ) satisfies estimates

|RN (x, η, ξ)| ≤ A|η|N (1 + |η|)−M (1 + |ξ|)m−N

for large M and for 2|η| ≤ |ξ|, and

|RN (x, η, ξ)| ≤ A|η|N (1 + |η|)−M for all large M and N.

The last estimate is used in the region 2|η| ≥ |ξ| and, similar to the proof of thecomposition formula in Theorem 2.5.1, we can complete the proof in the case ofc(x, y, ξ) compactly supported in y.

To treat the general case where c(x, y, ξ) is not necessarily compactly sup-ported in y, we reduce the analysis to the case when c(x, y, ξ) vanishes for y awayfrom y = x. For this, we can use the same argument as in the proof of the com-position formula in Theorem 2.5.1, where the estimate for the remainder from theintegration by parts argument becomes∫

Rn

∫Rn

(1 + |η|)m−2N1

1 + |η − ξ|)2N2

1(1 + |x− y|)2N1

dy dη = O((1 + |ξ|)m−N ),

for large N1, N2. The proof is complete. �Exercise 2.5.9. Justify the following alternative argument. We know by Theorem2.5.6 that the symbol of T can be defined by the formula

a(x, ξ) = e−2πix·ξ (T ( e2πiy·ξ)

)(x)


but we do not know that a ∈ Sm(Rn × Rn). Let now T = c(X, Y,D) be theoperator with amplitude symbol c. Then Theorem 2.5.8 says that we can write

c(X, Y,D)u(x) =∫

Rn

∫Rn

e2πi(x−y)·ξ a(x, ξ) u(y) dy dξ,

for all u ∈ C∞0 (Rn). To show this, we write the Fourier inversion formula

u(y) =∫

Rn

e2πiy·ξ u(ξ) dξ,

and, justifying the application of T to it and using the formula for the symbolfrom Theorem 2.5.6, we get

(Tu)(x) =∫

Rn

T ( e2πiy·ξ)(x) u(ξ) dξ =∫

Rn

e2πix·ξ a(x, ξ) u(ξ) dξ.

For the asymptotic expansion, we write

(T e2πi〈·,ξ〉)(x) =∫

Rn

∫Rn

e2πi(x−y)·η c(x, y, η) e2πiy·ξ dy dη

= e2πix·ξ∫

Rn

∫Rn

e−2πiy·η c(x, x + y, ξ + η) dy dη,

where we changed variables y �→ y + x and η �→ η + ξ, as well as recalculated thephase in new variables as

(x− (y + x)) · (η + ξ) + (y + x) · ξ = −y · η + x · ξ.

Now, using the Taylor expansion and estimates for the remainder, we can obtainTheorem 2.5.8 again.

Exercise 2.5.10. Prove the (ρ, δ) version of Theorem 2.5.8. Namely, if c ∈ Amρ,δ(R

n)is an amplitude and 0 ≤ δ < ρ ≤ 1, then there exists a symbol a ∈ Sm

ρ,δ(Rn × Rn)

such that a(X, D) = c(X, Y,D). Moreover, the asymptotic expansion for a is givenby formula (2.15), but with Sm−N (Rn × Rn) replaced by S

m−(ρ−δ)Nρ,δ (Rn × Rn).

Exercise 2.5.11. Let

Af(x) =∫

Rn

∫Rn

e2πi(x−y)·ξ c(x, y, ξ) f(y) dy dξ

be an operator with amplitude c ∈ Am(Rn). Prove that if f is smooth in someneighbourhood of a point x ∈ Rn, then Au is also smooth in some neighbourhood ofthe same point x. How can one reformulate this in terms of the singular supports?

Exercise 2.5.12. Let Bu(x) =∫

Rn K(x, y) u(y) dy be an operator from C∞0 (Rn)to C∞(Rn) with kernel K = K(x, y) ∈ S(Rn ×Rn). Show that B is an amplitudeoperator of the form (2.13) with amplitude c ∈ A−∞(Rn).


We now use Theorem 2.5.8 to obtain the asymptotic formula for the adjointof a pseudo-differential operator.

Theorem 2.5.13 (Adjoint operator). Let a ∈ Sm(Rn × Rn). Then there exists asymbol a∗ ∈ Sm(Rn × Rn) such that a(X, D)∗ = a∗(X, D), where a(X, D)∗ is theL2–adjoint operator of a(X, D). Moreover, we have the asymptotic expansion

a∗(x, ξ)−∑|α|<N

(2πi)−|α|

α!∂α

ξ ∂αx a(x, ξ) ∈ Sm−N (Rn × Rn),

for all N ≥ 0, where a(x, ξ) is the complex conjugate of a(x, ξ).

Proof. As we have already calculated in (2.7), we have a∗(X, D) = c(X, Y,D),which is an operator with amplitude c(x, y, ξ) = a(y, ξ). Applying Theorem 2.5.8,we obtain the statement of Theorem 2.5.13. �

Exercise 2.5.14. Prove the (ρ, δ) version of Theorem 2.5.13 for 0 ≤ δ < ρ ≤ 1.Namely, show that if a(X, D) ∈ Ψm

ρ,δ(Rn × Rn) then a(X, D)∗ ∈ Ψm

ρ,δ(Rn × Rn)

and the asymptotic formula for its symbol is as in Theorem 2.5.13, withSm−N (Rn × Rn) replaced by S

m−(ρ−δ)Nρ,δ (Rn × Rn).

Definition 2.5.15 (Transpose operators). For u ∈ S ′(Rn), define the transpose T t

of a linear operator T : S(Rn)→ S(Rn) by the formula

(T tu)(ϕ) := u(Tϕ)


Exercise 2.5.16 (Properties of the transpose). Prove that if T is a continuouslinear operator from S(Rn) → S(Rn), then T t is a continuous linear operatorS ′(Rn) → S ′(Rn). In particular, show that if u ∈ S ′(Rn) then T tu ∈ S ′(Rn).Prove that the adjoint operator T ∗ satisfies

(T ∗u)(ψ) = u(Tψ)

for all ψ ∈ S(Rn). Then also prove that

T tu = T ∗u.

Here u ∈ S ′(Rn) is the complex conjugate of u, defined by u(ϕ) = u(ϕ), for allϕ ∈ S(Rn), and u(ϕ) is the complex conjugate of u(ϕ) ∈ C.

Remark 2.5.17. Recalling the canonical identification of functions with distribu-tions in Remark 1.3.7 we see that if u ∈ S(Rn), then the transpose T t of T satisfiesthe relation ∫

Rn

(T tu)(x) ϕ(x) dx =∫

Rn

u(x) (Tϕ)(x) dx.


This is the difference with the adjoint operator T ∗ which would satisfy∫Rn

(T ∗u)(x) ϕ(x) dx =∫

Rn

u(x) (Tϕ)(x) dx.

Exercise 2.5.18 (Asymptotic expansion of the transpose). Let now T be a pseudo-differential operator with symbol a ∈ Sm(Rn × Rn). Prove that T t is also a pseudo-differential operator of order m, with symbol having an asymptotic expansion

symbol of T ta ∼

∑α

(2πi)−|α|

α!∂α

ξ ∂αx [a(x,−ξ)] .

Formulate and prove the (ρ, δ)-version of this result.

We finish with a brief discussion of properly supported operators.Remark 2.5.19 (Proper mappings). Let X ⊂ Rn and Y ⊂ Rm be open sets. Werecall that if a mapping Φ : X → Y is continuous then the images of compact setsin X under Φ are compact in Y . This is not true in general if instead of “images”we take “preimages”. Thus, a continuous mapping Φ : X → Y is called proper iffor every compact set K ⊂ Y its preimage Φ−1(K) = {x ∈ X : Φ(x) ∈ K} ⊂ X iscompact.

Definition 2.5.20 (Properly supported operators). A pseudo-differential operatorT is called properly supported if the projections π1(x, y) := x, π2(x, y) := y fromsupp K to Rn are proper mappings. Here suppK ⊂ Rn×Rn is the support of theintegral kernel of T .

Exercise 2.5.21. Prove that T is properly supported if and only if T ∗ is properlysupported.

Proposition 2.5.22. Let A ∈ Ψm(Rn × Rn). Then there exists a properly supportedamplitude operator B ∈ OpAm(Rn) such that A−B ∈ Ψ−∞(Rn × Rn).

Proof. Let A ∈ Ψm(Rn × Rn) be an operator with symbol σA. By Theorem 2.3.1the singular support of its integral kernel K(x, y) is contained in the diagonal {x =y}. Let Ω be a neighbourhood of the diagonal {x = y} of Rn × Rn such that π1 :Ω→ Rn and π2 : Ω→ Rn are proper. Let χ ∈ C∞(Rn×Rn) be such that χ(x, x) =1 for all x ∈ Rn and such that suppχ = Ω. Let B be an operator with kernelχK, i.e., Bf(x) :=

∫Rn χ(x, y) K(x, y) f(y) dy. Then B is properly supported.

Moreover, B is an amplitude operator with amplitude b(x, y, ξ) = χ(x, y)σA(x, ξ) ∈Am(Rn) and A − B ∈ Ψ−∞(Rn × Rn) because its integral kernel is smooth (seeExercise 2.5.12). �Corollary 2.5.23. Because every amplitude operator is a pseudo-differential oper-ator by Theorem 2.5.8 we see that a pseudo-differential operator differs from aproperly supported operator by a smoothing operator: if A ∈ Ψm(Rn × Rn), thenthere exists a properly supported pseudo-differential operator B ∈ Ψm(Rn × Rn)such that A−B ∈ Ψ−∞(Rn × Rn).


Remark 2.5.24. The advantage of working with properly supported operators isthat they take compactly supported functions to compactly supported functions(or distributions). For example, a linear continuous operator T : C∞0 (Rn) →C∞0 (Rn) can be extended to an operator D′(Rn)→ D′(Rn) by duality.

2.5.2 Changes of variables

It is clear from the definition of the symbol class Sm(Rn × Rn) that it is locallyinvariant under smooth changes of variables, i.e., if we take a local change of thevariable x in the symbol from symbol class Sm(Rn × Rn), it will still belong tothe same symbol class Sm(Rn × Rn). We will now investigate what happens withpseudo-differential operators when we make a change of variable in space.

Let U, V ⊂ Rn be open bounded sets and let τ : V → U be a surjectivediffeomorphism. For f ∈ C∞0 (V ), we define its pullback by the change of variablesτ by

(τf)(x) = f(τ−1(x)).

It easily follows that the new function satisfies τf ∈ C∞0 (U). Let now a ∈Sm(Rn × Rn) be a symbol of order m with compact support with respect to x,and assume that this support is contained in U . Then we will show that thereexists a symbol b ∈ Sm(Rn × Rn) with compact support with respect to x whichis contained in V such that τ−1a(X, D)τ = b(X, D). In other words, we have[

a(X, D)(f ◦ τ−1)](τ(x)) = b(X, D)f(x)

for all f ∈ C∞0 (V ) and all x ∈ V . More precisely, we have the following expressionfor the “main part” of b:

Proposition 2.5.25. We have b(x, ξ) = a(τ(x),[

∂τ∂x

]′ξ) modulo Sm−1(Rn × Rn),

where[

∂τ∂x

]′=

((∂τ∂x

)T)−1

.

Proof. We can write

(τ−1a(X, D)τf)(x)

=∫

Rn

∫Rn

e2πi(τ(x)−y)·ξ a(τ(x), ξ) f(τ−1(y)) dy dξ

=∫

Rn

∫Rn

e2πi(τ(x)−τ(y))·ξ a(τ(x), ξ) f(y) |det∂τ

∂y| dy dξ, (2.16)

where we changed variables y �→ τ(y) to get the last equality. Now we will arguethat the main contribution in this integral comes from variables y close to x.Indeed, let us insert a cut-off function χ(x, y) in the integral, where χ is a smoothfunction supported in the set where |y − x| is small and where χ(x, x) = 1 for allx. The remaining integral∫

Rn

∫Rn

e2πi(τ(x)−τ(y))·ξ (1− χ(x, y)) a(τ(x), ξ) f(y) |det∂τ

∂y| dy dξ


has a smooth kernel since we can integrate by parts any number of times withrespect to ξ to see that the symbol of this operator belongs to symbol classesS−N (Rn × Rn) for all N ≥ 0. Let us now analyse what happens for y close to x.By the mean value theorem, for y sufficiently close to x, we have

τ(x)− τ(y) = Lx,y(x− y), (2.17)

where Lx,y is an invertible linear mapping which is smooth in x and y, and satisfies

Lx,x =∂τ

∂x(x).

Using (2.17) in (2.16), we get∫Rn

∫Rn

e2πiLx,y(x−y)·ξ χ(x, y) a(τ(x), ξ) |det∂τ

∂y| f(y) dy dξ

=∫

Rn

∫Rn

e2πi(x−y)·ξ χ(x, y) a(τ(x), L′x,yξ) |det∂τ

∂y| |det L−1

x,y| f(y) dy dξ,

where we changed variables LTx,yξ �→ ξ and where LT

x,y is the transpose matrix ofLx,y and L′x,y = (LT

x,y)−1. Thus, we get an operator with the amplitude c definedby

c(x, y, ξ) = χ(x, y) a(τ(x), L′x,yξ) |det∂τ

∂y| |det L−1

x,y|.

Applying the asymptotic expansion in Theorem 2.5.8, we see that the first termof this expansion is given by

c(x, y, ξ)|y=x = χ(x, x) a(τ(x), L′x,xξ) |det∂τ

∂x| |det L−1

x,x|

= a(τ(x),[∂τ

∂x

]′ξ),


2.5.3 Principal symbol and classical symbols

We see in Proposition 2.5.25 that the equivalence class modulo Sm−1(Rn × Rn)has some meaning for the changes of variables. In fact, we may notice that thetransformation

(x, ξ) �→ (τ(x),[∂τ

∂x

]′ξ)

is the same as the change of variables in the cotangent bundle T ∗Rn of Rn which isinduced by the change of variables x �→ τ(x) in Rn. This observation allows one tomake an invariant geometric interpretation of the class of symbols in Sm(Rn × Rn)modulo terms of order Sm−1(Rn × Rn). We note that we can use the asymptotic


expansion for amplitudes in the proof of Proposition 2.5.25 to find also lower-order terms in the asymptotic expansion of b(x, ξ), but these terms will not havesuch nice invariant interpretation. Without going into detail, let us just mentionthat Proposition 2.5.25 allows one to introduce a notion of a principal symbol of apseudo-differential operator with symbol in Sm(Rn × Rn) as the equivalence classof this symbol modulo the subclass Sm−1(Rn × Rn), and this principal symbol isdefined on the cotangent bundle of Rn. This construction can be further carriedout on manifolds leading to many remarkable applications, in particular to thosein geometry and index theory.

We will not pursue this path further in this chapter, but we will clarify thenotion of such equivalent classes for the so-called classical symbols which forma class of symbols that plays a very important role in applications to partialdifferential equations. First, we define homogeneous functions/symbols.

Definition 2.5.26 (Homogeneous symbols). We will say that a symbol

ak = ak(x, ξ) ∈ Sk(Rn × Rn)

is positively homogeneous (or simply homogeneous) of order k if for all x ∈ Rn wehave

ak(x, λξ) = λkak(x, ξ) for all λ > 1 and all |ξ| > r,

for some r > 0 (independent of λ, x and ξ).

We note that we exclude small ξ from this definition because if we assumedthe property ak(x, λξ) = λkak(x, ξ) for all ξ ∈ Rn, such a function ak would notbe in general smooth at ξ = 0.

Definition 2.5.27 (Classical symbols). We will say that a symbol a ∈ Smcl (R

n × Rn)is a classical symbol of order m if a ∈ Sm(Rn × Rn) and if there is an asymptoticexpansion a ∼ ∑∞

k=0 am−k, where each am−k is positively homogeneous of orderm− k, and if a−∑N

k=0 am−k ∈ Sm−N−1(Rn × Rn), for all N ≥ 0.

Now, for a classical symbol a ∈ Smcl (R

n × Rn) the principal symbol, i.e.,its equivalence class modulo Sm−1(Rn × Rn) can be easily found. This is simplythe first term am in the asymptotic expansion in its definition. We will discussasymptotic sums in more detail in Section 2.5.5, namely we will show that if westart with the asymptotic sum

∑∞k=0 am−k, we can in turn interpret it as a symbol

from Sm(Rn × Rn). In the following exercises, one may take classical symbols toavoid unnecessary technicalities.

Exercise 2.5.28. For operators A ∈ Ψm1(Rn × Rn) and B ∈ Ψm2(Rn × Rn), findthe principal symbol of the commutator [A,B] = AB −BA.

Exercise 2.5.29. Let am ∈ Sm(Rn × Rn) be the principal symbol of A ∈ Ψm(Rn×Rn). Assume that am is real valued, i.e., am(x, ξ) ∈ R for all x, ξ ∈ Rn. Show that[A,A∗] = AA∗ −A∗A ∈ Ψ2m−2(Rn × Rn).


Exercise 2.5.30. Assume that A ∈ Ψm(Rn × Rn) is self-adjoint, i.e., that A = A∗.Show that the principal symbol of A is real valued.

Exercise 2.5.31 (Euler’s identity). Prove that f ∈ C1(Rn\0) is positively homoge-neous of order k ∈ R for all ξ = 0, i.e., f satisfies f(tξ) = tkf(ξ) for all t > 0 andξ = 0, if and only if f satisfies Euler’s identity:

∇f(ξ) · ξ = kf(ξ)

for all ξ = 0. Consequently, show that a symbol ak = ak(x, ξ) ∈ Sk(Rn × Rn) ispositively homogeneous of order k if and only if function

n∑j=1

ξj∂ak

∂ξj− kak

has a bounded support with respect to ξ uniformly with respect to x.

2.5.4 Calculus proof of L2-boundedness

In this section we will give a short proof of the fact that pseudo-differential opera-tors of order zero are bounded on L2(Rn) which was given in Theorem 2.4.2. Theproof will rely on the following lemma that is a special case of Young’s inequalityfor σ-finite measures in Corollary C.5.20.

Lemma 2.5.32 (Schur’s lemma). Let T be an integral operator of the form

Tu(x) =∫

Rn

K(x, y)u(y) dy

with kernel K ∈ L1loc(R

n × Rn) satisfying

supx∈Rn

∫Rn

|K(x, y)| dy ≤ C, supy∈Rn

∫Rn

|K(x, y)| dx ≤ C.

Then T is bounded on L2(Rn).

Calculus proof of Theorem 2.4.2. Let us now give an alternative proof of Theorem2.4.2 based on the calculus of pseudo-differential operators. Let T ∈ Ψ0(Rn × Rn)be a pseudo-differential operator of order zero with symbol a ∈ S0(Rn × Rn) andprincipal symbol σa (for simplicity one may assume that a is a classical symbolhere). Then its adjoint satisfies T ∗ ∈ Ψ0(Rn × Rn) and hence also the compositionT ∗T ∈ Ψ0(Rn × Rn) by Theorem 2.5.1. The operator T ∗T has a bounded principalsymbol |σa(x, ξ)|2 by the composition formula, so that |σa(x, ξ)|2 < M for someconstant 0 < M < ∞. Then the function p(x, ξ) =

√M − |σa(x, ξ)|2 is well

defined and it is easy to check that p ∈ S0(Rn × Rn). Let P ∈ Ψ0(Rn × Rn) be


the pseudo-differential operator with symbol p. By the calculus again we have theidentity

T ∗T = M − P ∗P + R,

for some pseudo-differential operator R ∈ Ψ−1(Rn × Rn). Then

||Tu||2L2 = 〈Tu, Tu〉 = 〈T ∗Tu, u〉= M ||u||2L2 − ||Pu||2L2 + 〈Ru, u〉 ≤M ||u||2L2 + 〈Ru, u〉,

so that T is bounded on L2(Rn) if R is. The boundedness of R on L2(Rn) can beproved by induction. Indeed, using the estimate

||Ru||2L2 = 〈Ru, Ru〉 = 〈R∗Ru, u〉 ≤ C||R∗Ru||L2 ||u||L2

we see that R ∈ Ψ−1(Rn × Rn) is bounded on L2(Rn) if R∗R ∈ Ψ−2(Rn × Rn)is bounded on L2(Rn). Continuing this argument we can reduce the question ofL2-boundedness to the boundedness of pseudo-differential operators S ∈ Ψm(Rn×Rn) for some sufficiently negative m < 0. We can now use Theorem 2.3.1 withN = 0 for x close to y to show that the integral kernel K(x, y) of S is boundedfor |y − x| ≤ 1 while it decreases fast for large |x− y| if we take sufficiently largeN . Therefore, we can use Lemma 2.5.4 to conclude that S must be bounded onL2(Rn) thus completing the proof. �

2.5.5 Asymptotic sums

Our objective in Section 2.6 will be to show how the constructed calculus of pseudo-differential operators can be applied to “easily” solve so-called elliptic partial dif-ferential equations. However, in order to carry out this application we will needanother very useful construction.

Proposition 2.5.33 (Asymptotic sums). Let aj ∈ Smj (Rn × Rn), j = 0, 1, 2, . . . ,where the sequence mj of orders satisfies m0 > m1 > m2 > · · · and mj → −∞ asj →∞. Then there exists a symbol a ∈ Sm0(Rn × Rn) such that

a ∼∞∑

j=0

aj = a0 + a1 + a2 + · · · ,

which means that we have

a−k−1∑j=0

aj ∈ Smk(Rn × Rn),

for all k ∈ N.


Proof. Let us fix a function χ ∈ C∞(Rn) such that χ(ξ) = 1 for all |ξ| ≥ 1 and suchthat χ(ξ) = 0 for all |ξ| ≤ 1/2. Then, for some sequence τj increasing sufficientlyfast and to be chosen later, we define

a(x, ξ) =∞∑

j=0

aj(x, ξ)χ(

ξ

τj

).

We note that this sum is well defined pointwise because it is in fact locally finitesince χ

(ξτj

)= 0 for |ξ| < τj/2. In order to show that a ∈ Sm0(Rn × Rn) we first

take a sequence τj such that the inequality∣∣∣∣∂βx∂α

ξ

[aj(x, ξ)χ

(ξ

τj

)]∣∣∣∣ ≤ 2−j(1 + |ξ|)mj+1−|α| (2.18)

is satisfied for all |α|, |β| ≤ j. We first show that function ξα∂αξ χ

(ξτj

)is uniformly

bounded in ξ for each j. Indeed, we have

ξα∂αξ χ

(ξ

τj

)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩0, |ξ| < τj/2,

bounded by C

∣∣∣∣( ξ

τj

)α∣∣∣∣ , τj/2 ≤ |ξ| ≤ |τ |,

0, τj < |ξ|,

so that∣∣∣ξα∂α

ξ χ(

ξτj

)∣∣∣ ≤ C is uniformly bounded for all ξ, for any given j. Usingthis fact, we can also estimate∣∣∣∣∂β

x∂αξ

[aj(x, ξ)χ

(ξ

τj

)]∣∣∣∣ =

∣∣∣∣∣ ∑α1+α2=α

cα1α2∂βx∂α1

ξ aj(x, ξ)∂α2ξ χ

(ξ

τj

)∣∣∣∣∣≤

∑α1+α2=α

|cα1α2 |(1 + |ξ|)mj−|α1|(1 + |ξ|)−|α2|

≤ C(1 + |ξ|)mj−|α|

=[C(1 + |ξ|)−1

](1 + |ξ|)mj+1−|α|.

Now, the left-hand side in estimate (2.18) is zero for |ξ| < τj/2, so we may assumethat |ξ| ≥ τj/2. Hence we can have

C(1 + |ξ|)−1 ≤ C(1 + |τj/2|)−1 < 2−j

if we take τj sufficiently large. This implies that we can take the sum of ∂βx∂α

ξ –derivatives in the definition of a(x, ξ) and (2.18) implies that a ∈ Sm0(Rn × Rn).Finally, to show the asymptotic formula, we can write

a−k−1∑j=0

aj =∞∑

j=k

aj(x, ξ)χ(

ξ

τj

),

2.6. Applications to partial differential equations 287

and so ∣∣∣∣∣∣∂βx∂α

ξ

⎡⎣a−k−1∑j=0

aj

⎤⎦∣∣∣∣∣∣ ≤ C(1 + |ξ|)mk−|α|.

In this argument we fix α and β first, and then use the required estimates for allj ≥ |α|, |β|. This shows that a−∑k−1

j=0 aj ∈ Smk(Rn × Rn) finishing the proof. �

Exercise 2.5.34. Prove that Proposition 2.5.33 remains valid in (ρ, δ) classes forall ρ and δ.

2.6 Applications to partial differential equations

The main question in the theory of partial differential equations is how to solve theequation Au = f for a given partial differential operator A and a given functionf . In other words, how to find the inverse of A, i.e., an operator A−1 such that

A ◦A−1 = A−1 ◦A = I (2.19)

is the identity operator (on some space of functions where everything is well de-fined). In this case function u = A−1f gives a solution to the partial differentialequation Au = f .

First of all we can observe that if the operator A is an operator with variablecoefficients in most cases it is impossible or very hard to find an explicit formulafor its inverse A−1 (even when it exists). However, in many questions in the theoryof partial differential equations one is actually not so much interested in having aprecise explicit formula for A−1. Indeed, in reality one is mostly interested not inknowing the solution u to the equation Au = f explicitly but rather in knowingsome fundamental properties of u. One of the most important properties is theposition and the strength of singularities of u. Thus, the question becomes whetherwe can say something about singularities of u knowing singularities of f = Au. Inthis case we do not need to solve equation Au = f exactly but it is sufficient toknow its solution modulo the class of smooth functions. Namely, instead of A−1 in(2.19) one is interested in finding an “approximate” inverse of A modulo smoothfunctions, i.e., an operator B such that u = Bf solves the equation Au = fmodulo smooth functions, i.e., if (BA − I)f and (AB − I)f are smooth for allfunctions f from some class. Recalling that operators in Ψ−∞(Rn × Rn) have sucha property, we have the following definition, which applies to all pseudo-differentialoperators A:

Definition 2.6.1 (Parametrix). Operator B is called the right parametrix of Aif AB − I ∈ Ψ−∞(Rn × Rn). Operator C is called the left parametrix of A ifCA− I ∈ Ψ−∞(Rn × Rn).


Remark 2.6.2 (Left or right parametrix?). In fact, the left and right parametrixare closely related. Indeed, by definition we have AB − I = R1 and CA− I = R2

with some R1, R2 ∈ Ψ−∞(Rn × Rn). Then we have

C = C(AB −R1) = (CA)B − CR1 = B + R2B − CR1.

If A,B,C are pseudo-differential operators of finite orders, the composition for-mula in Theorem 2.5.1 implies that R2B,CR1 ∈ Ψ−∞(Rn × Rn), i.e., C − B isa smoothing operator. Thus, we will be mainly interested in the right parametrixB because u = Bf immediately solves the equation Au = f modulo smoothfunctions.

We also note that since we work here modulo smoothing operators (i.e.,operators in Ψ−∞(Rn × Rn)), parametrices are obviously not unique – finding oneof them is already very good because any two parametrices differ by a smoothingoperator.

2.6.1 Freezing principle for PDEs

The following freezing principle provides a good and well-known motivation (see,e.g., [118]) for the use of the symbolic analysis in finding parametrices. Supposewe want to solve the following equation for an unknown function u = u(x):

(Au)(x) :=∑

1≤i,j≤n

aij(x)∂2u

∂xi∂xj(x) = f(x),

where the matrix {aij(x)}ni,j=1 is real valued, smooth, symmetric and positive

definite. If we want to proceed in analogy to the Laplace equation in Remark1.1.17, we should look for the inverse of the operator A. In the case of an operatorwith variable coefficients this may turn out to be difficult, so we may look for anapproximate inverse B such that AB = I + E, where the error E is small in somesense. To be able to argue similar to Remark 1.1.17, we “freeze” the operator Aat x0 to get the constant coefficients operator

Ax0 =∑

1≤i,j≤n

aij(x0)∂2

∂xi∂xj.

Now, Ax0 has the exact inverse which is the operator of multiplication by⎛⎝−4π2∑

1≤i,j≤n

aij(x0)ξiξj

⎞⎠−1

on the Fourier transform side. To avoid a singularity at the origin, we introducea cut-off function χ ∈ C∞ which is 0 near the origin and 1 for large ξ. Then we


define

(Bx0f)(x) =∫

Rn

e2πix·ξ

⎛⎝−4π2∑

1≤i,j≤n

aij(x0)ξiξj

⎞⎠−1

χ(ξ) f(ξ) dξ.

Consequently, we can readily see that

(Ax0Bx0f)(x) =∫

Rn

e2πix·ξ χ(ξ) f(ξ) dξ

= f(x) +∫

Rn

e2πix·ξ (χ(ξ)− 1) f(ξ) dξ.

It follows that Ax0Bx0 = I + Ex0 , where

(Ex0f)(x) =∫

Rn

e2πix·ξ (χ(ξ)− 1) f(ξ) dξ

is an operator of multiplication by a compactly supported function on the Fouriertransform side. Writing it as a convolution with a smooth test function we canreadily see that it is a smoothing operator.

Exercise 2.6.3. Prove this.

Now, we can “unfreeze” the point x0 expecting that the inverse B will beclose to Bx0 for x close to x0, and define

(Bf)(x) = (Bxf)(x)

=∫

Rn

e2πix·ξ

⎛⎝−4π2∑

1≤i,j≤n

aij(x)ξiξj

⎞⎠−1

χ(ξ) f(ξ) dξ.

This does not yield a parametrix yet, but it will be clear from the compositionformula that we still have AB = I + E1 with error E1 ∈ Ψ−1(Rn × Rn) being“smoothing of order 1”. We can then set up an iterative procedure to improvethe approximation of the inverse operator relying on the calculus of the appearingoperators, and to find a parametrix for A. This will be done in Theorem 2.6.7.

2.6.2 Elliptic operators

We will now show how we can use the calculus to “solve” elliptic partial differentialequations. First, we recall the notion of ellipticity.

Definition 2.6.4 (Elliptic symbols). A symbol a ∈ Sm(Rn × Rn) is called ellipticif for some A > 0 it satisfies

|a(x, ξ)| ≥ A|ξ|m


for all |ξ| ≥ n0 and all x ∈ Rn, for some n0 > 0. We also say that the symbol ais elliptic in U ⊂ Rn if the above estimate holds for all x ∈ U . Pseudo-differentialoperators with elliptic symbols are also called elliptic.

Exercise 2.6.5. Show that the constant n0 is not essential in this definition. Namely,for an elliptic symbol a ∈ Sm(Rn × Rn) show that there exists a symbol b ∈Sm(Rn × Rn) satisfying

|b(x, ξ)| ≥ c(1 + |ξ|)m

for all x, ξ ∈ Rn, such that b differs from a by a symbol in S−∞(Rn × Rn).

Now, let L = a(X, D) be an elliptic pseudo-differential operator with symbola ∈ Sm(Rn × Rn) (which is then also elliptic by definition). Let us introduce acut-off function χ ∈ C∞(Rn) such that χ(ξ) = 0 for small ξ, e.g., for |ξ| ≤ 1, andsuch that χ(ξ) = 1 for large ξ, e.g., for |ξ| > 2. The ellipticity of a(x, ξ) assuresthat it can be inverted pointwise for |ξ| ≥ 1, so we can define the symbol

b(x, ξ) = χ(ξ) [a(x, ξ)]−1.

Since a ∈ Sm(Rn × Rn) is elliptic, we easily see that b ∈ S−m(Rn × Rn). If wetake P0 = b(X, D) then by the composition Theorem 2.5.1 we obtain

LP0 = I + E1, PL = I + E2,

for some E1, E2 ∈ Ψ−1(Rn × Rn). Thus, we may view P0 as a good first approxi-mation for a parametrix of L. In order to find a parametrix of L, we need to modifyP0 in such a way that E1 and E2 would be in Ψ−∞(Rn × Rn). This constructioncan be carried out in an iterative way. Indeed, we now show that ellipticity isequivalent to invertibility in the algebra Ψ∞(Rn × Rn)/Ψ−∞(Rn × Rn):

Theorem 2.6.6 (Elliptic⇐⇒ ∃ Parametrix). Operator A ∈ Ψm(Rn × Rn) is ellipticif and only if there exists B ∈ Ψ−m(Rn × Rn) such that BA ∼ I ∼ AB moduloΨ−∞(Rn × Rn).

Proof. Let σA and σB denote symbols of A and B, respectively. Assume first thatA ∈ Ψm(Rn × Rn) and B ∈ Ψ−m(Rn × Rn) satisfy BA = I − T and AB =I − T ′ with T, T ′ ∈ Ψ−∞(Rn × Rn). Then 1 − σBA = σT ∈ S−∞(Rn × Rn)and consequently by Theorem 2.5.1 we have 1 − σBσA ∈ S−1(Rn × Rn), so that|1−σBσA| ≤ C〈ξ〉−1. Hence 1− |σB | · |σA| ≤ C〈ξ〉−1, or equivalently |σB | · |σA| ≥1 − C〈ξ〉−1. If we choose n0 > C, then |σB(x, ξ)| · |σA(x, ξ)| ≥ 1 − C〈n0〉−1 > 0for any |ξ| ≥ n0. Thus, σA(x, ξ) = 0 for |ξ| > n0 and

1|σA(x, ξ)| ≤ C|σB(x, ξ)| ≤ C〈ξ〉−m

.

Hence A is elliptic of order m. This yields the first part of the proof.


Conversely, assume that A and σA(x, ξ) are elliptic. We will construct thesymbol b as an asymptotic sum

b ∼ b0 + b1 + b2 + · · ·

and then use Proposition 2.5.5 to justify this infinite sum. Then we take operatorsBj with symbols bj and the operator B with symbol b will be the parametrix forA. We will also work with |ξ| ≥ n0 since small ξ are not relevant for symbolicconstructions. Moreover, once we have the left parametrix, we also have the rightone in view of Remark 2.6.2.

First, we take b0 = 1/σA which is well defined for |ξ| ≥ n0 in view of theellipticity of σA. Then we have

σB0A = 1− e0, b0 ∈ S−m, with e0 = 1− σB0A ∈ S−1.

Then we take b1 = e0/σA ∈ S−m−1 so that we have

σ(B0+B1)A = 1− e0 + σB1A = 1− e1, with e1 = e0 − σB1A ∈ S−2.

Inductively, we define bj = ej−1/σA ∈ S−m−j and we have

σ(B0+B1+···+Bj)A = 1− ej , with ej = ej−1 − σBjA ∈ S−j−1.

Now, Proposition 2.5.5 shows that b ∈ S−m and it satisfies σBA = 1 − e withe ∈ S−∞ by its construction, completing the proof. �

We now give a slightly more general statement which is useful for otherpurposes as well. It is a consequence of Theorem 2.6.6 and composition Theorem2.5.1.

Corollary 2.6.7 (Local parametrix). Let a ∈ Sm(Rn × Rn) be elliptic on an openset U ⊂ Rn, i.e., there exists some A > 0 such that |a(x, ξ)| ≥ A|ξ|m for allx ∈ U and all |ξ| ≥ 1. Let c ∈ Sl be a symbol of order l whose support with respectto x is a compact subset of U . Then there exists a symbol b ∈ Sl−m such thatb(X, D)a(X, D) = c(X, D)− e(X, D) for some symbol e ∈ S−∞.

We also have the following local version of this for partial differential opera-tors:

Corollary 2.6.8 (Parametrix for elliptic differential operators). Let

L =∑|α|≤m

aα(x)∂αx

be an elliptic partial differential operator in an open set U ⊂ Rn. Let χ1, χ2, χ3 ∈C∞0 (Rn) be such that χ2 = 1 on the support of χ1 and χ3 = 1 on the support ofχ2. Then there is an operator P ∈ S−m(Rn × Rn) such that

P (χ2L) = χ1I + Eχ3,

for some E ∈ Ψ−∞(Rn × Rn).


Proof. We take a(X, D) = χ2L and c(X, D) = χ1I in Corollary 2.6.7. Thena(X, D) is elliptic on the support of χ2 and we can take P = b(X, D) with b ∈ S−m

from Corollary 2.6.7. �

We will now apply this result to obtain a statement on the regularity ofsolution to elliptic partial differential equations. We assume that the order mbelow is an integer which is certainly true when L is a partial differential operator.However, if we take into account the discussion from the next section, we will seethat the statements below are still true for any m ∈ R.

Theorem 2.6.9 (A-priori estimate). Let L ∈ Ψm be an elliptic pseudo-differentialoperator in an open set U ⊂ Rn and let Lu = f in U . Assume that f ∈ (L2

k(U))loc.Then u ∈ (L2

m+k(U))loc.

This theorem shows that if u is a solution of an elliptic partial differentialequation Lu = f then there is local gain of m derivatives for u compared to f ,where m is the order of the operator L.

Proof. Let χ1, χ2, χ3 ∈ C∞0 (U) be non-zero functions such that χ2 = 1 on thesupport of χ1 and χ3 = 1 on the support of χ2. Then, similar to the proof ofCorollary 2.6.8 we have

P (χ2L) = χ1I + Eχ3,

with some P ∈ Ψ−m. Since f ∈ (L2k)loc we have P (χ2f) ∈ (L2

m+k)loc. Also,E(χ3u) ∈ C∞ so that ||χ3E(χ3u)||L2

k≤ ||χ3u||L2 for any k. Summarising and

usingP (χ2f) = χ1u + E(χ3u),

we obtain||χ1u||L2

k+m≤ C

(||χ2f ||L2

k+ ||χ3u||L2

),

which implies that u ∈ (L2m+k)loc in U . �

Remark 2.6.10. We can observe from the proof that properties of solution u bythe calculus and the existence of a parametrix are reduced to the fact that pseudo-differential operators in Ψ−m map L2

k to L2k+m. In fact, in this way many properties

of solutions to partial differential equations are reduced to questions about generalpseudo-differential operators. In the following statement for now one can think ofm and k being integers or zeros such that m ≤ k, but if we adopt the definition ofSobolev spaces from Definition 2.6.15, it is valid for all m, k ∈ R. We will prove itcompletely in the case of p = 2, and in the case p = 2 we will show how to reduceit to the Lp-boundedness of pseudo-differential operators.

Theorem 2.6.11 (Lpk-continuity). Let T ∈ Sm(Rn × Rn) be a pseudo-differential

operator of order m ∈ R, let 1 < p < ∞, and let k ∈ R. Then T extends toa bounded linear operator from the Sobolev space Lp

k(Rn) to the Sobolev spaceLp

k−m(Rn).


We will prove this statement in the next section. As an immediate conse-quence, by the same argument as in the proof of this theorem, we also obtain

Corollary 2.6.12 (Local Lpk-continuity). Let L ∈ Ψm be an elliptic pseudo-differ-

ential operator in an open set U ⊂ Rn, let 1 < p < ∞, m, k ∈ R, and let Lu = fin U . Assume that f ∈ (Lp

k(U))loc. Then u ∈ (Lpm+k(U))loc.

Let us briefly discuss an application of the established a priori estimates.

Definition 2.6.13 (Harmonic functions). A distribution f ∈ D′(Rn) is called har-monic if Lf = 0, where L = ∂2

∂x21

+ · · ·+ ∂2

∂x2n

is the usual Laplace operator.

Taking real and imaginary parts of holomorphic functions, we see that Liou-ville’s theorem D.6.2 for holomorphic functions follows from

Theorem 2.6.14 (Liouville’s theorem for harmonic functions). Every harmonicfunction f ∈ L∞(Rn) is constant.

Proof. Since L is elliptic, by Theorem 2.6.9 it follows from the equation Lf = 0that f ∈ C∞(Rn). Taking the Fourier transform of Lf = 0 we obtain −4π2|ξ|2f =0 which means that supp f ⊂ {0}. By Exercise 1.4.15 it follows that f =∑|α|≤m aα∂αδ. Taking the inverse Fourier transform we see that f(x) must be

a polynomial. Finally, the assumption that f is bounded implies that f must beconstant. �

2.6.3 Sobolev spaces revisited

Up to now we defined Sobolev spaces Lpk assuming that the index k is an integer.

In fact, using the calculus of pseudo-differential operators we can show that thesespaces can be defined for all k ∈ R thus allowing one to measure the regularity offunctions much more precisely. In the following discussion we assume the statementon the Lp-continuity of pseudo-differential operators from Theorem 2.6.22.

We recall from Definition 1.5.6 that for an integer k ∈ N we defined theSobolev space Lp

k(Rn) as the space of all f ∈ Lp(Rn) such that their distributionalderivatives satisfy ∂α

x f ∈ Lp(Rn), for all 0 ≤ |α| ≤ k. This space is equipped witha norm ||f ||Lp

k=

∑|α|≤k ||∂α

x f ||Lp (or with any equivalent norm) for 1 ≤ p < ∞,with a modification for p =∞.

Let L = ∂2

∂x21

+ · · ·+ ∂2

∂x2n

be the Laplace operator, so that its symbol is equal

to 4π2|ξ|2. Let s ∈ R be a real number and let us consider operators (I −L)s/2 ∈Ψs(Rn × Rn) which are pseudo-differential operators with symbols a(x, ξ) = (1 +4π2|ξ|2)s/2.

Definition 2.6.15 (Sobolev spaces). We will say f is in the Sobolev space Lps(R

n),i.e., f ∈ Lp

s(Rn), if (I − L)s/2f ∈ Lp(Rn). We equip this space with the norm

||f ||Lps

:= ||(I − L)s/2f ||Lp .


Proposition 2.6.16. If s ∈ N is an integer, the space Lps(R

n) coincides with thespace Lp

k(Rn) with k = s, with equivalence of norms.

Proof. We will use the index k for both spaces. Since operator (I − L)k/2 is apseudo-differential operator of order k, by Theorem 2.6.11 we get that it is boundedfrom Lp

k to Lp, i.e., we have

||(I − L)k/2f ||Lp ≤ C∑|α|≤k

||∂αx f ||Lp .

Conversely, let Pα be a pseudo-differential operator defined by Pα = ∂αx (I−L)−k/2,

i.e., a pseudo-differential operator with symbol pα(x, ξ) = (2πiξ)α(1+4π2|ξ|2)−k/2,independent of x. If |α| ≤ k, we get that pα ∈ S|α|−k ⊂ S0, so that Pα ∈S0(Rn × Rn) for all |α| ≤ k. By Theorem 2.6.22 operators Pα are bounded onLp(Rn). Therefore, we obtain∑

|α|≤k

||∂αx f ||Lp =

∑|α|≤k

||Pα(I − L)k/2f ||Lp ≤ C||(I − L)k/2f ||Lp ,

completing the proof. �Exercise 2.6.17 (Sobolev embedding theorem). Prove that if s > k + n/2 thenHs(Rn) ⊂ Ck(Rn) and the inclusion is continuous. This gives a sharper version ofExercise 1.5.11.

Exercise 2.6.18 (Distributions as Sobolev space functions). Recall from Exercise1.4.14 that if u ∈ E ′(Rn) then u is a distribution of some finite order m. Provethat if s < −m− n/2 then u ∈ Hs. Contrast this with Exercise 2.6.17.

Exercise 2.6.19. Prove that

S(Rn) =⋂s∈R

〈x〉−sHs(Rn) and S ′(Rn) =

⋃s∈R

〈x〉sHs(Rn).

Note that the equalities fail without weights: for example, show that we havesin x

x ∈ ⋂k∈N0

Hk(R) but sin xx ∈ S(R). The situation on the torus will be somewhat

simpler, see Corollary 3.2.12.

Finally, let us justify Theorem 2.6.11 However, we will assume without proofthat pseudo-differential operators of order zero are bounded on Lp(Rn) for all1 < p <∞, see Theorem 2.6.22.

Proof of Theorem 2.6.11. Let f ∈ Lps(R

n). By definition this means that (I −L)s/2f ∈ Lp(Rn). Then we can write using the calculus of pseudo-differentialoperators (composition Theorem 2.5.1):

(I − L)(s−μ)/2Tf = (I − L)(s−μ)/2T (I − L)−s/2(I − L)s/2f ∈ Lp(Rn)

since operator (I−L)(s−μ)/2T (I−L)−s/2 is a pseudo-differential operator of orderzero and is, therefore, bounded on Lp(Rn) by Theorem 2.6.22 if p = 2 and byTheorem 2.4.2 if p = 2. �


Remark 2.6.20. It is often very useful to conclude something about properties offunctions in one Sobolev space knowing about their properties in another Sobolevspace. One instance of such a conclusion will be used in the proof of Theorem 4.2.3on the Sobolev boundedness of operators on the L2-space on the torus. A generalBanach space setting for such conclusions will be presented in Section 3.5. Herewe present without proof another instance of this phenomenon:

Theorem 2.6.21 (Rellich’s theorem). Let (fk)∞k=1⊂Hs(Rn) be a uniformly boundedsequence of functions: there exists C such that ||fk||Hs(Rn) ≤ C for all k. Assumethat all functions fk are supported in a fixed compact set. Then there exists asubsequence of (fk)∞k=1 which converges in Hσ(Rn) for all σ < s.

Remarks on Lp-continuity of pseudo-differential operators. Let a∈S0(Rn×Rn).Then by Theorem 2.3.1 the integral kernel K(x, y) of pseudo-differential operatora(X, D) satisfies estimates

|∂αx ∂β

y K(x, y)| ≤ Aαβ |x− y|−n−|α|−|β|

for all x = y. In particular, for α = β = 0 this gives

|K(x, y)| ≤ A|x− y|−n for all x = y. (2.20)

Moreover, if we use it for α = 0 and |β| = 1, we get∫|x−z|≥2δ

|K(x, y)−K(x, z)| dx ≤ A if |x− z| ≤ δ, for all δ > 0. (2.21)

Now, if we take a general integral operator T of the form

Tu(x) =∫

Rn

K(x, y)u(y) dy,

properties (2.20) and (2.21) of the kernel are the starting point of the so-calledCalderon–Zygmund theory of singular integral operators. In particular, one canconclude that such operators are of weak type (1, 1), i.e., they satisfy the estimate

μ{x ∈ Rn : |Tu(x)| > λ} ≤ ||u||L1

λ,

(see Definition 1.6.3 and the discussion following it for more details). Since we alsoknow from Theorem 2.4.2 that a(X, D) ∈ Ψ0(Rn × Rn) are bounded on L2(Rn)and since we also know from Proposition 1.6.4 that this implies that a(X, D) isof weak type (2, 2), we get that pseudo-differential operators of order zero areof weak types (1, 1) and (2, 2). Then, by Marcinkiewicz’ interpolation Theorem1.6.5, we conclude that a(X, D) is bounded on Lp(Rn) for all 1 < p < 2. By thestandard duality argument, this implies that a(X, D) is bounded on Lp(Rn) alsofor all 2 < p <∞.


Since we also have the boundedness of L2(Rn), we obtain

Theorem 2.6.22. Let T ∈ Ψ0(Rn × Rn). Then T extends to a bounded operatorfrom Lp(Rn) to Lp(Rn), for all 1 < p <∞.

We note that there exist different proofs of this theorem. On one hand, itfollows automatically from the Calderon–Zygmund theory of singular integral op-erators which include pseudo-differential operators considered here, if we viewthem as integral operators with singular kernels. There are many other proofsthat can be found in monographs on pseudo-differential operators. Another alter-native and more direct method is to reduce the Lp-boundedness to the questionof uniform boundedness of Fourier multipliers in Lp(Rn) which then follows fromHormander’s theorem on Fourier multipliers. However, in this monograph we de-cided not to immerse ourselves in the Lp-world since our aims here are different.We can refer to [91] and to [92] for more information on the Lp-boundedness ofgeneral Fourier integral operators in (ρ, δ)-classes with real and complex phasefunctions, respectively.

Chapter 3

Periodic and Discrete Analysis

In this chapter we will review basics of the periodic and discrete analysis whichwill be necessary for development of the theory of pseudo-differential operatorson the torus in Chapter 4. Our aim is to make these two chapters accessibleindependently for people who choose periodic pseudo-differential operators as astarting point for learning about pseudo-differential operators on Rn. This maybe a fruitful idea in the sense that many technical issues disappear on the torusas opposed to Rn. Among them is the fact that often one does not need to worryabout convergence of the integrals in view of the torus being compact. Moreover,the theory of distributions on the torus is much simpler than that on Rn, at leastin the form required for us. The main reason is that the periodic Fourier transformtakes functions on Tn = Rn/Zn to functions on Zn where, for example, tempereddistributions become pointwise defined functions on the lattice Zn of polynomialgrowth at infinity. Also, on the lattice Zn there are no questions of regularity sinceall the objects are defined on a discrete set. However, there are many parallelsbetween Euclidean and toroidal theories of pseudo-differential operators, so lookingat proofs of similar results in different chapters may be beneficial. In many caseswe tried to avoid overlaps by presenting a different proof or by giving a differentexplanation.

Therefore, we also try to make the reading self-contained and elementary,avoiding cross-references to other chapters unless they increase the didactic valueof the material.

Yet, being written for people working with analysis, this chapter only brieflystates the related notations and facts of more general function analysis. Supple-mentary material is, of course, referred to. The reader should have a basic knowl-edge of Banach and Hilbert spaces (the necessary background material is providedin Chapter B); some familiarity with distributions and point set topology defi-nitely helps (this material can be found in Chapter A and in Chapter 1 if neces-sary). A word of warning has to be said: in order to use the theory of periodicpseudo-differential operators as a tool, there is no demand to dwell deeply on theseprerequisites. One is rather encouraged to read the appropriate theory only when

298 Chapter 3. Periodic and Discrete Analysis

it is encountered and needed, and that is why we present a summary of necessarythings here as well.

We will use the following notation in the sequel. Triangles � and � willdenote the forward and backward difference operators, respectively. The Laplacianwill be denoted by L to avoid any confusion. The Dirac delta at x will be denotedby δx and the Kronecker delta at ξ will be denoted by δξ,η.

As is common, R and C are written for real and complex numbers, re-spectively, Z stands for the integers, while N = Z+ := {n ∈ Z | n ≥ 1} andN0 := Z+ ∪ {0} are the sets of positive integers and nonnegative integers, respec-tively. We would also like to draw the reader’s attention to the notation |α| and‖ξ‖ in (3.2) and (3.3), respectively, that we will be using in this chapter as wellas in Chapters 4 and 5. This is especially of relevance in these chapters as bothmulti-indices α ∈ Nn

0 and frequencies ξ ∈ Zn are integers.

3.1 Distributions and Fourier transforms on Tn and Zn

We fix the notation for the torus as Tn = (R/Z)n = Rn/Zn. Often we may identifyTn with the cube [0, 1)n ⊂ Rn, where we identify the measure on the torus withthe restriction of the Euclidean measure on the cube. Functions on Tn may bethought as those functions on Rn that are 1-periodic in each of the coordinatedirections. We will often simply say that such functions are 1-periodic (instead ofZn-periodic). More precisely, on the Euclidean space Rn we define an equivalencerelation

x ∼ ydefinition⇐⇒ x− y ∈ Zn,

where the equivalence classes are

[x] = {y ∈ Rn : x ∼ y}= {x + k : k ∈ Zn} .

A point x ∈ Rn is naturally mapped to a point [x] ∈ Tn, and usually there is noharm in writing x ∈ Tn instead of the actual [x] ∈ Tn. We may identify functionson Tn with Zn-periodic functions on Rn in a natural manner, f : Tn → C beingidentified with g : Rn → C satisfying

g(x) = f([x])

for all x ∈ Rn. In such a case we typically even write g = f and g(x) = f(x), andwe might casually say things like

• “f is periodic”,

• “g ∈ C∞(Tn)” when actually “g ∈ C∞(Rn) is periodic”,

• etc.

3.1. Distributions and Fourier transforms on Tn and Zn 299

The reader has at least been warned. Moreover, the one-dimensional torus T1 =R1/Z1 is isomorphic to the circle

S1 ={z ∈ R2 : ‖z‖ = 1

}= {(cos(t), sin(t)) : t ∈ R}

by the obvious mapping

[t] �→ (cos(2πt), sin(2πt)) ,

so we may identify functions on T1 with functions on S1.Remark 3.1.1 (What makes T1 and Tn special?). At this point, we must emphasizehow fundamental the study on the one-dimensional torus T1 = R1/Z1 is. First,smooth Jordan curves, especially the one-dimensional sphere S1, are diffeomorphicto T1. Secondly, the theory on the n-dimensional torus Tn = Rn/Zn sometimes re-duces to the case of T1. Furthermore, compared to the theory of pseudo-differentialoperators on Rn, the case of Tn is beautifully simple. This is due to the fact thatTn is a compact Abelian group – whereas Rn is only locally compact – on whichthe powerful aid of Fourier series is at our disposal. However, the results on Rn

and Tn are somewhat alike. Many general results concerning series on the torusand their properties can be found in, e.g., [155].

To make this chapter more self-contained, let us also briefly review the multi-index notation. A vector α = (αj)n

j=1 ∈ Nn0 is called a multi-index. If x = (xj)n

j=1 ∈Rn and α ∈ Nn

0 , we write xα := xα11 · · ·xαn

n . For multi-indices, α ≤ β meansαj ≤ βj for all j ∈ {1, . . . , n}. We also write β! := β1! · · ·βn! and(

α

β

):=

α!β! (α− β)!

=(

α1

β1

)· · ·

(αn

βn

),

so that

(x + y)α =∑β≤α

(α

β

)xα−β yβ . (3.1)

For α ∈ Nn0 and x ∈ Rn we shall write

|α| :=n∑

j=1

αj , (3.2)

‖x‖ :=

⎛⎝ n∑j=1

x2j

⎞⎠1/2

, (3.3)

∂αx := ∂α1

x1· · · ∂αn

xn,

where ∂xj= ∂

∂xjetc. We will also use the notation Dxj

= −i2π∂xj= −i2π ∂

∂xj,

where i =√−1 is the imaginary unit. We have chosen the notation ‖x‖ for the


Euclidean distance in this chapter, to contrast it with |α| used for multi-indices.We also denote

〈x〉 := (1 + ‖x‖2)1/2.

Exercise 3.1.2. Prove (3.1).

Exercise 3.1.3. Show that (∑n

j=1xj)m =∑|α|=m

m!α! xα, where x ∈ Rn and m ∈ N0.

Definition 3.1.4 (Periodic functions). A function f : Rn → Y is 1-periodic iff(x+k) = f(x) for every x ∈ Rn and k ∈ Zn. We shall consider these functions tobe defined on Tn = Rn/Zn = {x + Zn|x ∈ Rn}. The space of 1-periodic m timescontinuously differentiable functions is denoted by Cm(Tn), and the test functionsare the elements of the space C∞(Tn) :=

⋂m∈Z+ Cm(Tn) .

Remark 3.1.5. The natural inherent topology of C∞(Tn) is induced by the semi-norms that one gets by demanding the following convergence: uj → u if and onlyif ∂αuj → ∂αu uniformly, for all α ∈ Nn

0 . Thus, e.g., by [89, 1.46] C∞(Tn) is aFrechet space, but it is not normable as it has the Heine–Borel property.

Let S(Rn) denote the space of the Schwartz test functions from Definition1.1.11, and let S ′(Rn) be its dual, i.e., the space of the tempered distributionsfrom Definition 1.3.1. The integer lattice Zn plays an important role in periodicand discrete analysis.

Definition 3.1.6 (Schwartz space S(Zn)). Let S(Zn) denote the space of rapidlydecaying functions Zn → C. That is, ϕ ∈ S(Zn) if for any M < ∞ there exists aconstant Cϕ,M such that

|ϕ(ξ)| ≤ Cϕ,M 〈ξ〉−M

holds for all ξ ∈ Zn. The topology on S(Zn) is given by the seminorms pk, wherek ∈ N0 and pk(ϕ) := supξ∈Zn〈ξ〉k |ϕ(ξ)| .Exercise 3.1.7 (Tempered distributions S ′(Zn)). Show that the continuous linearfunctionals on S(Zn) are of the form

ϕ �→ 〈u, ϕ〉 :=∑ξ∈Zn

u(ξ) ϕ(ξ),

where functions u : Zn → C grow at most polynomially at infinity, i.e., there existconstants M <∞ and Cu,M such that

|u(ξ)| ≤ Cu,M 〈ξ〉M

holds for all ξ ∈ Zn. Such distributions u : Zn → C form the space S ′(Zn).Note that compared to S ′(Rn), distributions in S ′(Zn) are pointwise well-definedfunctions (!) on the lattice Zn.


To contrast Euclidean and toroidal Fourier transforms, they will be denotedby FRn and FTn , respectively. Let FRn : S(Rn)→ S(Rn) be the Euclidean Fouriertransform defined by

(FRn)f(ξ) :=∫

Rn

e−2πix·ξ f(x) dx.

Mapping FRn : S(Rn)→ S(Rn) is a bijection, and its inverse F−1Rn is given by

f(x) =∫

Rn

e2πix·ξ (FRnf)(ξ) dξ,

see Theorem 1.1.21. As is well known, this Fourier transform can be uniquelyextended to FRn : S ′(Rn) → S ′(Rn) by duality, see Definition 1.3.2. We refer toSection 1.1 for further details concerning the Euclidean Fourier transform.

Definition 3.1.8 (Toroidal/periodic Fourier transform). Let

FTn = (f �→ f) : C∞(Tn)→ S(Zn)

be the toroidal Fourier transform defined by

f(ξ) :=∫

Tn

e−i2πx·ξ f(x) dx. (3.4)

Then FTn is a bijection and its inverse F−1Tn : S(Zn)→ C∞(Tn) is given by

f(x) =∑ξ∈Zn

ei2πx·ξ f(ξ),

so that for h ∈ S(Zn) we have(F−1

Tn h)(x) :=

∑ξ∈Zn

ei2πx·ξ h(ξ).

Remark 3.1.9 (Notations i2πx · ξ vs 2πix · ξ). We note that in the case of thetoroidal Fourier transform we write i2πx · ξ in the exponential with i in front toemphasize that 2πx is now a periodic variable, and also to distinguish it from theEuclidean Fourier transform in which case we usually write 2πi in the exponential.We will write FTn instead of f in this and the next chapters only if we want toemphasize that we want to take the periodic Fourier transform.

Exercise 3.1.10 (Two Fourier inversion formulae). Prove that the Fourier trans-form FTn : C∞(Tn) → S(Zn) is a bijection, that FTn : C∞(Tn) → S(Zn) andF−1

Tn : S(Zn)→ C∞(Tn) are continuous, and that

FTn ◦ F−1Tn : S(Zn)→ S(Zn) and F−1

Tn ◦ FTn : C∞(Tn)→ C∞(Tn)

are identity mappings on S(Zn) and C∞(Tn), respectively.


Let us study an example of periodic distributions, the space L2(Tn).

Definition 3.1.11 (Space L2(Tn)). Space L2(Tn) is a Hilbert space with the innerproduct

(u, v)L2(Tn) :=∫

Tn

u(x) v(x) dx, (3.5)

where z is the complex conjugate of z ∈ C. The Fourier coefficients of u ∈ L2(Tn)are

u(ξ) =∫

Tn

e−i2πx·ξ u(x) dx (ξ ∈ Zn), (3.6)

and they are well defined for all ξ due to Holder’s inequality (Proposition 1.2.4)and compactness of Tn.

Remark 3.1.12 (Fourier series on L2(Tn)). The family {eξ : ξ ∈ Zn} defined by

eξ(x) := ei2πx·ξ (3.7)

forms an orthonormal basis on L2(Tn), which will be proved in Theorem 3.1.20.Thus the partial sums of the Fourier series

∑ξ∈Zn u(ξ) ei2πx·ξ converge to u in

the L2-norm, so that we shall identify u with its Fourier series representation:

u(x) =∑ξ∈Zn

u(ξ) ei2πx·ξ.

As before, we call u : Zn → C the Fourier transform of u.

As a consequence of the Plancherel identity on general compact topologicalgroups to be proved in Corollary 7.6.7 we obtain:Remark 3.1.13 (Plancherel’s identity). If u ∈ L2(Tn) then u ∈ �2(Zn) and

||u||�2(Zn) = ||u||L2(Tn).

Exercise 3.1.14. Give a simple direct proof of Remark 3.1.13. (Hint: it is similarto the proof on Rn but simpler.)

Exercise 3.1.15. Show that S(Zn) is dense in �2(Zn).

Remark 3.1.16 (Functions eξ). We can observe that the functions eξ(x) = ei2πx·ξ

from (3.7) satisfy eξ(x + y) = eξ(x)eξ(y) and |eξ(x)| = 1 for all x ∈ Tn. Theconverse is also true, namely:

Theorem 3.1.17 (Unitary representations of Tn). If f ∈ L1(Tn) is such that wehave f(x + y) = f(x)f(y) and |f(x)| = 1 for all x, y ∈ Tn, then there exists someξ ∈ Zn such that f = eξ.

Remark 3.1.18. It is a nice exercise to show this directly and we do it below.However, we note that employing a more general terminology of Chapter 7, theconditions on f mean that f : Tn → U(1) is a unitary representation of Tn,


automatically irreducible since it is one-dimensional. Moreover, these conditionsimply that f is continuous, and hence f ∈ Tn, the unitary dual of f . Since functionseξ exhaust the unitary dual by the Peter–Weyl theorem (see, e.g., Remark 7.5.17),we obtain the result.

Proof of Theorem 3.1.17. We will prove the one-dimensional case since the generalcase of Tn follows from it if we look at functions f(τej) where ej is the jth unitbasis vector of Rn.

Thus, x ∈ T1, we can think of T as of periodic R, and we choose λ > 0 suchthat Λ =

∫ λ

0f(τ) dτ = 0. Such λ exists because otherwise we would have f = 0

a.e. by Corollary 1.5.17 of the Lebesgue differentiation theorem, contradicting theassumptions. Consequently we can write

f(x) = Λ−1

∫ λ

0

f(x)f(τ) dτ = Λ−1

∫ λ

0

f(x + τ) dτ = Λ−1

∫ x+λ

x

f(τ) dτ.

From this we can observe that f ∈ L1(R) implies that f is continuous at x. Sincethis is true for all x ∈ T we get f ∈ C1(T). By induction, we get that actuallyf ∈ C∞(T). Differentiating the equality above, we see that f satisfies the equation

f ′(x) = Λ−1(f(x + λ)− f(x)) = Λ−1(f(x)f(λ)− f(x)) = C0f(x),

with C0 = Λ−1(f(λ)−1). Solving this equation we find f(x) = f(0) eC0x. Recallingthat |f(0)| = 1 we get that |f(x)| = eReC0x. Since |f(x)| = 1 we see that ReC0 = 0,and thus C0 = i2πξ for some ξ ∈ R. Finally, the fact that f is periodic impliesthat ξ ∈ Z. �

Exercise 3.1.19. Work out the details of the extension of the proof from T1 to Tn.Also, show that the conclusion of Theorem 3.1.17 remains true if we replace Tn

by Rn and condition f ∈ L1(Tn) by f ∈ L1loc(R

n), but in this case ξ ∈ Rn doesnot have to be in the lattice Zn.

Theorem 3.1.20 (An orthonormal basis of L2(Tn)). The collection {eξ : ξ ∈ Zn}is an orthonormal basis of L2(Tn).

Remark 3.1.21. Let us make some general remarks first. From the general theory ofHilbert spaces we know that L2(Tn) has an orthonormal basis, which is countableby Theorem B.5.35 if we can check that L2(Rn) is separable. On the other hand,a more precise conclusion is possible from the general theory if we use that Tn isa group. Indeed, Theorem 3.1.17 (see also Remark 3.1.18) implies that Tn ∼= {eξ :ξ ∈ Zn}. Theorem 3.1.20 is then a special case of the Peter–Weyl theorem (see,e.g., Remark 7.5.17). However, at this point we give a more direct proof:

Proof of Theorem 3.1.20. It is easy to check the orthogonality property

(eξ, eη)L2(Tn) = 0 for ξ = η,


and the normality(eξ, eξ)L2(Tn) = 1 for all ξ ∈ Zn,

so the real issue is to show that we have the basis according to Definition B.5.34.We want to apply the Stone–Weierstrass theorem A.14.4 to show that the setE = span{eξ : ξ ∈ Zn} is dense in C(Tn). If we have this, we can use the densityof C(Tn) in L2(Tn), so that by Theorem B.5.32 it would be a basis. We note thatin fact the density of E in both C(Tn) and L2(Tn) is a special case of Theorem7.6.2 on general topological groups, but we give a direct short proof here. In viewof the Stone–Weierstrass theorem A.14.4 all we have to show is that E is aninvolutive algebra separating the points of Tn. It is clear that E separates points.Finally, from the identity eξeη = eξ+η it follows that E is an algebra, which is alsoinvolutive because of the identity eξ = e−ξ. �Exercise 3.1.22. Show explicitly how E separates the points of Tn, as well as verifythe orthonormality statement in the proof.

Definition 3.1.23 (Spaces Lp(Tn)). For 1 ≤ p < ∞ let Lp(Tn) be the space of allu ∈ L1(Tn) such that

||u||Lp(Tn) :=(∫

Tn

|u(x)|p dx

)1/p

<∞.

For p =∞, let L∞(Tn) be the space of all u ∈ L1(Tn) such that

||u||L∞(Tn) := esssupx∈Tn |u(x)| <∞.

These are Banach spaces by Theorem C.4.9.

Corollary 3.1.24 (Hausdorff–Young inequality). Let 1 ≤ p ≤ 2 and 1p + 1

q = 1. Ifu ∈ Lp(Tn) then u ∈ �q(Zn) and

||u||�q(Zn) ≤ ||u||Lp(Tn).

Proof. The statement follows by the Riesz–Thorin interpolation theorem C.4.18from the simple estimate ||u||�∞(Zn) ≤ ||u||L1(Tn) and Plancherel’s identity||u||�2(Zn) = ||u||L2(Tn) in Remark 3.1.13. �Definition 3.1.25 (Periodic distribution space D′(Tn)). The dual space D′(Tn) =L(C∞(Tn), C) is called the space of periodic distributions. For u ∈ D′(Tn) andϕ ∈ C∞(Tn), we shall write

u(ϕ) = 〈u, ϕ〉.For any ψ ∈ C∞(Tn),

ϕ �→∫

Tn

ϕ(x) ψ(x) dx

is a periodic distribution, which gives the embedding ψ ∈ C∞(Tn) ⊂ D′(Tn). Notethat the same argument also shows the embedding of the spaces Lp(Tn), 1 ≤ p ≤


∞, into D′(Tn). Due to the test function equality 〈∂αψ, ϕ〉 = 〈ψ, (−1)|α|∂αϕ〉, itis natural to define distributional derivatives by

〈∂αf, ϕ〉 := 〈f, (−1)|α|∂αϕ〉.

The topology of D′(Tn) = L(C∞(Tn), C) is the weak∗-topology.

Remark 3.1.26 (Trigonometric polynomials). The space TrigPol(Tn) of trigono-metric polynomials on the torus is defined by

TrigPol(Tn) := span{eξ : ξ ∈ Zn}.

Thus, f ∈ TrigPol(Tn) is of the form

f(x) =∑ξ∈Zn

f(ξ)ei2πx·ξ,

where f(ξ) = 0 for only finitely many ξ ∈ Zn. In the proof of Theorem 3.1.20we showed that TrigPol(Tn) is dense in both C(Tn) and in L2(Tn) in the corre-sponding norms. Now, the set of trigonometric polynomials is actually also dense inC∞(Tn), so that a distribution is characterised by evaluating it at the vectors eξ forall ξ ∈ Zn. We note that there exist linear mappings u ∈ L(span{eξ | ξ ∈ Zn}, C)that do not belong to L(C∞(Tn), C), but for which the determination of theFourier coefficients u(ξ) = u(eξ) makes sense.

Definition 3.1.27 (Fourier transform on D′(Tn)). By dualising the inverse F−1Tn :

S(Zn) → C∞(Tn), the Fourier transform is extended uniquely to the mappingFTn : D′(Tn)→ S ′(Zn) by the formula

〈FTnu, ϕ〉 := 〈u, ι ◦ F−1Tn ϕ〉, (3.8)

where u ∈ D′(Tn), ϕ ∈ S(Zn), and ι is defined by (ι ◦ ψ)(x) = ψ(−x).

Exercise 3.1.28. Prove that if u ∈ D′(Tn) then FTnu ∈ S ′(Zn). Note that byExercise 3.1.7 it means in particular that FTnu is defined pointwise on Zn.

Exercise 3.1.29 (Compatibility). Check that extension (3.8) when restricted toC∞(Tn), is compatible with the definition (3.4). Here, the inclusion C∞(Tn) ⊂D′(Tn) is interpreted in the standard way by

〈u, ϕ〉 = u(ϕ) =∫

Tn

u(x) ϕ(x) dx.

Remark 3.1.30 (Notice: different spaces). Observe that spaces of functions wherethe toroidal Fourier transform FTn acts are different: one is the space of functionson the torus C∞(Tn) while the other is the space of functions on the lattice S(Zn).That is why one has to be more careful on the torus, e.g., compared to the Fouriertransform for distributions on Rn in Definition 1.3.2. This difference will be evenmore apparent in the case of compact Lie groups in Chapter 10.


Remark 3.1.31 (Bernstein’s theorem). The Fourier transform can be studied onother spaces on the torus. For example, let Λs(T) be the space of Holder continuousfunctions of order 0 < s < 1 on the one-dimensional torus T1, defined as

Λs(T) :=

{f ∈ C(T) : sup

x,h∈T

|f(x + h)− f(x)||h|s <∞.

}.

Then Bernstein’s theorem holds: if f ∈ Λs(T) with s > 12 , then f ∈ �1(Z). We

refer to [35] for further details on the Holder continuity on the torus.

Working on the lattice it is always useful to keep in mind the following:

Definition 3.1.32 (Dirac delta comb). The Dirac delta comb δZn : S(Rn) → C isdefined by

〈δZn , ϕ〉 :=∑

x∈Zn

ϕ(x),

and the sum here is absolutely convergent.

Exercise 3.1.33. Prove that δZn ∈ S ′(Rn).

We recall that the Dirac delta δx ∈ S ′(Rn) at x is defined by δx(ϕ) = ϕ(x)for all ϕ ∈ S(Rn). It may be not surprising that we obtain the Dirac delta combby summing up Dirac deltas over the integer lattice:

Proposition 3.1.34. We have the convergence∑x∈Zn: |x|≤j

δxS′(Rn)−−−−→j→∞

δZn .

Proof. Let us denote Pj :=∑

x∈Zn: |x|≤j δx. If ϕ ∈ S(Rn) then

|〈Pj − δZn , ϕ〉| ≤∑

x∈Zn: |x|>j

|ϕ(x)| ≤∑

x∈Zn: |x|>j

cM 〈x〉−M −−−→j→∞

0

for M large enough (e.g., M = n + 1), proving the claim. �

Another sequence converging to the Dirac delta comb will be shown in Propo-sition 4.6.8.

3.2 Sobolev spaces Hs(Tn)

Fortunately, we have rich structures to work on. The periodic Sobolev spacesHs(Tn) that we introduce in Definition 3.2.2 are actually Hilbert spaces (and inSection 3.5 we prove several auxiliary theorems about continuity and extensionsin Banach spaces that apply in our situation). Here we shall deal with periodicfunctions and distributions on Rn and we shall pursue another more applicable

3.2. Sobolev spaces Hs(Tn) 307

definition of distributions: a Hilbert topology will be given for certain distributionsubspaces, which are the Sobolev spaces. It happens that every periodic distribu-tion belongs to some of these spaces.

Thus, we are attempting to create spaces which include L2(Tn) as a specialcase and which would pay attention to smoothness properties of distributions. Togive an informal motivation, assume that u ∈ L2(Tn) also satisfies ∂αu ∈ L2(Tn)for some α ∈ Nn

0 . Then writing ∂αu in a Fourier series we have

∂αu =∑ξ∈Zn

(i2πξ)α u(ξ) eξ,

with eξ as in (3.7), from which by Parseval’s equality we obtain∫Tn

|∂αu(x)|2 dx = (2π)2|α|∑ξ∈Zn

|ξαu(ξ)|2 ;

with α = 0 this is just the L2-norm. Let us define

〈ξ〉 := (1 + ‖ξ‖2)1/2,

where we recall the notation ‖ξ‖ for the Euclidean norm in (3.3).Remark 3.2.1. This function will be used for measuring decay rates, and otherpossible analogues for (ξ �→ 〈ξ〉) : Zn → R+ would be 1 + ‖ξ‖, or a function equalto ‖ξ‖ for ξ = 0 and to 1 for ξ = 0. The idea here is to get a function ξ �→ 〈ξ〉, whichbehaves asymptotically like the norm ξ �→ ‖ξ‖ when ‖ξ‖ → ∞, and which satisfiesa form of Peetre’s inequality (see Proposition 3.3.31), thus vanishing nowhere.

Definition 3.2.2 (Sobolev spaces Hs(Tn)). For u ∈ D′(Tn) and s ∈ R we definethe norm ‖ · ‖Hs(Tn) by

‖u‖Hs(Tn) :=

⎛⎝ ∑ξ∈Zn

〈ξ〉2s |u(ξ)|2⎞⎠1/2

. (3.9)

The Sobolev space Hs(Tn) is then the space of 1-periodic distributions u for which‖u‖Hs(Tn) <∞. For them, we will formally write their Fourier series representation∑

ξ∈Zn u(ξ) ei2πx·ξ, and in Remark 3.2.5 we give a justification for this. Thus,such u will be also called 1-periodic distributions, represented by the formal series∑

ξ∈Zn u(ξ) ei2πx·ξ. Note that in the definition (3.9) we again take an advantageof Tn: compared to Rn the distributions on the lattice Zn take pointwise values,see Exercise 3.1.7.

Exercise 3.2.3. For example, the 1-periodic Dirac delta δ is expressed by δ(x) =∑ξ∈Zn ei2πx·ξ, or by

(δ(ξ)

)ξ∈Zn

, where δ(ξ) ≡ 1. Show that δ belongs to Hs(Tn)

if and only if s < −n/2.


Exercise 3.2.4. For the function eξ(x) = ei2πx·ξ show that ‖eξ‖Hs(Tn) = 〈ξ〉s.Remark 3.2.5. One can readily see that the union

⋃s∈R Hs(Tn) is the dual of

C∞(Tn) in its uniform topology from Remark 3.1.5 (see Corollary 3.2.12). Forthe details concerning this duality we refer, e.g., to [11, Theorem 6.1]). Henceour definition of the 1-periodic distributions in Definition 3.2.2 coincides with the“official” one in view of the equality

D′(Tn) = L(C∞(Tn), C) =⋃s∈R

Hs(Tn). (3.10)

Proposition 3.2.6 (Sobolev spaces are Hilbert spaces). For every s ∈ R, the Sobolevspace Hs(Tn) is a Hilbert space with the inner product

(u, v)Hs(Tn) :=∑ξ∈Zn

〈ξ〉2s u(ξ) v(ξ).

Proof. The spaces H0(Tn) and Hs(Tn) are isometrically isomorphic by the canon-ical isomorphism ϕs : H0(Tn)→ Hs(Tn), defined by

ϕsu(x) :=∑ξ∈Zn

〈ξ〉−s u(ξ) ei2πx·ξ.

Indeed, ϕs is a linear isometry between Ht(Tn) and Ht+s(Tn) for every t ∈ R,and it is true that ϕs1ϕs2 = ϕs1+s2 and ϕ−1

s = ϕ−s. Then the completeness ofL2(Tn) = H0(Tn) is transferred to that of Hs(Tn) for every s ∈ R. �

Exercise 3.2.7. For k ∈ N0 the traditional Sobolev norm ‖ · ‖′k is defined by

‖u‖′k :=

⎛⎝ ∑|α|≤k

∫Tn

|∂αu(x)|2 dx

⎞⎠1/2

.

Show that‖u‖Hk(Tn) ≤ ‖u‖′k ≤ Ck ‖u‖Hk(Tn),

and try to find the best possible constant Ck <∞. This resembles Definitions 1.5.6and 2.6.15 in the case of Rn, with the equivalence of norms proved in Proposition2.6.16.

Definition 3.2.8 (Banach and Hilbert dualities). We can define different dualitiesbetween Sobolev spaces. The Sobolev space H−s(Tn) is the dual space of Hs(Tn)via the Banach duality product 〈·, ·〉 defined by

〈u, v〉 :=∑ξ∈Zn

u(ξ) v(−ξ),

3.3. Discrete analysis toolkit 309

where u ∈ Hs(Tn) and v ∈ H−s(Tn). Note that 〈u, v〉 =∫

Tn u(x) v(x) dx, whens = 0. Accordingly, the L2- (or H0-) inner product

(u, v)H0(Tn) =∫

Tn

u(x) v(x) dx

is the Hilbert duality product, and Hs(Tn) and H−s(Tn) are duals of each otherwith respect to this duality. If A is a linear operator between two Sobolev spaces,we shall denote its Banach and Hilbert adjoints by A(∗B) and A(∗H), respectively.Often, the Banach adjoint is called the transpose of the operator A and is denotedby At. Then the Hilbert adjoint is simply called the adjoint and denoted by A∗. Forthe relation between Banach and Hilbert adjoints see Definition 2.5.15, Exercise2.5.16 and Remark 2.5.15.

Exercise 3.2.9 (Trigonometric polynomials are dense). Prove that the trigonomet-ric polynomials (and hence also C∞(Tn)) are dense in every Hs(Tn).

Exercise 3.2.10 (Embeddings are compact). Prove that the inclusion ι : Ht(Tn) ↪→Hs(Tn) is compact for s < t.

Exercise 3.2.11 (An embedding theorem). Let m ∈ N0 and s > m + n/2. Provethat Hs(Tn) ⊂ Cm(Tn).

As a corollary, we get

Corollary 3.2.12. We have the equality⋂

s∈R Hs(Tn) = C∞(Tn). By the dualityin Definition 3.2.8 it is related to (3.10) in Remark 3.2.5. Note that the situationon Rn is somewhat more complicated, see Exercise 2.6.19.

Definition 3.2.13 (Biperiodic Sobolev spaces). The biperiodic Sobolev spaceHs,t(Tn × Tn) (s, t ∈ R) is the subspace of biperiodic distributions having thenorm ‖ · ‖s,t defined by

‖v‖s,t :=

⎡⎣ ∑η∈Zn

∑ξ∈Zn

〈η〉2s 〈ξ〉2t |v(η, ξ)|2⎤⎦1/2

, (3.11)

wherev(η, ξ) =

∫Tn

∫Tn

e−η(x) e−ξ(y) v(x, y) dy dx (3.12)

are the Fourier coefficients. It is true that the family of C∞-smooth biperiodicfunctions satisfies C∞(Tn × Tn) =

⋂s,t∈R Hs,t(Tn × Tn). In an obvious manner

one relates all these definitions for 1-periodic spaces Tn = Rn/Zn.

3.3 Discrete analysis toolkit

In this section we provide tools for the study of periodic pseudo-differential opera-tors. In fact, some of the discrete results presented date back to the 18th and 19thcenturies, but seem to have been forgotten in the advent of modern numerical anal-


ysis. Global investigation of periodic functions also requires a special treatment,presented in the last subsection, as well as periodic Taylor series in Section 3.4.

Defining functions on the discrete space Zn instead of Rn, we lose the tra-ditional limit concepts of differential calculus. However, it is worth viewing differ-ences and sums as relatives to derivatives and integrals, and what we shall comeup with is a theory that quite nicely resembles differential calculus. Therefore itis no wonder that this theory is known as the calculus of finite differences.

3.3.1 Calculus of finite differences

In this section we develop the discrete calculus which will be needed in the sequel.In particular, we will formulate and prove a discrete version of the Taylor expansionformula on the lattice Zn. Let us first list some conventions that will be spottedin the formulae: a sum over an empty index set is 0 (empty product is 1), 0! = 1,and heretically 00 = 1. When the index set is known from the context, we mayeven leave it out.

Definition 3.3.1 (Forward and backward differences �αξ and �α

ξ ). Let σ : Zn → Cand 1 ≤ i, j ≤ n. Let δj ∈ Nn

0 be defined by

(δj)i :=

{1, if i = j,

0, if i = j.

We define the forward and backward partial difference operators �ξjand �ξj

,respectively, by

�ξj σ(ξ) := σ(ξ + δj)− σ(ξ),

�ξjσ(ξ) := σ(ξ)− σ(ξ − δj),

and for α ∈ Nn0 define

�αξ := �α1

ξ1· · ·�αn

ξn,

�α

ξ := �α1

ξ1· · ·�αn

ξn.

Remark 3.3.2 (Classical relatives). Several familiar formulae from classical analy-sis have discrete relatives: for instance, it can be easily checked that these differenceoperators commute, i.e., that

�αξ�β

ξ = �βξ�α

ξ = �α+βξ

for all multi-indices α, β ∈ Nn0 . Moreover,

�αξ (sϕ + tψ)(ξ) = s�α

ξ ϕ(ξ) + t�αξ ψ(ξ),

where s and t are scalars.


Exercise 3.3.3. Prove these formulae.

Proposition 3.3.4 (Formulae for �αξ and �α

ξ ). Let φ : Zn → C. We have

�αξ φ(ξ) =

∑β≤α

(−1)|α−β|(

α

β

)φ(ξ + β),

�α

ξ φ(ξ) =∑β≤α

(−1)|β|(

α

β

)φ(ξ − β).

Proof. Let us introduce translation operators Ej := (I +�ξj), acting on functions

φ : Zn → C byEjφ(ξ) := (I +�ξj

)φ(ξ) = φ(ξ + δj).

Let Eα := Eα11 · · ·Eαn

n . An application of the binomial formula is enough:

�αξ φ(ξ) = (E − I)αφ(ξ)

=∑β≤α

(α

β

)(−1)|α−β| Eβφ(ξ)

=∑β≤α

(−1)|α−β|(

α

β

)φ(ξ + β).

The backward difference equality is left for the reader to prove as Exercise 3.3.5.�

Exercise 3.3.5. Notice that Ej�ξj= �ξj = �ξj

Ej . Complete the proof of Propo-sition 3.3.4.

The discrete Leibniz formula is complicated enough to have a proof of its own,and it can be compared with the Leibniz formula on Rn in Theorem 1.5.10, (iv).

Lemma 3.3.6 (Discrete Leibniz formula). Let φ, ψ : Zn → C. Then

�αξ (φψ)(ξ) =

∑β≤α

(α

β

) (�β

ξ φ(ξ))�α−β

ξ ψ(ξ + β). (3.13)

Proof. (Another proof idea, not using induction, can be found in [117, p. 11] and[52, p. 16].) First, we have an easy check

�ξj(ϕψ)(ξ) = (ϕψ)(ξ + δj)− (ϕψ)(ξ)

= ϕ(ξ) (ψ(ξ + δj)− ψ(ξ)) + (ϕ(ξ + δj)− ϕ(ξ)) ψ(ξ + δj)= ϕ(ξ) �ξj

ψ(ξ) +(�ξj

ϕ(ξ))

ψ(ξ + δj).


We use this and induction on α ∈ Nn0 :

�α+δj

ξ (ϕψ)(ξ) = �ξj�α

ξ (ϕψ)(ξ)

= �ξj

∑β≤α

(α

β

) (�β

ξ ϕ(ξ))�α−β

ξ ψ(ξ + β)

=∑β≤α

(α

β

) [(�β

ξ ϕ(ξ))�α+δj−β

ξ ψ(ξ + β)

+(�β+δj


ξ ψ(ξ + β + δj)]

(�)=

∑β≤α+δj

[(α

β

)+

(α

β − δj

)] (�β


ξ ψ(ξ + β)

=∑

β≤α+δj

(α + δj

β

) (�β


ξ ψ(ξ + β).

In (�) above, we used the convention that(αγ

)= 0 if γ ≤ α or if γ ∈ Nn

0 . The proofis complete. �Exercise 3.3.7. Verify that(

α

β

)+

(α

β − δj

)=

(α + δj

β

)in the proof of (3.13).

Remark 3.3.8 (Discrete product rule – notice the shifts). Notice the shift in (3.13)in the argument of ψ. For example, already the product rule becomes

�ξj(ϕψ)(ξ) = ϕ(ξ) �ξj

ψ(ξ) +(�ξj

ϕ(ξ))

ψ(ξ + δj).

The shift is caused by the difference operator �ξ, and it is characteristic to thecalculus of finite differences – in classical Euclidean analysis it is not present.This shift will have its consequences for the whole theory of pseudo-differentialoperators on the torus in Chapter 4, especially for the formulae in the calculus.

Exercise 3.3.9. Prove the following form of the discrete Leibniz formula:

�αξ (ϕ(ξ) ψ(ξ)) =

∑β≤α

(α

β

) (�β


ξ ψ(ξ + α).

As it is easy to guess, in the calculus of finite differences, sums correspondto integrals of classical analysis, and the theory of series (presented, e.g., in [66])serves as an integration theory. Assuming convergence of the following series, itholds that ∑

ξ

(sϕ(ξ) + tψ(ξ)) = s∑

ξ

ϕ(ξ) + t∑

ξ

ψ(ξ),


and when a ≤ b on Z1, we have an analogue of the fundamental theorem ofcalculus:

b∑ξ=a

�ξψ(ξ) = ψ(b + 1)− ψ(a).

Difference and partial difference equations (cf. differential and partial differential)are handled in several books concerning combinatorics or difference methods (e.g.,[52]), but various mean value theorems have no straightforward interpretation here,since the functions are usually defined only on a discrete set of points (although onecan use some suitable interpolation; we refer to Theorem 3.3.39 and Section 4.5).Integration by parts can be, however, translated for our purposes:

Lemma 3.3.10 (Summation by parts). Assume that ϕ, ψ : Zn → C. Then∑ξ∈Zn

ϕ(ξ) �αξ ψ(ξ) = (−1)|α|

∑ξ∈Zn

(�ξ

αϕ(ξ)

)ψ(ξ) (3.14)

provided that both series are absolutely convergent.

Proof. Let us check the case |α| = 1:∑ξ∈Zn

ϕ(ξ) �ξjψ(ξ) =

∑ξ∈Zn

(ψ(ξ + δj)− ψ(ξ)) ϕ(ξ)

=∑ξ∈Zn

ψ(ξ) (−ϕ(ξ) + ϕ(ξ − δj))

= (− 1)1∑ξ∈Zn

ψ(ξ) �ξjϕ(ξ).

For any α ∈ Nn0 the result is obtained recursively. �

Exercise 3.3.11. Complete the proof of (3.14) for |α| ≥ 2.

3.3.2 Discrete Taylor expansion and polynomials on Zn

The usual polynomials θ �→ θα do not behave naturally with respect to differences:typically�γ

θθα = cαγ θα−γ for any constant cαγ . Thus let us introduce polynomialsθ �→ θ(α) to cure this defect:

Definition 3.3.12 (Discrete polynomials). For θ ∈ Zn and α ∈ Nn0 , we define

θ(α) = θ(α1)1 · · · θ(αn)

n , where θ(0)j = 1 and

θ(k+1)j = θ

(k)j (θj − k) = θj(θj − 1) . . . (θj − k). (3.15)

Exercise 3.3.13. Show that

�γθθ(α) = α(γ) θ(α−γ),

in analogy to the Euclidean case where ∂γθ θα = α(γ) θα−γ .


Remark 3.3.14. Difference operators lessen the degree of a polynomial by 1. Inthe literature on numerical analysis the polynomials θ �→ θ(α) appear sometimesin a concealed form using the binomial coefficients:

θ(α) = α!(

θ

α

).

Next, let us consider “discrete integration”.

Definition 3.3.15 (Discrete integration). For b ≥ 0, let us write

Ibk :=

∑0≤k<b

and I−bk := −

∑−b≤k<0

. (3.16)

In the sequel, we adopt the notational conventions

Iθk1

Ik1k2· · · Ikα−1

kα1 =

⎧⎪⎨⎪⎩1, if α = 0,

Iθk1

1, if α = 1,

Iθk1

Ik1k2

1, if α = 2,

and so on.

Remark 3.3.16. One can think of Iθξ · · · as a discrete version of the one-dimensional

integral∫ θ

0· · ·dξ; in this discrete context, the difference �ξ takes the role of the

differential operator d/dξ.

Lemma 3.3.17 (Discrete “fundamental theorem of calculus” 1D). If θ ∈ Z andα ∈ N0 then

Iθk1

Ik1k2· · · Ikα−1

kα1 =

1α!

θ(α). (3.17)

Proof. We observe simple equalities k(0) ≡ 1, �kk(i) = i k(i−1) and Ibk�kk(i) =

b(i), from which (3.17) follows by induction. �

Remark 3.3.18. We note that Lemma 3.3.17 can be viewed as a discrete trivialversion of the fundamental theorem of calculus:∫ θ

0

f ′(ξ) dξ = f(θ)− f(0)

for smooth enough f : R→ C corresponds to

Iθξ�ξf(ξ) = f(θ)− f(0)

for f : Z→ C.


Lemma 3.3.17 immediately implies its multidimensional version:

Corollary 3.3.19 (Discrete “fundamental theorem of calculus”). If θ ∈ Zn andα ∈ Nn

0 thenn∏

j=1

Iθj

k(j,1)Ik(j,1)k(j,2) · · · I

k(j,αj−1)

k(j,αj)1 =

1α!

θ(α), (3.18)

where∏n

j=1 Ij means I1I2 · · · In, where Ij := Iθj

k(j,1)Ik(j,1)k(j,2) · · · I

k(j,αj−1)

k(j,αj).

Exercise 3.3.20. Work out the details for the proof of Corollary 3.3.19.

Now we are about to present a combinatorial tool of uttermost importance: adiscrete version of the classical Taylor polynomial expansion theorem. The resultis exactly what one might expect, as differences correspond to derivatives, andpolynomials θ �→ θ(α) replace θ �→ θα; also the error estimate seems familiar. Inthe sequel, the word “series” will also be used in those cases, where the summationindex set is finite.

Theorem 3.3.21 (Discrete Taylor expansion on Zn). Let p : Zn → C. Then we canwrite

p(ξ + θ) =∑|α|<M

1α!

θ(α) �αξ p(ξ) + rM (ξ, θ),

with the remainder rM (ξ, θ) satisfying∣∣�ωξ rM (ξ, θ)

∣∣ ≤ CM max|α|=M, ν∈Q(θ)

∣∣∣θ(α)�α+ωξ p(ξ + ν)

∣∣∣ , (3.19)

where Q(θ) := {ν ∈ Zn : |νj | ≤ |θj | for all j = 1, . . . , n}.Remark 3.3.22. Notice that the estimate above resembles closely the Lagrangeform of the error in the traditional Taylor expansion theorem for f ∈ C∞(Rn):

f(x + h) =∑|α|<M

1α!

∂αf(x) hα + RM (x, h),

RM (x, h) =∑|α|=M

1α!

Rα(x, h) hα,

|Rα(x, h)| ≤ supθ|∂αf(θ)| ,

where supθ is taken over the segment between x and x + h.

Exercise 3.3.23. Let |β| < M = |α|. Regarding Theorem 3.3.21, show that{�α

θ rM (ξ, θ) = �αθ p(ξ + θ),

�βξ rM (ξ, θ)

∣∣∣θ=0

= 0.

Formally, this involves a discrete Cauchy problem.


The essential ideas for proving Theorem 3.3.21 are most transparent in thecase of n = 1:

Proof of Theorem 3.3.21 for dimension n = 1. We claim that the remainder is

rM (ξ, θ) = Iθk1

Ik1k2· · · IkM−1

kM�M

ξ p(ξ + kM ). (3.20)

This is easily verified for

r1(ξ, θ) = p(ξ + θ)− p(ξ) = Iθk1�ξp(ξ + k1).

We proceed by induction. Thus

rM+1(ξ, θ) = rM (ξ, θ)− 1M !

θ(M) �Mξ p(ξ)

(3.20), (3.17)= Iθ

k1Ik1k2· · · IkM−1

kM�M

ξ (p(ξ + kM )− p(ξ))

= Iθk1

Ik1k2· · · IkM−1

kMIkM

kM+1�M+1

ξ p(ξ + kM+1).

Applying (3.17) to (3.20), we obtain∣∣�ωξ rM (ξ, θ)

∣∣ ≤ 1M !

∣∣∣θ(M)∣∣∣ max

ν∈{0,...,θ}

∣∣∣�M+ωξ p(ξ + ν)

∣∣∣ . (3.21)

Hence the proof of the one-dimensional discrete Taylor theorem is complete. �

Remark 3.3.24. The discrete Taylor formula is not a new invention: it can be ob-tained from the Newton interpolation formula, but this direct proof is beautifullysimple, assumes the function to be defined on a discrete space only, and exposes afundamental structure of the calculus of finite differences. In fact, Niels Norlund’s([85, p. 11]) Newtonsche Interpolationsformel and Steffensen’s ([117, p. 22]) New-ton’s interpolation-formula with finite differences are equivalent to Theorem 3.3.21,but they are disguised beyond recognition. Already George Boole presented theRemainder in the Generalised form of Taylor’s (sic) theorem in [15, p. 146]. How-ever, Boole’s RN does not easily give a nice approximation for |�l

ηRN (ξ, η)|; inany case it reveals the connection to differential calculus better than later books.Charles Jordan’s ([60, p. 75, 164]) generalised Newton series masks the Taylor-likeness under the binomial coefficients, and the convergence of an infinite series isassumed, thus avoiding the study of the remainder. In the modern literature, thesituation is even worse as the Newton interpolation formula is usually spoiled inmyriad ways destroying its Taylor appearance. However, Newton’s (or Gregory’s)formula in [52, p. 9] is a welcome exception, but again it expresses the error interms of derivatives.

Numerical analysis, especially interpolation theory, may yield useful tools fordiscrete purposes. Briefly, almost everything that you can do in classical analysisis in some sense legal when working with differences.


Now we prove the multidimensional discrete version of the Taylor expansionformula.

Proof of Theorem 3.3.21 for the n-dimensional Taylor expansion. We encouragethe reader to compare this general proof to the earlier deduction of the low-dimensional case n = 1. For 0 = α ∈ Nn

0 , let us denote mα := min{j : αj = 0}.For θ ∈ Zn and i ∈ {1, . . . , n}, let us define ν(θ, i, k) ∈ Zn by

ν(θ, i, k) := (θ1, . . . , θi−1, k, 0, . . . , 0),

i.e.,

ν(θ, i, k)j =

⎧⎪⎨⎪⎩θj , if 1 ≤ j < i,

k, if j = i,

0, if i < j ≤ n.

We claim that the remainder can be written in the form

rM (ξ, θ) =∑|α|=M

rα(ξ, θ), (3.22)

where for each α, we have

rα(ξ, θ) =n∏

j=1

Iθj

k(j,1)Ik(j,1)k(j,2) · · · I

k(j,αj−1)

k(j,αj)�α

ξ p(ξ + ν(θ, mα, k(mα, αmα))); (3.23)

recall (3.16) and (3.18). The proof of (3.23) is by induction. The first remainderterm r1 is of the claimed form, since

r1(ξ, θ) = p(ξ + θ)− p(ξ) =n∑

i=1

rδi(ξ, θ),

whererδi(ξ, θ) = Iθi

k �δi

ξ p(ξ + ν(θ, i, k));

here rδi is of the form (3.23) for α = δi, m(α) = i and αmα = 1. So suppose thatthe claim (3.23) is true up to order |α| = M . Then

rM+1(ξ, θ) = rM (ξ, θ)−∑|α|=M

1α!

θ(α) �αξ p(ξ)

=∑|α|=M

(rα(ξ, θ)− 1

α!θ(α) �α

ξ p(ξ))

=∑|α|=M

n∏j=1

Iθj

k(j,1)Ik(j,1)k(j,2) · · · I

k(j,αj−1)

k(j,αj)

× �αξ [p(ξ + ν(θ, mα, k(mα, αmα

)))− p(ξ)] ,


where we used (3.23) and (3.18) to obtain the last equality. Combining this withthe equality

p(ξ + ν(θ, mα, k))− p(ξ) =mα∑i=1

Iν(θ,mα,k)i

� �δi

ξ p(ξ + ν(θ, i, �)),

we get

rM+1(ξ, θ) =∑|α|=M

n∏j=1

Iθj

k(j,1)Ik(j,1)k(j,2) · · · I

k(j,αj−1)

k(j,αj)

mα∑i=1

Iν(θ,mα,k(mα,αmα ))i

�(i)

�α+δi

ξ p(ξ + ν(θ, i, �(i)))

=∑

|β|=M+1

n∏j=1

Iθj

k(j,1)Ik(j,1)k(j,2) · · · I

k(j,βj−1)

k(j,βj)

�βξ p(ξ + ν(θ, mβ , k(mβ , βmβ

)));

the last step here is just simple tedious book-keeping. Thus the induction proof of(3.23) is complete. Finally, let us prove estimate (3.19). By (3.23), we obtain∣∣�ω

ξ rM (ξ, θ)∣∣

=

∣∣∣∣∣∣∑|α|=M

�ωξ rα(ξ, θ)

∣∣∣∣∣∣=

∣∣∣∣∣∣∑|α|=M

n∏j=1

Iθj

k(j,1)Ik(j,1)k(j,2) · · · I

k(j,αj−1)

k(j,αj)�α+ω

ξ p(ξ + ν(θ, mα, k(mα, αmα)))

∣∣∣∣∣∣≤

∑|α|=M

1α!

∣∣∣θ(α)∣∣∣ max

ν∈Q(θ)

∣∣∣�α+ωξ p(ξ + ν)

∣∣∣ ,

where in the last step we used (3.18). The proof is complete. �Remark 3.3.25 (Remainder). If n ≥ 2, there are many alternative forms of remain-ders rα(ξ, θ). This is due to the fact that there may be many different shortestdiscrete step-by-step paths in the space Zn from ξ to ξ + θ; in the proof above, wechose just one such path, traveling via the points

ξ, ξ + θ1δ1, . . . , ξ +j∑

i=1

θiδi, . . . , ξ + θ.

But if n = 1, there is just one shortest discrete path from ξ ∈ Z to θ ∈ Z, and inthat case

rM (ξ, θ) = Iθk1

Ik1k2· · · IkM−1

kM�M

ξ p(ξ + kM ).

In Theorem 3.3.21 we estimated the remainder over the discrete box Q(θ), but anestimate over a discrete path would have been enough.


Notice also that the discrete Taylor theorem presented above implies thefollowing smooth Taylor result:

Corollary 3.3.26 (Discrete =⇒ continuous). Let p ∈ C∞(Rn) and

rM (ξ, θ) := p(ξ + θ)−∑|α|<M

1α!

θα

(∂

∂ξ

)α

p(ξ).

Then we have∣∣∂ωξ rM (ξ, θ)

∣∣ ≤ cM max|α|=M, ν∈QRn (θ)

∣∣∣θα ∂α+ωξ p(ξ + ν)

∣∣∣ , (3.24)

where QRn(θ) := {ν ∈ Rn : min(0, θj) ≤ νj ≤ max(0, θj)}.Remark 3.3.27. It is evident from the proof that, in the remainder estimatesabove, the cubes Q(θ) ⊂ Zn and QRn(θ) ⊂ Rn could be replaced by (discrete,respectively continuous) paths from 0 to θ; e.g., on Rn, the cube could be replacedby the straight line from 0 to θ.

3.3.3 Several discrete inequalities

In the symbolic analysis of periodic pseudo-differential operators we frequentlyneed the inequalities of Young and Peetre, and that is why we present themseparately here. Here �p = �p(Zn), 1 ≤ p < ∞, is the space of those complex“sequences” f = (f(ξ))ξ∈Zn (i.e., mappings f : Zn → C), for which the norm

‖f‖�p :=

⎡⎣ ∑ξ∈Zn

|f(ξ)|p⎤⎦1/p

is finite. For p =∞ the expression of the norm is modified into

‖f‖�∞ = supξ∈Zn

|f(ξ)|.

We will write �p for �p(Zn) here.Looking back to the definition of the periodic Sobolev spaces Hs(Tn), it

becomes evident that the exponent p = 2 is the significant one, as ‖ · ‖H0(Tn) =‖ · ‖�2 . Naturally, the Holder’s and Minkowski’s inequalities hold true in discretecases.

Lemma 3.3.28 (Discrete Holder’s inequality). Let 1 ≤ p, q ≤ ∞ be conjugate, i.e.,1p + 1

q = 1. Let f ∈ �p and g ∈ �q. Then

‖fg‖�1 ≤ ‖f‖�p‖g‖�q .


Of course, the general Holder inequality in Theorem C.4.4 implies our lemma,but this special case is so much easier to prove that it is noteworthy to study itindependently:

Exercise 3.3.29. Find a simple proof of Holder’s inequality in Lemma 3.3.28.

Lemma 3.3.30 (Discrete Young’s inequality). Assume that h : Zn × Zn → C is afunction satisfying

C1 := supξ∈Zn

∑η∈Zn

|h(η, ξ)| <∞, C2 := supη∈Zn

∑ξ∈Zn

|h(η, ξ)| <∞.

Let 1 ≤ p ≤ ∞. For any sequence f ∈ �p let us define g : Zn → C by g(η) =∑ξ∈Zn h(η, ξ) f(ξ). Then

‖g‖�p ≤ C1/p1 C

1/q2 ‖f‖�p ,

where q is the conjugate exponent of p, i.e., 1p + 1

q = 1.

Proof. By the discrete Holder’s inequality (Lemma 3.3.28) we have∑ξ∈Zn

|h(η, ξ)| |f(ξ)| =∑ξ∈Zn

[|h(η, ξ)|1/p |f(ξ)|

] [|h(η, ξ)|1/q

]

≤

⎡⎣ ∑ξ∈Zn

|h(η, ξ)| |f(ξ)|p⎤⎦1/p ⎡⎣ ∑

ξ∈Zn

|h(η, ξ)|

⎤⎦1/q

≤

⎡⎣ ∑ξ∈Zn

|h(η, ξ)| |f(ξ)|p⎤⎦1/p

C1/q2 .

Using this we get

‖g‖p�p =

∑η∈Zn

∣∣∣∣∣∣∑ξ∈Zn

h(η, ξ) f(ξ)

∣∣∣∣∣∣p

≤∑

η∈Zn

⎡⎣ ∑ξ∈Zn

|h(η, ξ) f(ξ)|

⎤⎦p

≤ Cp/q2

∑η∈Zn

∑ξ∈Zn

|h(η, ξ)| |f(ξ)|p

= Cp/q2

∑ξ∈Zn

|f(ξ)|p∑

η∈Zn

|h(η, ξ)|

≤ C1Cp/q2

∑ξ∈Zn

|f(ξ)|p

=(C

1/p1 C

1/q2

)p

‖f‖p�p . �


Recall that 〈ξ〉 = (1 + ‖ξ‖2)1/2, with ‖ξ‖ as in (3.3). Working with pseudo-differential operators, some form of the elementary inequality of Jaak Peetre hasto be derived at some stage. The most important variant for us is the following:

Proposition 3.3.31 (Peetre’s inequality). For all s ∈ R and ξ, η ∈ Rn, we have

〈ξ + η〉s ≤ 2|s|〈ξ〉s〈η〉|s|.

Proof. We have 〈ξ + η〉 ≤ 〈η〉+ 〈ξ〉 ≤ 2〈η〉〈ξ〉, so that

〈ξ + η〉t ≤ 2t〈η〉t〈ξ〉t,

when t ≥ 0. Thus also〈ξ〉t ≤ 2t〈−η〉t〈ξ + η〉t

implying〈ξ + η〉−t ≤ 2t〈η〉t〈ξ〉−t.


Exercise 3.3.32. Let ξ, η ∈ Rn. Prove the following kindred Peetre inequalities:

(1 + ‖ξ + η‖)s ≤ 2|s| (1 + ‖ξ‖)|s| (1 + ‖η‖)s, (3.25)

ξ + ηs ≤ 2|s| ξ|s| ηs,

where ω := max {1, ‖ω‖}. This latter form of the weight is used, e.g., in [102].

3.3.4 Linking differences to derivatives

In the light of contemporary analysis, the words of Charles Jordan ([60]) havenot expired: “The importance of Stirling’s numbers in Mathematical Calculus hasnot yet been fully recognised, and they are seldom used. This is especially due tothe fact that different authors have reintroduced them under different definitionsand notations, often not knowing, or not mentioning, that they deal with the samenumbers. . . ”. Many properties of these numbers can be found in [2], and they canbe defined in various ways. We shall explain their relation to combinatorics, yetwe start with another approach, since our purpose is to establish a connectionbetween difference and differential calculi:

Definition 3.3.33 (Stirling numbers). Let x ∈ R and j, k ∈ N0 such that j ≤ k.The Stirling numbers S

(j)k of the first kind are defined by the formula

x(k) =k∑

j=0

S(j)k xj .


The uniqueness of S(j)k is obvious. Stirling numbers

{kj

}of the second kind are a

sort of dual for the first kind:

xk =k∑

j=0

{kj

}x(j).

For j < 0 and j > k, it is natural to extend these definitions by

S(j)k := 0 and

{kj

}:= 0.

For multi-indices α, β ∈ Zn, let

S(β)α := S(β1)

α1· · ·S(βn)

αn,{

αβ

}:=

{α1

β1

}· · ·

{αn

βn

}.

Lemma 3.3.34. Let j, k ∈ N0. Then

S(j)k =

1j!

(ddx

)j

x(k)∣∣∣x=0

,

{kj

}=

1j!�j

ξ ξk∣∣ξ=0

.

Proof. The first formula is a direct consequence of the definition of S(j)k . Of course,

so is the second one, but since we do not have as profound an experience ofdifference calculus, we go through a simple calculation:

�jξ ξk

∣∣ξ=0

= �jξ

k∑i=0

{ki

}ξ(i)

∣∣∣ξ=0

=k∑

i=0

{ki

}i(j) δi,j = j!

{kj

}. �

Exercise 3.3.35. Let x ∈ Rn and α, β ∈ Nn0 . Show that

x(α) =∑β≤α

S(β)α xβ , S(β)

α =1β!

∂βxx(α)

∣∣∣x=0

,

xα =∑β≤α

{αβ

}x(β),

{αβ

}=

1β!�β

ξ ξα∣∣∣ξ=0

.

Lemma 3.3.36 (Recursion formulae for Stirling numbers). Stirling numbers areuniquely determined by the recursions⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

S(0)k = δ0,k,

{k0

}= δ0,k, k ∈ N0,

S(k)k = 1,

{kk

}= 1, k ∈ N0,

S(j)k = 0,

{kj

}= 0, j < 0 or j > k,

S(j)k+1 = S

(j−1)k − kS

(j)k ,

{k + 1

j

}=

{k

j − 1

}+ j

{kj

}, j ≥ 1, k ≥ 0,

where δp,q is the Kronecker delta, defined by δp,p = 1 and by δp,q = 0 for p = q.


Proof. From x(0) = 1 = S(0)0 and x(k+1) = x(x−1) · · · (x−k) we see that it has to

be S(0)k = δ0,k and that S

(k)k = 1 for every k ∈ N0. The statement S

(j)k = 0 =

{kj

}when j < 0 or j > k simply rephrases a part of the extended definition for Stirlingnumbers. Suppose that x(k) =

∑kj=0 S

(j)k xj . Then

k+1∑j=0

S(j)k+1x

j = x(k+1) = (x− k)x(k)

= S(k)k xk+1 − k S

(0)k +

k∑j=1

[S

(j−1)k − k S

(j)k

]xj

=k+1∑j=0

[S

(j−1)k − k S

(j)k

]xj .

The case of the first kind is thus concluded. No doubt{

kk

}= 1 for every k ∈ N0,

and clearly{

k0

}= δ0,k. Assume that xk =

∑kj=0

{kj

}x(j). Then

k+1∑j=0

{k + 1

j

}x(j) = xk+1 = x xk = x

k∑j=0

{kj

}x(j)

=k∑

j=0

(x− j + j){

kj

}x(j)

=k∑

j=0

[{kj

}x(j+1) + j

{kj

}x(j)

]

=k+1∑j=0

[{k

j − 1

}+ j

{kj

}]x(j),

so that we can calculate{

kj

}by recursion. �

The general solution of the difference equation S(j)k+1 = S

(j−1)k − k S

(j)k is

unknown (see [60, p. 143]), but for the second kind there is a closed form withoutrecursion (it is easily obtained by applying Proposition 3.3.4 on Lemma 3.3.34):

{kj

}=

1j!

j∑i=0

(−1)j−i

(j

i

)ik.


There are combinatorial ideas behind the Stirling numbers, as explained, e.g., in

[24]. The following exercise collects these ideas, using notations(

kj

),

[kj

],

{kj

}of [67]:

Exercise 3.3.37 (Combinatorial background). Let j, k ∈ N0 such that j ≤ k, andlet S be a set with exactly k elements. Show that S has(

k

j

)=

k!j! (k − j)!

subsets of exactly j elements (as usual, read: “k choose j”). Moreover, prove that[kj

]:= (−1)k−jS

(j)k

is the number of permutations of S with precisely j cycles. Finally, show that{kj

}is the number of ways to partition S into j non-empty subsets, i.e., the number ofthe equivalence relations on S having j equivalence classes (read: “k quotient j”).Hint: exploit the recursions in Lemma 3.3.36.

In the following matrices some of the Stirling numbers are presented. Theindex j is used for rows and k for columns:

(S

(j)k

)5

j,k=0=

⎛⎜⎜⎜⎜⎜⎜⎝1 0 0 0 0 00 1 −1 2 −6 240 0 1 −3 11 −500 0 0 1 −6 350 0 0 0 1 −100 0 0 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎠ ,

({kj

})5

j,k=0

=

⎛⎜⎜⎜⎜⎜⎜⎝1 0 0 0 0 00 1 1 1 1 10 0 1 3 7 150 0 0 1 6 250 0 0 0 1 100 0 0 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎠ .

Such matrices are inverses to each other:

Lemma 3.3.38. Assume that i, j, N ∈ N0 such that 0 ≤ i, j ≤ N . Then

N∑k=0

S(i)k

{jk

}= δi,j =

N∑k=0

{ki

}S

(k)j .


Proof. Due to the symmetry, it suffices to prove only that the first sum equals δi,j :

xj =j∑

k=0

{jk

}x(k) =

N∑k=0

{jk

}x(k)

=N∑

k=0

{jk

} k∑i=0

S(i)k xi =

N∑k=0

{jk

} N∑i=0

S(i)k xi

=N∑

i=0

xiN∑

k=0

S(i)k

{jk

}. �

In the sequel, we shall present two alternative definitions for periodic pseudo-differential operators. To build a bridge between these approaches (in Lemma4.7.1), we have to know how to approximate differences by derivatives. This prob-lem is considered, for example, in [60, p. 164–165, 189–192], where the error es-timates are neglected, as well as in the treatise of the subject in [15]. FrancisHildebrand ([52, p. 123–125]) makes a notice on the estimates, but does not calcu-late them, and there the approximation is thoroughly presented only for degreesj = 1, 2, and without a connection to the Stirling numbers. The finest account isby Steffensen in [117, p. 60-70], where the presentation of Markoff’s formulae isgeneral with error terms for any degree, but still it lacks the Stirling numbers.

The following theorem is in one dimension, so here ϕ(k) denotes the usualkth derivative of ϕ.

Theorem 3.3.39 (Approximating differences by derivatives). There exist constantsc�N,j , c

dN,j > 0 for any N ∈ N0 and j < N such that for every ϕ ∈ C∞(R1) and

ξ ∈ R1 the following inequalities hold:∣∣∣∣∣∣�jξϕ(ξ)−

N−1∑k=j

j!k!

{kj

}ϕ(k)(ξ)

∣∣∣∣∣∣ ≤ c�N,j maxη∈[0,j]

∣∣∣ϕ(N)(ξ + η)∣∣∣ , (3.26)

∣∣∣∣∣∣ϕ(j)(ξ)−N−1∑k=j

j!k!

S(j)k �k

ξϕ(ξ)

∣∣∣∣∣∣ ≤ cdN,j max

η∈[0,N−1]

∣∣∣ϕ(N)(ξ + η)∣∣∣ . (3.27)

Proof. Note that in the following we cannot apply the discrete Taylor series, be-cause its remainder is defined only on Z1 with respect to the variable η. Theclassical Taylor series does not have this disadvantage:

ϕ(ξ + η) =N−1∑k=0

1k!

ϕ(k)(ξ) ηk +1

N !ϕ(N)(θ(η)) ηN .

We use the Lagrange form of the error term. Here θ(η) is some point in the segmentconnecting ξ and ξ + η. Assume that N > j. Applying �j

η at η = 0, and using


Lemma 3.3.34 we get

�jξϕ(ξ) = �j

η

[N−1∑k=0

1k!

ϕ(k)(ξ) ηk +1

N !ϕ(N)(θ(η)) ηN

]∣∣∣∣∣η=0

=N−1∑k=j

j!k!

{kj

}ϕ(k)(ξ) +

1N !�j

η

[ϕ(N)(θ(η)) ηN

]∣∣∣η=0

. (3.28)

Using the Leibniz formula on the remainder term, we see that its absolute value ismajorised by c�N,j

∣∣ϕ(N)(θj)∣∣ for some θj ∈ [ξ, ξ+j], and hence (3.26) is true. For the

latter inequality (3.27), the “orthogonality” of Stirling numbers (Lemma 3.3.38),and (3.28) are essential:

N−1∑k=i

i!k!

S(i)k �k

ξϕ(ξ) =N−1∑k=i

i!k!

S(i)k

N−1∑j=k

k!j!

{jk

}ϕ(j)(ξ)

+N−1∑k=i

i!k!

S(i)k

1N !�k

η

[ϕ(N)(θ(η)) ηN

]∣∣∣η=0

=N−1∑j=i

i!j!

ϕ(j)(ξ)j∑

k=i

S(i)k

{jk

}

+N−1∑k=i

i!k!

S(i)k

1N !�k

η

[ϕ(N)(θ(η)) ηN

]∣∣∣η=0

= ϕ(i)(ξ) +N−1∑k=i

i!k!

S(i)k

1N !�k

η

[ϕ(N)(θ(η)) ηN

]∣∣∣η=0

,

where the absolute value of the remainder part is estimated above by somecdN,j

∣∣ϕ(N)(θN )∣∣ (cf. the proof of (3.26)). �

Inequality (3.27) is not actually needed in this work, but as a dual statementto (3.26) it is justified. Note that in (3.26) the maximum of |ϕ(N)(ξ + η)| is takenover the interval η ∈ [0, j], whereas in (3.27) over η ∈ [0, ξ − 1].

Exercise 3.3.40. Let α, β ∈ Nn0 , ξ ∈ Zn and ϕ ∈ C∞(Rn). Estimate∣∣∣∣∣∣�β

ξ ϕ(ξ)−∑|α|<N

β!α!

{αβ

}∂αϕ(ξ)

∣∣∣∣∣∣ and

∣∣∣∣∣∣∂βϕ(ξ)−∑|α|<N

β!α!

S(β)α �α

ξ ϕ(ξ)

∣∣∣∣∣∣in the manner of Theorem 3.3.39.

3.4. Periodic Taylor expansion 327

3.4 Periodic Taylor expansion

For the global analysis on the torus, the ordinary Taylor series is useless due to thelack of periodicity. We are now going to present a tool to fill in this gap. It mustbe admitted, though, that this approach is not a necessary one, as we can studythe theory of periodic pseudo-differential operators without it. Nevertheless, theperiodic Taylor series provides an appropriate alternative, and one never knowswhen it is gravely needed.

Definition 3.4.1 (Discrete modification of partial derivatives). For α ∈ Nn0 , let us

introduce an abbreviation for partial derivatives,

Dαx := Dα1

x1· · ·Dαn

xn,

D(α)x := D(α1)

x1· · ·D(αn)

xn,

where for k ∈ N0,

Dkxl

:=(

1i2π

∂

∂xl

)k

,

D(k)xl

:=k−1∏j=0

(1

i2π

∂

∂xl− j

), (3.29)

in the spirit of Stirling numbers. We interpret D0x = I = D

(0)x .

Theorem 3.4.2 (Periodic Taylor expansion for T1). Any a ∈ C∞(T1) has theperiodic Taylor representation

a(x) =N−1∑j=0

1j!

(ei2πx − 1

)jD(j)

z a(z)∣∣∣z=0

+ aN (x)(ei2πx − 1

)N,

where aN ∈ C∞(T1).

Proof. For j ∈ N0, we define the functions aj by

a0(x) := a(x), aj+1(x) :=

{aj(x)−aj(0)

ei2πx−1 , if x = 0,

Dxaj(x), if x = 0.

Inductively we obtain that aj+1 is in C∞(T1). Thus

aj(x) = aj(0) + aj+1(x)(ei2πx − 1

),

and recursively

a(x) =N−1∑j=0

(ei2πx − 1

)jaj(0) + aN (x)

(ei2πx − 1

)N, (3.30)

so we have to prove that aj(0) = 1j! D

(j)x a(x)

∣∣∣x=0

.


Clearly expression D(j)x

(ei2πx − 1

)k∣∣∣x=0

vanishes, if j < k. It vanishes alsowhen j > k, since(

1i2π

∂

∂x− k

) (ei2πx − 1

)k= k ei2πx

(ei2πx − 1

)k−1 − k(ei2πx − 1

)k

= k(ei2πx − 1

)k−1

implies that [k∏

m=1

(1

i2π

∂

∂x−m

)] (ei2πx − 1

)k= k!.

We use this instantly:

D(k)x

(ei2πx − 1

)k∣∣∣x=0

=(

1i2π

∂

∂x− k + k

) k−1∏j=1

(1

i2π

∂

∂x− j

) (ei2πx − 1

)k∣∣∣x=0

= k!.

Hence we get D(j)x

(ei2πx − 1

)k∣∣∣x=0

= j! δj,k, so that by applying D(j)x to both sides

of the equality (3.30) we obtain D(j)x a(x)

∣∣∣x=0

= j! aj(0) proving the claim. �

As an immediate consequence of Theorem 3.4.2 we get the biperiodic Taylorseries:

Corollary 3.4.3 (Biperiodic Taylor expansion). Any function a ∈ C∞(T1×T1) hasthe Taylor representation

a(x, y) =N−1∑j=0

1j!

(ei2π(y−x) − 1

)j

D(j)z a(z, y)

∣∣∣z=x

+aN (x, y)(ei2π(y−x) − 1

)N

,

where aN ∈ C∞(T1 × T1).

We shall return to the biperiodic expansion in the study of the amplitudesof periodic pseudo-differential operators (see Lemma 4.3.4 and Theorem 4.4.5).Instead of such biperiodic series, we may use also the following multidimensionalperiodic expansion, another corollary to Theorem 3.4.2:

Theorem 3.4.4 (Periodic Taylor expansion for Tn). Any a ∈ C∞(Tn) has theperiodic Taylor representation

a(x) =∑|α|<N

1α!

(ei2πx − 1

)αD(α)

z a(z)∣∣∣z=0

+∑|α|=N

aα(x)(ei2πx − 1

)α,

where aα ∈ C∞(Tn) and

(ei2πx − 1)α := (ei2πx1 − 1)α1 · · · (ei2πxn − 1)αn .

3.5. Appendix: on operators in Banach spaces 329

Proof. We define the functions aα ∈ C∞(Tn) with the aid of Theorem 3.4.2: leta0 := a, and if αk > 0 for each k ∈ {1, . . . , n} then

a(α1,0,0,...,0)(x) := aα1(x1, x2, . . . , xn),

a(α1,α2,0,...,0)(x) :=(a(α1,0,0,...,0)

)α2

(0, x2, . . . , xn),

...a(α1,...,αk,0,...,0)(x) :=

(a(α1,...,αk−1,0,...,0)

)αk

(0, . . . , 0, xk, . . . , xn),

...a(α1,...,αn)(x) :=

(a(α1,...,αn−1,0)

)αn

(0, . . . , 0, xn).

Then we obtain

a(x) =N−1∑α1=0

(ei2πx1 − 1

)α1a(α1,0,...,0)(0, x2, . . . , xn)

+a(N,0,...,0)(x)(ei2πx1 − 1

)N

=∑

α1+α2<N

(ei2πx1 − 1

)α1 (ei2πx2 − 1

)α2

×a(α1,α2,0,...,0)(0, 0, x3, . . . , xn)

+∑

α1+α2=N

a(α1,α2,0,...,0)(x)(ei2πx1 − 1

)α1 (ei2πx2 − 1

)α2

= . . . ,

which iteratively leads to

a(x) =∑|α|<N

(ei2πx − 1

)αaα(0) +

∑|α|=N

aα(x)(ei2πx − 1

)α.

Observing that D(β)x

(ei2πx − 1

)α∣∣∣x=0

= β! δα,β , this implies that we have

D(β)x a(x)

∣∣∣x=0

= β! aβ(0),

leading to the claimed expansion. �

3.5 Appendix: on operators in Banach spaces

Elsewhere, we will need several functional analysis results to be used in the analysisof periodic Sobolev spaces and we present them here. The reader is encouragedto skip this section at the first reading and consult it only when a reference isencountered in some proof.


Let A be an operator defined on a set X, and let W ⊂ X. Then the restrictionof A to W is the operator A|W , defined on W and satisfying A|W (w) = A(w)(where w ∈ W ). An evaluation is occasionally written as A(x)|x=x0 := A(x0).If U is a family of subsets of X, the restriction of U to W ⊂ X means U|W :={U ∩W |U ∈ U}.

The topology τX of a set X is the family of the open subsets of X. Assumethat X is a vector space with R or C as the scalar field K, and that K has the usualtopology induced by the absolute value norm. Elements of the theory of topologicalvector spaces are presented in Chapters A and C, but here it is enough to recognisethe outlines concerning Banach and Hilbert spaces, the most important topologicalvector spaces.

Let X and Y be vector spaces. The convergence of the sequence (uj)j∈Z+ ⊂ X

to u ∈ X in the topology τX is marked by ujτX−−→ u. The closure of a subset U ⊂ X

in this topology is clX(U), or when the topology is known from the context, cl(U)or merely U . From now on, the notation τX is reserved for the vector space topologyof X. The set of all linear operators with X as the domain, and the range in Y isdenoted by L(X, Y ), and if the spaces are equipped with vector space topologiesτX , τY , then L(X, Y ) is the set of continuous linear mappings with respect to thesetopologies. The norm of X, if there exists such, is ‖ · ‖X , and the operator normbetween normed spaces X and Y is ‖ · ‖L(X,Y ). As before, τE refers to the vectorspace topology of space E.

Theorem 3.5.1. Let E, F , and G be Banach spaces such that F ⊂ G and τG|F ⊂τF . If A ∈ L(E,G) maps E into F , then A ∈ L(E,F ).

Proof. (Note that in this case L(E,F ) ⊂ L(E,G) is topologically trivial, as theinclusion F ↪→ G is continuous.) Take a sequence (uj)j∈Z+ ⊂ E such that uj

τE−−→u ∈ E and Auj

τF−−→ v ∈ F . Now AujτG−−→ v, since τG|F ⊂ τF . On the other hand,

AujτG−−→ Au ∈ G, because A ∈ L(E,G). Hence it has to be v = Au. The proof is

now completed by the Closed Graph Theorem (Theorem B.4.34). �

The periodic Sobolev spaces Hs(Tn) (s ∈ R) are nested in a chain so thats < t implies Hs(Tn) ⊃ Ht(Tn), with τHt(Tn) finer than the restriction of τHs(Tn)

to Ht(Tn). The continuous linear operators are most conveniently defined firston the trigonometric polynomials, which are dense in every Hs(Tn), and thenextended to the Sobolev spaces using the next theorem:

Theorem 3.5.2. Let Ei, Fj (i, j = 1, 2) be Banach spaces such that E1 ⊂ E2 andF1 ⊂ F2, τE2 |E1 ⊂ τE1 and τF2 |F1 ⊂ τF1 . Assume that X ⊂ E1 is dense in E1 andE2, and that A ∈ L(X, F1) is continuous as a map E1 → F1 and E2 → F2. ThenA can be uniquely extended to a map A ∈ L(E2, F2) satisfying A|E1 ∈ L(E1, F1).

Proof. We first show that A can be extended to a continuous operator A = A1 ∈L(E1, F1). Assume that (uj)j∈Z+ ⊂ X : uj

τE1−−→ u ∈ E1. Now (Auj)j∈Z+ ⊂ F1

is a Cauchy sequence, because A is continuous from X to F1 with respect to τE1

3.5. Appendix: on operators in Banach spaces 331

and τF1 . Since F1 is complete, (Auj)j∈Z+ converges to some v ∈ F1, and we defineAu := v. Uniqueness and linearity of this extension are trivial. Let us denote‖A‖E1|X ,F1 := supu∈X:‖u‖E1≤1 ‖Au‖F1 . Then

‖Au‖F1 = limj→∞

‖Auj‖F1

≤ ‖A‖E1|X ,F1 limk→∞

‖uj‖E1

= ‖A‖E1|X ,F1 ‖u‖E1 ,

so that A ∈ L(E1, F1) with ‖A‖L(E1,F1) ≤ ‖A‖E1|X ,F1 . Indeed, ‖A‖L(E1,F1) =‖A‖E1|X ,F1 . The proof that A = A2 ∈ L(E2, F2) follows exactly the same pattern.

Note that A = A1 is also in L(E1, F2), and that uj

τE2−−→ u, since τF2 |F1 ⊂ τF1 andτE2 |E1 ⊂ τE1 . Thus

‖(A1 −A2)u‖F2

≤ ‖A1(u− uj)‖F2 + ‖(A1 −A2)uj‖F2 + ‖A2(uj − u)‖F2

≤ ‖A1‖L(E1,F2) ‖u− uj‖E1 + 0 + ‖A2‖L(E2,F2) ‖uj − u‖E2

−−−→j→∞

0.

Hence A1 = A2|E1 . �

Chapter 4

Pseudo-differential Operators on Tn

Pseudo-differential operators on the torus Tn = Rn/Zn, or the periodic pseudo-differential operators, are studied next. The presentation is written in a way for areader to be able to compare and to contrast it to the general theory of pseudo-differential operators on the Euclidean space from Chapter 2.

However, while in Rn in Chapter 2 we aimed at avoiding technicalities byrestricting ourselves to symbols of type Sm

1,0, here we will also discuss symbols oftypes Sm

ρ,δ. One reason for this is that no comprehensive treatment of operators onTn seems to be available in the literature so we may treat a more general situationalso to exhibit the dependence of some results on the values of parameters ρ and δ.

In Chapter 3 a sound basis for the development of the theory of periodicpseudo-differential operator was founded, where the pieces of information havebeen known practically for decades, and in some cases, for centuries; however, thesefragments of wisdom have been scattered widely apart in the field of mathematics.

We will see how closely periodic pseudo-differential operators are tied to thegeneral pseudo-differential operators, the main difference actually being that inthe periodic case, the theory appears to be more crystallised. This is definitelygood news for those who want to grasp the ideas of any pseudo-differential theory.

In 1979 Agranovich [3] proposed, crediting L.R. Volevich, a global definitionof pseudo-differential operators on the unit circle S1, called the periodic pseudo-differential operators. Of course, the definition was readily generalisable for anytorus Tn. Due to the group structure of Tn, by exploiting the Fourier series repre-sentation these new operators admitted globally defined symbols instead of merelocal analysis. We also note here that a similar representation of operators hasbeen already used by Petrovski in [86] in the analysis of the Cauchy problem forsystems of partial differential equations.

It is a non-trivial fact, however, that the definitions of pseudo-differential op-erators on a torus given by Agranovich and Hormander are equivalent. Agranovichproved this in [4] in the special case of classical operators, and later without somedetails in [5] in the case of the Hormander (1, 0)-operators. Another treatise ofthe classical operators was presented in [103]. A complete proof was provided by

334 Chapter 4. Pseudo-differential Operators on Tn

McLean [76] for all the Hormander (ρ, δ)-classes. McLean proved equivalence ofthe global and local definitions by directly studying charts of the tori. Anotherproof of this type was given in [79] for the (1, 0)-class. In the sequel, we giveone more approach, based on extension and periodisation techniques, providingthe equality of (ρ, δ)-symbol classes (Corollary 4.6.13), also yielding an explicitrelation between operators (Theorem 4.6.12).

Periodic integral operators are a major source of applications for the periodicpseudo-differential operator theory. Unfortunately, there is not much room fordiscussing periodic integral operators here, except for an application in Section4.11. Important further results on this subject are fast methods of solving periodicintegral equations presented in [142] and [102], which certainly is recommendedfor further reading on these topic. Other applications to the numerical analysis ofperiodic equations in mechanics and aerodynamics can be found in, e.g., [144, 145].Numerical aspects of Fourier transforms on general compact groups can be foundin, e.g., [75].

From the point of view of these applications, a theory of pseudo-differentialoperators expressed in terms of Fourier coefficients and discrete operations is ap-pealing. Periodic pseudo-differential operators were briefly considered, e.g., in [34],and certain aspects studied in [6, 7], [142], [102] and [138]. We note that analysisof vector fields has an obvious embedding into the theory of pseudo-differentialoperators on the torus, and thus, for example, questions of global hypoellipticityand solvability of vector fields (e.g., [42], [43], [44], [13], etc.) obtain a more fun-damental ground. Some aspects of the analysis presented in this chapter appearedin [97, 98].

In this chapter we develop the foundations of the theory of pseudo-differentialoperators on the torus Tn. This includes toroidal quantization of operators, toroi-dal symbol classes, toroidal amplitudes, asymptotic expansions, symbolic calculus,boundedness on L2(Tn) and on the Sobolev spaces Hs(Tn), questions of ellipticityand regularity. Section 4.11 gives an application to periodic integral equations.

In Sections 4.12 we consider toroidal wave front sets relating them to thestandard Hormander wave front sets in Rn. In Section 4.13 we introduce Fourierseries operators and study their compositions with pseudo-differential operatorsin terms of the toroidal symbols. In Section 4.14 we establish the boundedness ofFourier series operators in L2(Tn) and Hs(Tn), and in Section 4.15 we discuss anapplication to the Cauchy problem for hyperbolic partial differential equations.

Fourier series operators considered here are analogues of the Fourier integraloperators on the torus and we study them in terms of the toroidal quantization.The main new difficulty here is that while pseudo-differential operators do notmove the wave front sets of distributions, this is no longer the case for Fourier seriesoperators. Thus, we are forced to make extensions of functions from the integerlattice to the Euclidean space on the frequency side in Theorem 4.13.11. However,the other composition formula in Theorem 4.13.8 is still expressed entirely in thetoroidal language.

4.1. Toroidal symbols 335

4.1 Toroidal symbols

In this section we discuss the quantization of operators that we will call toroidalquantization and the corresponding classes of toroidal symbols.

4.1.1 Quantization of operators on Tn

First, we discuss the ideas informally.

Informal discussion. The main informal underlying idea here is that for a givenlinear operator A : C∞(Tn) → D′(Tn), if we study how it maps the functionseξ = (x �→ ei2πx·ξ) for all ξ ∈ Zn, then A would be completely determined. Tocollect this kind of information we define the symbol σA of A by testing A on thewaves eξ yielding Aeξ(x) = ei2πx·ξσA(x, ξ), i.e.,

σA(x, ξ) := e−i2πx·ξAeξ(x). (4.1)

Conversely, a function σ : Tn × Zn → C defines a linear operator Op(σ) :C∞(Tn)→ D′(Tn) by

Op(σ)u(x) :=∑ξ∈Zn

ei2πx·ξ σ(x, ξ) u(ξ), (4.2)

provided that there are some reasonable restrictions on σ(x, ξ). Clearly σ is thesymbol of Op(σ).Example. A partial differential operator A =

∑|α|≤M aj(x) (∂/∂x)α, where aα ∈

C∞(Tn), has the symbol

σA(x, ξ) =∑|α|≤M

aj(x) (i2πξ)α,

a polynomial of degree M in ξ. This observation motivates the concept of periodicpseudo-differential operators.

Thus, given a continuous linear operator A : C∞(Tn) → C∞(Tn), we canconsider its toroidal quantization

Aϕ(x) =∑ξ∈Zn

ei2πx·ξ σA(x, ξ) f(ξ) dy,

where its toroidal symbol σA ∈ C∞(Tn × Zn) will be uniquely determined by theformula

σA(x, ξ) = e−i2πx·ξAeξ(x),

where eξ(x) := ei2πx·ξ. We note that for periodic pseudo-differential operators onRn this could be just another quantization of the same class, as several quan-tizations on Rn already exist, including the Kohn-Nirenberg quantization fromChapter 1.


More formal presentation. We will now make this more precise.Remark 4.1.1 (Periodic Schwartz kernel). For ψ,ϕ∈C∞(Tn), let ψ⊗ϕ∈C∞(T2n)be defined by (ψ ⊗ ϕ)(x, y) := ψ(x)ϕ(y). If A : C∞(Tn)→ D′(Tn) is a continuouslinear operator then one can verify that

〈KA, ψ ⊗ ϕ〉 := 〈Aϕ, ψ〉

defines the periodic Schwartz distributional kernel KA ∈ D′(T2n) of the operatorA ∈ L(C∞(Tn),D′(Tn)); a common informal notation is

Aϕ(x) =∫

Tn

KA(x, y) ϕ(y) dy.

The convolution kernel kA ∈ D′(T2n) of A is related to the Schwartz kernel by

KA(x, y) = kA(x, x− y),

i.e., we have

Aϕ(x) =∫

Tn

kA(x, x− y) ϕ(y) dy

in the sense of distributions. We write kA(x)(y) := kA(x, y).

We now define symbols of operators on the torus.

Definition 4.1.2 (Toroidal symbols of operators on Tn). Let eξ(x) = ei2πx·ξ. Thetoroidal symbol of a linear continuous operator A : C∞(Tn)→ C∞(Tn) at x ∈ Tn

and ξ ∈ Zn is defined by

σA(x, ξ) := kA(x)(ξ) = FTn(kA(x))(ξ).

HenceσA(x, ξ) =

∫Tn

kA(x, y) e−i2πy·ξ dy = 〈kA(x), e−ξ〉.

By the Fourier inversion formula the convolution kernel can be regained from thesymbol:

kA(x, y) =∑ξ∈Zn

ei2πy·ξ σA(x, ξ) (4.3)

in the sense of distributions. We now show that an operator A can be representedby its symbol:

Theorem 4.1.3 (Quantization of operators on Tn). Let σA be the toroidal symbolof a continuous linear operator A : C∞(Tn)→ C∞(Tn). Then

Af(x) =∑ξ∈Zn

ei2πx·ξ σA(x, ξ) f(ξ) (4.4)

for every f ∈ C∞(Tn) and x ∈ Tn.


Proof. Let us define a convolution operator Ax0 ∈ L(C∞(Tn)) by the kernelkA(x0, y) = kx0(y), i.e., by

Ax0f(x) :=∫

Tn

f(y) kx0(x− y) dy = (f ∗ kx0)(x).

ThusσAx0

(x, ξ) = kx0(ξ) = σA(x0, ξ),

so that we have

Ax0f(x) =∑ξ∈Zn

ei2πx·ξ Ax0f(ξ)

=∑ξ∈Zn

ei2πx·ξ σA(x0, ξ) f(ξ),

where we used that f ∗ kx0 = f kx0 by the same calculation as in Theorem 1.1.30,(iii). This implies the result, because Af(x) = Axf(x). �Theorem 4.1.4 (Formula for the toroidal symbol). Let σA be the toroidal symbolof a continuous linear operator A : C∞(Tn)→ C∞(Tn). Then for all x ∈ Tn andξ ∈ Zn we have

σA(x, ξ) = e−i2πx·ξ(Aeξ)(x) = eξ(x)(Aeξ)(x). (4.5)

Proof. First we observe that

eξ(η) =∫

Tn

e−i2πx·η ei2πx·ξ dx = δξ,η

is Kronecker’s delta. We can now calculate

e−i2πx·ξ(Aeξ)(x)(4.4)= e−i2πx·ξ ∑

η∈Zn

ei2πx·η σA(x, η) eξ(η) = σA(x, ξ),

completing the proof. �Remark 4.1.5 (Mapping σ = (A �→ σA)). The mapping σ = (A �→ σA) can beviewed as a linear mapping: σsA+tB = sσA+tσB . However, σ is not multiplicative,as usually σAB = σAσB . It should be emphasized that the symbol of a linearoperator is unique (A = 0 ⇔ σA(x, ξ) = 0), and this fact will be used every nowand then.

4.1.2 Toroidal symbols

We will now proceed in the direction opposite to that of the proceeding sectionand discuss classes of symbols that correspond to the toroidal quantization ofoperators.


Definition 4.1.6 (Space C∞(Tn × Zn)). We will write a ∈ C∞(Tn × Zn) whenfunction a(·, ξ) is smooth on Tn for all ξ ∈ Zn.

Definition 4.1.7 (Toroidal symbol class Smρ,δ(T

n × Zn)). Let m ∈ R, 0 ≤ δ, ρ ≤ 1.Then the toroidal symbol class Sm

ρ,δ(Tn × Zn) consists of those functions a(x, ξ)

which are smooth in x for all ξ ∈ Zn, and which satisfy toroidal symbol inequalities∣∣�αξ ∂β

xa(x, ξ)∣∣ ≤ Caαβm 〈ξ〉m−ρ|α|+δ|β| (4.6)

for every x ∈ Tn, for every α, β ∈ Nn0 , and for all ξ ∈ Zn. Here�α

ξ are the differenceoperators introduced in Definition 3.3.1. The constants Caαβm are independent ofx ∈ Tn and ξ ∈ Zn. The class Sm

1,0(Tn×Zn) (the smallest of the (ρ, δ) classes) will

be often denoted by writing simply Sm(Tn × Zn). Furthermore, we define

S−∞(Tn × Zn) :=⋂

m∈R

Sm(Tn × Zn),

S∞ρ,δ(Tn × Zn) :=

⋃m∈R

Smρ,δ(T

n × Zn).

Exercise 4.1.8. Show that for any ρ and δ we have⋂m∈R

Smρ,δ(T

n × Zn) = S−∞(Tn × Zn).

Definition 4.1.9 (Toroidal pseudo-differential operators). If a ∈ Smρ,δ(T

n×Zn), wedenote by a(X, D) = Op(a) the corresponding toroidal pseudo-differential operatordefined by

Op(a)f(x) = a(X, D)f(x) :=∑ξ∈Zn

ei2πx·ξ a(x, ξ) f(ξ). (4.7)

The series (4.7) converges if, for example, f ∈ C∞(Tn). The set of operators Op(a)of the form (4.7) with a ∈ Sm

ρ,δ(Tn×Zn) will be denoted by Op(Sm

ρ,δ(Tn×Zn)), or

by Ψmρ,δ(T

n×Zn). If an operator A satisfies A ∈ OpSmρ,δ(T

n×Zn), we will denoteits toroidal symbol by σA = σA(x, ξ), x ∈ Tn, ξ ∈ Zn. Naturally, σa(X,D)(x, ξ) =a(x, ξ). We also write

Op(S−∞(Tn × Zn)) :=⋂

m∈R

Op(Sm(Tn × Zn)),

Op(S∞ρ,δ(Tn × Zn)) :=

⋃m∈R

Op(Smρ,δ(T

n × Zn)).

Remark 4.1.10 (Toroidal Smρ,δ(T

n ×Zn) vs Euclidean Smρ,δ(T

n ×Rn)). To contrastthis with Euclidean (Hormander’s) symbol classes, we write b ∈ Sm

ρ,δ(Rn × Rn) if

b ∈ C∞(Rn × Rn) and if∣∣∂αξ ∂β

x b(x, ξ)∣∣ ≤ Cbαβm 〈ξ〉m−ρ|α|+δ|β|


holds for every x ∈ Rn, for every α, β ∈ Nn0 , and for all x, ξ ∈ Rn. If in addition

b(·, ξ) is 1-periodic for every ξ we will write b ∈ Smρ,δ(T

n×Rn) and call it a Euclideansymbol on the torus. The corresponding (Euclidean) pseudo-differential operatoron the torus is then given by

b(X, D)f(x) =∫

Rn

∫Tn

ei2π(x−y)·ξ b(x, ξ) f(y) dy dξ.

Remark 4.1.11 (Topology on Smρ,δ(T

n × Zn)). The set Smρ,δ(T

n × Zn) of symbolshas a natural topology. Let us consider the functions pm

αβ : Smρ,δ(T

n × Zn) → R(α, β ∈ Nn

0 , m ∈ R) defined by

pmαβ(σ) := sup

⎧⎨⎩∣∣∣�α

ξ ∂βxσ(x, ξ)

∣∣∣〈ξ〉m−ρ|α|+δ|β| : (x, ξ) ∈ Tn × Zn

⎫⎬⎭ .

Now{

pmαβ : α, β ∈ Nn

0

}is a countable family of seminorms, and they define

a Frechet topology on Smρ,δ(T

n × Zn). Due to the bijective correspondence ofOp(Sm

ρ,δ(Tn × Zn)) and Sm

ρ,δ(Tn × Zn), this directly topologises the set of oper-

ators. These spaces are not normable, and the topologies have but a marginalrole.Remark 4.1.12. On Tn, Hormander’s usual (ρ, δ) class of pseudo-differential op-erators OpSm

ρ,δ(Rn ×Rn) of order m ∈ R which are 1-periodic in x coincides with

the class OpSmρ,δ(T

n × Zn), i.e.,

OpSmρ,δ(T

n × Rn) = OpSmρ,δ(T

n × Zn),

see Corollary 4.6.13. This fact was originally proved in [76] by studying localcoordinate charts, but the techniques of the extension of symbols and periodisationof operators developed here in addition give a precise relation between actualsymbols, see, e.g., Theorem 4.5.3, and yield the relation between the correspondingoperators which is then given in Theorem 4.6.12.

Proposition 4.1.13. Let f ∈ C∞(Tn). Then Op(a)f in (4.7) is well defined andOp(a)f ∈ C∞(Tn). Moreover, operator Op(a) : C∞(Tn) → C∞(Tn) is continu-ous.

Proof. Since f ∈ S(Zn), the series in (4.7) converges absolutely and Op(a)f ∈C∞(Tn). We can write

Op(a)f(x) =∑ξ∈Zn

ei2πx·ξ a(x, ξ) f(ξ)

=∑ξ∈Zn

∫Tn

ei2π(x−y)·ξ a(x, ξ) f(y) dy

=∑ξ∈Zn

〈ξ〉−2q a(x, ξ)∫

Tn

ei2π(x−y)·ξ(

I − Ly

4π2

)q

f(y) dy,


where Ly is the usual Laplacian with respect to y. Then, if we take q ∈ Z+ largeenough, the series converges absolutely. Consequently, if we have the convergencefj → f in C∞(Tn), we can pass to the limit in the series and in the integralby Lebesgue’s dominated convergence theorem (Theorem C.3.22) to see that alsoOp(a)fj → Op(a)f in C∞(Tn). �

Exercise 4.1.14. How large a q should be chosen for the sum here to convergeabsolutely, if σ ∈ Sm

ρ,δ(Tn × Zn)?

Remark 4.1.15 (Difference formula for symbols). We now mention a nice formulafor differences for symbols which follows immediately from Proposition 3.3.4:

�αξ ∂β

xσA(x, ξ) =∑γ≤α

(−1)|α−γ|(

α

γ

)∂β

xσA(x, ξ + γ). (4.8)

This formula shows that properties of toroidal symbols automatically imply certainproperties for differences. For example, decay properties of the symbol automati-cally imply similar decay properties for differences applied to the symbol. This isa considerable advantage over the Euclidean quantization in classes Sm(Tn × Rn)where such a property does not hold.

4.1.3 Toroidal amplitudes

Remark 4.1.16 (Alternative representation). By choosing appropriate q largeenough in the proof of Proposition 4.1.13, we see that Op(a)f can be differen-tiated arbitrarily many times. Writing out the Fourier transform as an integralsuggests the notation

Op(a)f(x) =:∑ξ∈Zn

∫Tn

ei2π(x−y)·ξ a(x, ξ) f(y) dy. (4.9)

It should be noted that here the right-hand side is not meant to be read as anintegral operator, but rather as an operator defined by the formal integration byparts, so that we can perform operations like the exchange of summation andintegral, etc. The Schwartz kernel K(x, y) of (4.9) defined by the formal sumK(x, y) =

∑ξ∈Zn a(x, ξ) ei2π(x−y)·ξ may be singular only at the diagonal x = y;

anywhere else it is smooth (see Theorem 4.3.6).

Equation (4.9), however, inspires a possible generalisation: why do we notallow function a to depend on the variable y?

Definition 4.1.17 (Toroidal amplitudes). The classAmρ,δ(T

n) (or Smρ,δ(T

n×Tn×Zn))of toroidal amplitudes consists of the functions a(x, y, ξ) which are smooth in xand y for all ξ ∈ Zn and which satisfy∣∣�α

ξ ∂βx∂γ

y a(x, y, ξ)∣∣ ≤ Caαβγm 〈ξ〉m−ρ|α|+δ|β+γ| (4.10)


for every x, y ∈ Tn, for every α, β, γ ∈ Nn0 , and for all ξ ∈ Zn. The constants

Caαβγm < ∞ are independent of x, y, ξ. Such a function a is called a toroidalamplitude of order m ∈ R of type (ρ, δ). Formally we may also define

(Op(a)f)(x) :=∑ξ∈Zn

∫Tn

ei2π(x−y)·ξ a(x, y, ξ) f(y) dy (4.11)

for f ∈ C∞(Tn). Sometimes we denote Op(a) by a(X, Y,D).Clearly we can regard the symbols of periodic pseudo-differential operators as

a special class of amplitudes, namely the ones independent of the middle argument.The family of amplitudes of order m and type (ρ, δ) is denoted by Am

ρ,δ(Tn). We

also write Am(Tn) := Am1,0(T

n) as well as

A−∞(Tn) :=⋂

m∈R

Am(Tn) and A∞ρ,δ(Tn) :=

⋃m∈R

Amρ,δ(T

n).

In Elias Stein’s language (see [118, p.258]) amplitudes are called compound symbols.

Remark 4.1.18 (Amplitude operators). Formula (4.11) has to be interpreted as aresult of a formal integration by parts, being a short-hand writing for

Op(a)f(x) =∑ξ∈Zn

〈ξ〉−2q

∫Tn

ei2π(x−y)·ξ(

I − 14π2

Ly

)q

(a(x, y, ξ) f(y)) dy,

where Ly is the Laplacian in y. With this explanation we will take the libertyof changing the order of integration and summation, keeping in mind that it canbe justified by the integration by parts. Most of the time, we shall use the lesscumbersome notation of (4.11). This operator is called an amplitude operator ofdegree m.We define

Op(A−∞(Tn)) :=⋂

m∈R

Op(Am(Tn))

andOp(A∞ρ,δ(T

n)) :=⋃

m∈R

Op(Amρ,δ(T

n)).

Later on, Op(a) and Op(σ) for amplitudes a and symbols σ, respectively, willbe extended to operators acting on the Sobolev spaces Hs(Tn), s ∈ R. Symbolshave the advantage over amplitudes in showing directly how the extension works.We have considered the amplitudes as a generalisation of symbols, but fortunately(or unfortunately?), it turns out that the family of amplitude operators coincidesexactly with the family of periodic pseudo-differential operators. This is easy tobelieve, since on C∞(Tn) every linear operator is uniquely defined by its symbol.Of course, we do not know yet whether an amplitude operator has a symbolthat satisfies inequalities (4.6). The proof of this fact is a long one, and we aregoing to present it in small parts. Nevertheless, the concept of amplitudes is highlyjustified as a tool in the symbolic analysis. Moreover, amplitudes literally manifestthemselves in certain integral operators, a topic we will briefly discuss later.


In the following, symbols are considered as a subspace of amplitudes inde-pendent of y.

Definition 4.1.19 (Equivalence of amplitudes). We say that amplitudes a, a′ are

m(ρ, δ)-equivalent (m ∈ R), am,ρ,δ∼ a′, if a−a′ ∈ Am

ρ,δ(Tn); they are asymptotically

equivalent, a ∼ a′ (or a−∞∼ a′ if we need additional clarity), if a− a′ ∈ A−∞(Tn).

For the related operators we mark Op(a)m,ρ,δ∼ Op(a′) and Op(a) ∼ Op(a′) (or

Op(a) −∞∼ Op(a′) if we need additional clarity), respectively. It is obvious thatm(ρ,δ)∼ and ∼ are equivalence relations; the classes modulo ∼ are denoted by [·] forboth amplitudes and operators.

Remark 4.1.20. From the algebraic point of view, we could handle the amplitudes,symbols, and operators modulo the equivalence relation ∼, because the periodicpseudo-differential operators form a ∗-algebra with Op(S−∞(Tn × Zn)) as a sub-algebra (see Section 4.7). In fact, in [135] (in a similar setting of Rn) a “symbol”means the class [σ] = σ + S−∞. However, neither topological considerations norapplications support an extensive use of these equivalence classes.

4.2 Pseudo-differential operators on Sobolev spaces

In this section we show that domains of periodic pseudo-differential operators canbe extended to the Sobolev spaces. This will be important for the subsequentanalysis. Later, in Corollary 4.8.3 we will present conditions on the symbol thatensure that the corresponding operator is bounded between Sobolev spaces onthe torus. The proof will rely on the composition formulae for pseudo-differentialoperators which will be proved in Theorem 4.7.10. At the moment, also to showanother idea we will give a direct proof of the Sobolev boundedness for operatorswith symbols of type (0, 0).

When studying the orders of periodic pseudo-differential operators, estimatesfor the differences of the Fourier coefficients of symbols and amplitudes are useful,and therefore we present an auxiliary result on this subject:

Lemma 4.2.1. Let 0 ≤ δ, 0 ≤ ρ. Assume that σ ∈ Smρ,δ(T

n × Zn). Let σ be theFourier transform of the symbol with respect to x, i.e., the Fourier transform ofthe smooth function x �→ σ(x, ·). Then for every α ∈ Nn

0 and r ∈ N0 estimate∣∣�αξ σ(η, ξ)

∣∣ ≤ cr,α 〈η〉−r〈ξ〉m−ρ|α|+δr (4.12)

holds for all η, ξ ∈ Zn. Respectively, let a ∈ Amρ,δ(T

n) and let a be the Fouriertransform of (x, y) �→ a(x, y, ·). Then for every α ∈ Nn

0 and q, r ∈ N0 estimate∣∣�αξ a(λ, η, ξ)

∣∣ ≤ cq,r,α 〈λ〉−q〈η〉−r〈ξ〉m−ρ|α|+δ(q+r), (4.13)

holds for all λ, η, ξ ∈ Zn. The constants cr,α and cq,r,α are independent of λ,η and ξ.

4.2. Pseudo-differential operators on Sobolev spaces 343

Proof. Clearly (1− (2π)−2Lx

)qe−i2πx·η = 〈η〉2q e−i2πx·η,

where Lx is the Laplacian in x variables. Integrating by parts, we get

∣∣�αξ σ(η, ξ)

∣∣ =∣∣∣∣�α

ξ

∫Tn

e−i2πx·η σ(x, ξ) dx

∣∣∣∣= 〈η〉−2q

∣∣∣∣∫Tn

e−i2πx·η (1− (2π)−2Lx

)q�αξ σ(x, ξ) dx

∣∣∣∣≤ 〈η〉−2q

∫Tn

∣∣∣(1− (2π)−2Lx

)q�αξ σ(x, ξ)

∣∣∣ dx.

Since σ ∈ Smρ,δ(T

n × Zn) estimate (4.12) follows for even r = 2q. By taking ageometric mean we get it also for odd r. The proof for amplitudes is similar. �

Exercise 4.2.2. Prove (4.13).

We use Lemma 4.2.1 for the following:

Proposition 4.2.3 (Sobolev space boundedness for Sm0,0). Let A = Op σA be a

pseudo-differential operator with toroidal symbol σA ∈ Sm0,0(T

n × Zn). Then theoperator A extends to a bounded linear operator from Hs(Tn) to Hs−m(Tn) forevery s ∈ R.

Proof. We have A = Op(σA) with σA ∈ Sm0,0(T

n × Zn), i.e.,∣∣�αξ ∂β

xσA(x, ξ)∣∣ ≤ Cαβ 〈ξ〉m.

Let u ∈ C∞(Tn) and let us calculate the Fourier coefficients of Au:

Au(x) =∑ξ∈Zn

ei2πx·ξ σA(x, ξ) u(ξ)

=∑ξ∈Zn

ei2πx·ξ

⎡⎣ ∑η∈Zn

ei2πx·η σA(η, ξ)

⎤⎦ u(ξ)

=∑

η∈Zn

ei2πx·η

⎡⎣ ∑ξ∈Zn

σA(η − ξ, ξ) u(ξ)

⎤⎦ ,

and we note that both series are absolutely convergent. Therefore, we have

Au(η) =∑ξ∈Zn

σA(η − ξ, ξ) u(ξ).


Let us now estimate ‖Au‖Hs−m(Tn). We have

‖Au‖2Hs−m(Tn) =∑

η∈Zn

〈η〉2(s−m) |Au(η)|2

=∑

η∈Zn

∣∣∣∣∣∣∑ξ∈Zn

〈η〉s−m σA(η − ξ, ξ) u(ξ)

∣∣∣∣∣∣2

≤∑

η∈Zn

⎡⎣ ∑ξ∈Zn

〈η〉s−m |σA(η − ξ, ξ)| |u(ξ)|

⎤⎦2

,

which, by Peetre’s inequality (Proposition 3.3.31), can be estimated by

22|s−m| ∑η∈Zn

⎡⎣ ∑ξ∈Zn

〈η − ξ〉|s−m| 〈ξ〉−m|σA(η − ξ, ξ)|〈ξ〉s|u(ξ)|

⎤⎦2

.

Consequently, by the discrete Young’s inequality (Lemma 3.3.30), this can beestimated by

22|s−m|

⎡⎣ supη∈Zn

∑ξ∈Zn

〈η − ξ〉|s−m| 〈ξ〉−m|σA(η − ξ, ξ)|

⎤⎦×

⎡⎣ supξ∈Zn

∑η∈Zn

〈η − ξ〉|s−m| 〈ξ〉−m |σA(η − ξ, ξ)|

⎤⎦⎡⎣ ∑ξ∈Zn

〈ξ〉2s |u(ξ)|2⎤⎦ .

Applying (4.12) with α = 0 and ρ = δ = 0 we obtain

‖Au‖2Hs−m(Tn) ≤ 22|s−m|

⎡⎣ supη∈Zn

∑ξ∈Zn

〈η − ξ〉|s−m|−r Cr

⎤⎦×

⎡⎣ supξ∈Zn

∑η∈Zn

〈η − ξ〉|s−m|−r Cr

⎤⎦ ‖u‖2Hs(Tn)

= 22|s−m|

⎡⎣Cr

∑η∈Zn

〈η〉|s−m|−r

⎤⎦2

‖u‖2Hs(Tn).

If we take r large enough we obtain ‖Au‖Hs−m(Tn) ≤ C‖u‖2Hs(Tn). The desiredextension on Hs(Tn) is then obtained by Theorem 3.5.2 (or by a more directexplanation of the type given in Proposition 2.4.1 on how estimate (2.9) impliesthe boundedness). �

4.2. Pseudo-differential operators on Sobolev spaces 345

In particular, we see from the proof that it is enough to take r ∈ N such thatr > |m − s| + n for the proof to work. Combining this with Remark 4.2.9 belowon the symbol class Sm

0,0(Tn × Zn) we conclude:

Corollary 4.2.4 (How many derivatives are needed?). Let m, s ∈ R and let aninteger r ∈ N be such that r > |s−m|+n. Let a ∈ C∞(Tn × Zn) satisfy estimates∣∣∂β

xa(x, ξ)∣∣ ≤ Caβm 〈ξ〉m

for all multi-indices |β| ≤ r, all x ∈ Tn and all ξ ∈ Zn, with constants Caβm

independent of x and ξ. Then the operator Op(a) extends to a bounded linearoperator from Hs(Tn) to Hs−m(Tn).

We note a useful formula from the proof that does not require that the symbolof A is in a symbol class.

Corollary 4.2.5. Let A be a linear continuous operator from C∞(Tn) to C∞(Tn)and let u ∈ C∞(Tn). Then

Au(ξ) =∑

η∈Zn

σA(ξ − η, η) u(η),

where ξ ∈ Zn, σA is the toroidal symbol of A, and σA its Fourier transform in thespace variable.

Exercise 4.2.6. In the proof of Theorem 4.2.3, how are the constants Cr relatedto the symbol inequalities for σA?

Exercise 4.2.7. Explain how the discrete Young’s inequality (Lemma 3.3.30) wasexploited in the proof of Theorem 4.2.3.

Exercise 4.2.8. Suppose σA ∈ Smρ,δ(T

n × Zn) does not depend on the x-variable,i.e., σA(x, ξ) = σ(ξ). Find the norm ‖A‖L(Hs(Tn),Hs−m(Tn)).

Remark 4.2.9 (Characterisation of classes Sm0,0(T

n × Zn)). In view of Remark4.1.15 we see that in the case of Sm

0,0(Tn × Zn) we can drop difference condi-

tions from the definition of this class: we have a ∈ Sm0,0(T

n × Zn) if and only ifa ∈ C∞(Tn × Zn) satisfies ∣∣∂β


for all multi-indices β, all x ∈ Tn and all ξ ∈ Zn, with constants Caβm independentof x and ξ. From this point of view Proposition 4.2.3 will be equivalent to Corollary4.8.3 if we use the argument in the proof of Theorem 2.6.11 to change the orders ofSobolev spaces. However, the proof of Corollary 4.8.3 is different: it will follow bythe toroidal calculus from Theorem 4.8.1 which gives an estimate on the numberof x-derivatives that ensures the boundedness of operators on L2(Tn). In any case,Corollary 4.2.4 provides a result under weaker assumptions.


An analogous proof can be carried out for amplitudes. Let us denote theFourier transform of an amplitude a with respect to the first (or second) variableby a1 (or a2), respectively. The notation a is then reserved for the Fourier transformwith respect to both the first and the second variables.

Theorem 4.2.10. Let a ∈ Am0,0(T

n). Then Op(a) extends to a bounded linear oper-ator from Hs(Tn) to Hs−m(Tn), for any s ∈ R.

Proof. Let A = Op(a), a ∈ Am0,0(T

n), and u ∈ C∞(Tn). First, we want to find theFourier coefficients of Au. For this we have

Au(x) =∑ξ∈Zn

∫Tn

ei2π(x−y)·ξ a(x, y, ξ) u(y) dy

=∑ξ∈Zn

∫Tn

ei2π(x−y)·ξ

×

⎡⎣ ∑η∈Zn

ei2πx·(η−ξ) a1(η − ξ, y, ξ)

⎤⎦ ∑κ∈Zn

ei2πy·κ u(κ) dy

=∑

η∈Zn

ei2πx·η ∑κ∈Zn

∑ξ∈Zn

u(κ)∫

Tn

a1(η − ξ, y, ξ) e−i2πy·(ξ−κ) dy

=∑

η∈Zn

ei2πx·η ∑κ∈Zn

∑ξ∈Zn

u(κ) a(η − ξ, ξ − κ, ξ),

from which we obtain

Au(η) =∑

κ∈Zn

∑ξ∈Zn

u(κ) a(η − ξ, ξ − κ, ξ).

Then we estimate

‖Au‖2Hs−m(Tn) =∑

η∈Zn

〈η〉2(s−m) |Au(η)|2

=∑

η∈Zn

∣∣∣∣∣∣∑ξ∈Zn

∑κ∈Zn

〈η〉s−m a(η − ξ, ξ − κ, ξ) u(κ)

∣∣∣∣∣∣2

≤∑

η∈Zn

⎡⎣ ∑ξ∈Zn

∑κ∈Zn

〈η〉s−m |a(η − ξ, ξ − κ, ξ)| |u(κ)|

⎤⎦2

. (4.14)

Now, using Peetre’s inequality (Proposition 3.3.31) twice, we can estimate

〈η〉s−m ≤ 22|s−m|〈η − ξ〉|s−m| 〈ξ〉s−m

≤ 24|s−m|〈η − ξ〉|s−m| 〈ξ − κ〉|s| 〈κ〉s〈ξ〉−m.

4.3. Kernels of periodic pseudo-differential operators 347

Consequently, by (4.13) with ρ = δ = 0, we have

〈η〉s−m |a(η − ξ, ξ − κ, ξ)| ≤ 24|s−m| c2q,r,0〈η − ξ〉|s−m|−q 〈ξ − κ〉|s|−r 〈κ〉s.

Plugging this estimate into (4.14) we get

‖Au‖2Hs−m(Tn) ≤ C∑

η∈Zn

⎡⎣ ∑ξ∈Zn

∑κ∈Zn

〈η − ξ〉|s−m|−q 〈ξ − κ〉|s−m|−r 〈κ〉s |u(κ)|

⎤⎦2

Now, using the discrete Young’s inequality (Lemma 3.3.30) twice this can be esti-mated as

≤ C

⎡⎣ supξ∈Zn

∑η∈Zn

〈η − ξ〉|s−m|−q

⎤⎦ ∑ξ∈Zn

[ ∑κ∈Zn

〈ξ − κ〉|s|−r 〈κ〉s |u(κ)|]2

≤ C

⎡⎣ supξ∈Zn

∑η∈Zn

〈η − ξ〉|s−m|−q

⎤⎦⎡⎣ supκ∈Zn

∑ξ∈Zn

〈ξ − κ〉|s|−r

⎤⎦×

[ ∑κ∈Zn

〈κ〉2s |u(κ)|2]

= C

⎡⎣ ∑η∈Zn

〈η〉|s−m|−q

⎤⎦⎡⎣ ∑ξ∈Zn

〈ξ〉|s|−r

⎤⎦ ‖u‖2Hs(Tn).

Thus, if q and r are large enough we obtain ‖Au‖Hs−m(Tn) ≤ C‖u‖2Hs(Tn). Again,A can be extended to Hs(Tn) by Theorem 3.5.2. �

Exercise 4.2.11. Try to do Exercises 4.2.7 and 4.2.8 in the context of amplitudeoperators.

4.3 Kernels of periodic pseudo-differential operators

We start by describing Ψ−∞(Tn × Zn) = Op(S−∞(Tn × Zn))-operators conclu-sively, showing that there is every right to call them the infinitely smoothingperiodic pseudo-differential operators. Recall that by Theorem 3.5.1 a linear map-ping

A ∈ L(Hs(Tn),Ht1(Tn)) ∩ L(Hs(Tn),Ht2(Tn))

belongs to L(Hs(Tn),Ht1(Tn)), even when t1 > t2.


Theorem 4.3.1 (Smoothing). The following conditions are equivalent:

(i) A ∈ L(Hs(Tn),Ht(Tn)) for every s, t ∈ R.(ii) σA ∈ S−∞(Tn × Zn).(iii) There exists KA ∈ C∞(Tn × Tn) such that for all u ∈ C∞(Tn) we have

Au(x) =∫

Tn

KA(x, y) u(y) dy.

Proof. Assume that A satisfies (i). To obtain (ii), it is enough to prove

|∂βxσA(x, ξ)| ≤ cβ,r〈ξ〉−r for every r ∈ R,

because by Proposition 3.3.4 we have formula (4.8) which we recall here:

�αξ ∂β

xσA(x, ξ) =∑γ≤α

(−1)|α−γ|(

α

γ

)∂β

xσA(x, ξ + γ); (4.15)

reasoning why this is enough is left as Exercise 4.3.2. Recall that eξ(x) = ei2πx·ξ

so that ‖eξ‖Hs(Tn) = 〈ξ〉s. We now prepare another estimate:

‖e−ξf‖2H|β|+t(Tn) =∑

η∈Zn

〈η〉2|β|+2t|e−ξf(η)|2

≤ 22|β|+2|t| ∑η∈Zn

〈η + ξ〉2|β|+2|t|〈ξ〉2|β|+2t|f(η + ξ)|2

= 22|β|+2|t|〈ξ〉2|β|+2t‖f‖2H|β|+t(Tn),

where we applied Peetre’s inequality (Proposition 3.3.31). Finally, choosing t >n/2 and using the Sobolev embedding theorem (see, e.g., Exercise 2.6.17), we get∣∣∂β

xσA(x, ξ)∣∣ ≤

∑η∈Zn

(2π)|β|〈η〉|β| |σA(η, ξ)|

≤ Cβ,t ‖x �→ σA(x, ξ)‖H|β|+t(Tn)

= Cβ,t ‖e−ξAeξ‖H|β|+t(Tn)

≤ Cβ,t ‖e−ξI‖L(H|β|+t(Tn),H|β|+t(Tn))

×‖A‖L(Hs(Tn),H|β|+t(Tn)) ‖eξ‖Hs(Tn)

≤ 2|β|+tCβ,t ‖A‖L(Hs(Tn),H|β|+t(Tn)) 〈ξ〉s+|β|+t

where Cβ,t = (2π)|β|[∑

η∈Zn〈η〉−2t]1/2

. Since s ∈ R is arbitrary, we get (ii).

Let us now show that (ii) implies (iii). If σA ∈ S−∞(Tn × Zn), the Schwartzkernel

KA(x, y) :=∑ξ∈Zn

σA(x, ξ) ei2π(x−y)·ξ

4.3. Kernels of periodic pseudo-differential operators 349

is in C∞(Tn × Tn). Indeed, formally we can differentiate KA to obtain

∂αx ∂β

y KA(x, y) =∑ξ∈Zn

(−i2πξ)β∑γ≤α

(α

γ

)[∂γ

xσA(x, ξ)] ∂α−γx ei2π(x−y)·ξ;

this is justified, as the convergence of the resulting series is absolute, because|∂γ

xσA(x, ξ)| ≤ cγ,r〈ξ〉−r for any r ∈ R. This gives (iii).Finally, assume that (iii) holds. Define the amplitude a by a(x, y, ξ) :=

δ0,ξKA(x, y). Now a ∈ A−∞(Tn), since∣∣∣∂αx ∂β

y�γξ a(x, y, ξ)

∣∣∣ ≤ 2|γ|∣∣∂α

x ∂βy KA(x, y)

∣∣ χ[−|γ|,|γ|]n

≤ Crαβγ 〈ξ〉−r

for every r ∈ R, where χ[−|γ|,|γ|]n is the characteristic function of the cube[−|γ|, |γ|]n ⊂ Zn. On the other hand,

Op(a)u(x) =∑ξ∈Zn

∫Tn

ei2π(x−y)·ξ a(x, y, ξ) u(y) dy

=∫

Tn

KA(x, y) u(y) dy = Au(x).

Property (i) now follows by Theorem 4.2.10. �Exercise 4.3.2. In the proof above, based on (4.15), explain why it sufficed to prove∣∣∂β

xσA(x, ξ)∣∣ ≤ cβ,r〈ξ〉−r.

Because the inclusion of a Sobolev space into a strictly larger one is compact(see Exercise 3.2.10), we also obtain

Corollary 4.3.3. Operators from Op(S−∞(Tn × Zn)) are compact between anyspaces Hs(Tn),Ht(Tn).

Unlike in the case of symbols, the correspondence of amplitudes and ampli-tude operators is not bijective: several different amplitudes may define the sameoperator. As an example we are now going to study how the multiplication of anamplitude by (

ei2π(y−x) − 1)γ

:=n∏

j=1

(ei2π(yj−xj) − 1

)γj

(4.16)

affects the amplitude operator. Notice that this multiplier was encountered in thebiperiodic Taylor series (see Corollary 3.4.3 and Theorem 3.4.4).

Lemma 4.3.4. Let c ∈ Amρ,δ(T

n), and define

bγ(x, y, ξ) :=(ei2π(y−x) − 1

)γ

c(x, y, ξ),

where γ ∈ Nn0 . Then Op(bγ) = Op(�γ

ξ c) ∈ Op(Am−ρ|γ|

ρ,δ (Tn)).


Proof. First we note the identity

ei2π(x−y)·ξ(ei2π(y−x) − 1

)γ

= (−1)|γ|�γ

ξ ei2π(x−y)·ξ,

where �γ

ξ is the backward difference operator (see Definition 3.3.1) which we leaveas Exercise 4.3.5. Consequently, the summation by parts (see Lemma 3.3.10) yields

Op(bγ)u(x) =∑ξ∈Zn

∫Tn

ei2π(x−y)·ξ[(

ei2π(y−x) − 1)γ

c(x, y, ξ)]u(y) dy

=∫

Tn

⎡⎣ ∑ξ∈Zn

c(x, y, ξ) (−1)|γ| �γ

ξ ei2π(x−y)·ξ

⎤⎦u(y) dy

=∫

Tn

⎡⎣ ∑ξ∈Zn

ei2π(x−y)·ξ �γξ c(x, y, ξ)

⎤⎦u(y) dy.

Thus Op(bγ) = Op(�γξ c), and clearly �γ

ξ c ∈ Am−ρ|γ|ρ,δ (Tn), since c ∈ Am

ρ,δ(Tn).�

Exercise 4.3.5. Prove that for every γ ∈ Nn0 we have the identity

ei2π(x−y)·ξ(ei2π(y−x) − 1

)γ

= (−1)|γ|�γ

ξ ei2π(x−y)·ξ,

where �γ

ξ is the backward difference operator from Definition 3.3.1.

Surprising or not, but from the smoothness point of view the essential in-formation content of a periodic pseudo-differential operator is in the behavior ofits Schwartz kernel in any neighbourhood of the diagonal x = y. We note thatthis can be also seen from the local theory once we know the equality of operatorclasses Op(Am

ρ,δ(Tn)) and periodic operators in Op(Am

ρ,δ(Rn)). But here we give a

direct proof:

Theorem 4.3.6 (Schwartz kernel). Let 0 < ρ and δ < 1. Let A = Op(a) ∈Op(Am

ρ,δ(Tn)) be expressed in the form

Au(x) =∫

Tn

KA(x, y) u(y) dy,

where KA(x, y) =∑

ξ∈Zn ei2π(x−y)·ξ a(x, y, ξ). Then the Schwartz kernel KA is asmooth function outside the diagonal x = y.

Proof. Let j ∈ {1, . . . , n}. Take ψ ∈ C∞(Tn × Tn) such that xj = yj for every(x, y) ∈ supp(ψ). We have to prove that (x, y) �→ ψ(x, y) KA(x, y) belongs toC∞(Tn × Tn). Define

b(x, y, ξ) := ψ(x, y) a(x, y, ξ).

4.4. Asymptotic sums and amplitude operators 351

By Lemma 4.3.4, the amplitudes

(x, y, ξ) �→ b(x, y, ξ) and

(x, y, ξ) �→�k

ξjb(x, y, ξ)(

ei2π(yj−xj) − 1)k

give the same periodic pseudo-differential operator B := Op(b).Hence b ∈ Am−ρk

ρ,δ (Tn) for every k ∈ Nn0 , so that it is in A−∞(Tn). Theo-

rem 4.2.10 states that B is continuous between any Sobolev spaces, and then byTheorem 4.3.1 the kernel (x, y) �→ ψ(x, y) KA(x, y) belongs to C∞(Tn×Tn). �

Exercise 4.3.7. Derive the quantitative behavior of the kernel KA(x, y) near thediagonal x = y, similarly to Theorem 2.3.1.

4.4 Asymptotic sums and amplitude operators

The next theorem is a prelude to asymptotic expansions, which are the main toolin the symbolic analysis of periodic pseudo-differential operators.

Theorem 4.4.1 (Asymptotic sums of symbols). Let (mj)∞j=0 ⊂ R be a sequencesuch that mj > mj+1 −−−→

j→∞−∞, and σj ∈ S

mj

ρ,δ (Tn × Zn) for all j ∈ N0. Then

there exists a toroidal symbol σ ∈ Sm0ρ,δ (Tn × Zn) such that for all N ∈ N0,

σmN ,ρ,δ∼

N−1∑j=0

σj .

Proof. Choose a function ϕ ∈ C∞(Rn) satisfying ‖ξ‖ ≥ 1 ⇒ ϕ(ξ) = 1 and‖ξ‖ ≤ 1/2 ⇒ ϕ(ξ) = 0; otherwise ϕ can be arbitrary. Take a sequence (εj)∞j=0 ofpositive real numbers such that εj > εj+1 → 0 (j ∈ N0), and define ϕj ∈ C∞(Rn)by ϕj(ξ) := ϕ(εjξ). When |α| ≥ 1, the support set of �α

ξ ϕj is bounded, so thatby the discrete Leibniz formula (Lemma 3.3.6) we have∣∣�α

ξ ∂βx [ϕj(ξ)σj(x, ξ)]

∣∣ ≤ Cjαβ 〈ξ〉mj−ρ|α|+δ|β|

for some constant Cjαβ , since σj ∈ Smj

ρ,δ (Tn × Zn). This means that ((x, ξ) �→ϕj(ξ)σj(x, ξ)) ∈ S

mj

ρ,δ (Tn × Zn). Examining the support of �αξ ϕj , we see that

�αξ (ϕj(ξ)σj(x, ξ)) (where α ∈ Nn

0 ) vanishes for any fixed ξ ∈ Zn, when j is largeenough. This justifies the definition

σ(x, ξ) :=∞∑

j=0

ϕj(ξ) σj(x, ξ), (4.17)


and clearly σ ∈ Sm0ρ,δ (Tn). Furthermore,∣∣∣∣∣∣�α

ξ ∂βx

⎡⎣σ(x, ξ)−N−1∑j=0

σj(x, ξ)

⎤⎦∣∣∣∣∣∣≤

N−1∑j=0

∣∣�αξ ∂β

x {[ϕj(ξ)− 1] σj(x, ξ)}∣∣ +

∞∑j=N

∣∣�αξ ∂β

x [ϕj(ξ) σj(x, ξ)]∣∣ .

Recall that εj > εj+1 → 0, so that the∑N−1

j=0 part of the sum vanishes, whenever‖ξ‖ is large. Hence this part of the sum is dominated by CrNαβ 〈ξ〉−r for anyr ∈ R. The reader may verify that the

∑∞j=N part of the sum is majorised by

C ′Nαβ 〈ξ〉mN−ρ|α|+δ|β|. �Exercise 4.4.2. In the proof of Theorem 4.4.1. estimate the support of ξ �→�α

ξ ϕj(ξ) in terms of α and j. How large should ‖ξ‖ be for the∑N−1

j=0 part ofthe sum to vanish? Complete the proof by filling in the details. If necessary, con-sult the Euclidean version of this result which was proved in Proposition 2.5.33.

Definition 4.4.3 (Asymptotic expansions). The formal series∑∞

j=0 σj in Theo-rem 4.4.1 is called an asymptotic expansion of the symbol σ ∈ Sm0

ρ,δ (Tn × Zn) andit is presented in (4.17). In this case we denote

σ ∼∞∑

j=0

σj

(cf. a ∼ a′; a different but related meaning). Respectively,∑∞

j=0 Op(σj) is anasymptotic expansion of the operator Op(σ) ∈ Op(Sm0

ρ,δ (Tn × Zn)), denotedOp(σ) ∼∑∞

j=0 Op(σj). By altering ϕ ∈ C∞(Rn) and (εj)∞j=0 in the proof of The-orem 4.4.1 we get a (possibly) different symbol τ by (4.17). Nevertheless, σ ∼ τ ,which is enough in the symbol analysis of periodic pseudo-differential operators.We are often faced with asymptotic expansions σ ∼∑∞

j=0 σj , where

σj =∑

γ∈Nn0 : |γ|=j

σγ .

In such case we shall writeσ ∼

∑γ≥0

σγ .

Remark 4.4.4 (Principal symbol). Assume that in the asymptotic expansion σ ∼∑∞j=0 σj we have σ0 ∈ Sm0(Tn × Zn)\Sm1(Tn × Zn), i.e., σ0 is the most significant

term. It is then called the principal symbol of σ ∼ ∑∞j=0 σj . (In [130, p. 49] the

class σA + Sm−1(Rn) is by definition the principal symbol of the periodic pseudo-differential operator A ∈ Op(Sm(Rn)) when l < m implies that σA ∈ Sl; it isimportant due to its invariance under smooth changes of coordinates.)


Next we present an elementary result stating that amplitude operators aremerely periodic pseudo-differential operators, and we provide an effective way tocalculate the symbol modulo S−∞(Tn × Zn) from an amplitude: this theorem hasa fundamental status in the symbolic analysis. We give two alternative proofsfor it.

Theorem 4.4.5 (Symbols of amplitude operators). Let 0 ≤ δ < ρ ≤ 1. For ev-ery toroidal amplitude a ∈ Am

ρ,δ(Tn) there exists a unique toroidal symbol σ ∈

Smρ,δ(T

n × Zn) satisfying Op(a) = Op(σ), and σ has the following asymptotic ex-pansion:

σ(x, ξ) ∼∑γ≥0

1γ!�γ

ξ D(γ)y a(x, y, ξ)|y=x. (4.18)

Proof. As a linear operator in Sobolev spaces, Op(a) possesses the unique symbolσ = σOp(a) (or as an operator on C∞(Tn), see Definition 4.1.2), but at the momentwe do not yet know whether σ ∈ Sm

ρ,δ(Tn × Zn). By Theorem 4.1.4 the symbol is

computed from

σ(x, ξ) = e−i2πx·ξ Op(a) eξ(x)

= e−i2πx·ξ ∑η∈Zn

∫Tn

ei2π(x−y)·η a(x, y, η) ei2πy·ξ dy.

Now, we apply the discrete Taylor formula from Theorem 3.3.21 to obtain

σ(x, ξ) =∑

η∈Zn

∫Tn

ei2π(x−y)·(η−ξ) a(x, y, η) dy

=∑

η∈Zn

ei2πx·(η−ξ) a2(x, η − ξ, η)

=∑

η∈Zn

ei2πx·η a2(x, η, ξ + η)

=∑

η∈Zn

ei2πx·η ∑|γ|<N

1γ!�γ

ξ a2(x, η, ξ) η(γ) +

+∑

η∈Zn

ei2πx·η RN (x, η, ξ, η),

where RN (x, η, ξ, p) is the error term of the discrete Taylor expansion (Theorem3.3.21) of a2(x, η, ξ + p). Let us write

EN (x, ξ) :=∑

η∈Zn

ei2πx·η RN (x, η, ξ, η).


Notice that

D(γ)y a(x, y, ξ) = D(γ)

y

∑η∈Zn

ei2πy·η a2(x, η, ξ)

=∑

η∈Zn

ei2πy·η η(γ) a2(x, η, ξ),

which yields

σ(x, ξ) =∑|γ|<N

1γ!�γ

ξ D(γ)y a(x, y, ξ)|y=x + EN (x, ξ). (4.19)

All we need to show now is that EN ∈ Sm−Nρ,δ (Tn × Zn), and for this we have to

study the remainder RN . Recalling the form of the estimate for RN (see Theo-rem 3.3.21), noticing that for |γ| = N , |η(γ)| ≤ (|η|+N)N ≤ 2N 〈η〉NNN , applying(a close variant of) inequality (4.12), and using Peetre’s inequality (Proposition3.3.31), we get∣∣∣�α′

ξ ∂β′x RN (x, η, ξ, η)

∣∣∣ ≤ 1N !

max|γ|=N

∣∣∣η(γ)∣∣∣ max|ω|=N, ν∈Q(η)

∣∣∣�α′+ωξ ∂β′

x a2(x, η, ξ + ν)∣∣∣

≤ 1N !

2N 〈η〉NNN

× maxν∈Q(η)

[cr,β′,N+α′ 〈η〉−r〈ξ + ν〉m−ρN−ρ|α′|+δ|β′|+δr

].

Now, let us consider two cases. First, if |η| ≤ |ξ|/2, this can be estimated by

CN 〈η〉N−r〈ξ〉m−ρN−ρ|α′|+δ|β′|+δr,

and taking N = r this can be estimated by CN 〈ξ〉m−(ρ−δ)N−ρ|α′|+δ|β′| which wecan estimate by any 〈ξ〉−N ′

if we take N large enough. On the other hand, for theregion |η| > |ξ|/2, let us fix some (large) N ′ and take r = N + N ′. Then we canestimate

∣∣∣�α′ξ ∂β′

x RN (x, η, ξ, η)∣∣∣ by

CN 〈η〉N−r maxν∈Q(η)

[cr,β′,N+α′ 〈ξ + ν〉m−ρN−ρ|α′|+δ|β′|+δρ

]= 〈η〉−N ′

maxν∈Q(η)

[cr,β′,N+α′ 〈ξ + ν〉m−(ρ−δ)N−ρ|α′|+δ|β′|+δN ′]

.

Thus, if N is sufficiently large, the maximum term is bounded by a con-stant in view of ρ > δ, and the remaining term satisfies 〈η〉−N ′ ≤ C〈ξ〉−N ′

.Therefore, taking enough terms in the asymptotic expansion we can estimate�α

ξ ∂βxEN (x, ξ) by any power of 〈ξ〉−1 and since all the terms are in the necessary

symbol classes the estimate for the remainder is complete. Consequently σ belongsto Sm

ρ,δ(Tn × Zn) by equation (4.19), and Theorem 4.4.1 provides the asymptotic

expansion (4.18). �


Remark 4.4.6. Now we can compare the results above with the biperiodic Taylorseries. Applying Theorem 4.4.5 and Lemma 4.3.4, we get

a(x, y, ξ) ∼∞∑

γ=0

1γ!�γ

ξ D(γ)y a(x, y, ξ)|y=x

∼∞∑

γ=0

1γ!

(ei2π(y−x) − 1

)γ

D(γ)z a(x, z, ξ)|z=x,

reminding us of the series representation of Corollary 3.4.3.

Alternative proof for Theorem 4.4.5 on T1. We invoke the biperiodic Taylor ex-pansion for a(x, y, ξ) (see Corollary 3.4.3):

a(x, y, ξ) =N−1∑j=0

1j!

(ei2π(y−x) − 1

)D(j)

z a(x, z, ξ)|z=x

+aN (x, y, ξ)(ei2π(y−x) − 1

)N

.

Then we use Lemma 4.3.4 with bj(x, y, ξ) =(ei2π(y−x) − 1

)jcj(x, y, ξ), where

cj(x, y, ξ) = 1j!D

(j)y a(x, y, ξ)|y=x, to obtain

Op(bj) = Op(�jξcj) = Op

(1j!�j

ξD(j)y a(x, y, ξ)|y=x

).

By Lemma 4.3.4, the remainder aN (x, y, ξ)(ei2π(y−x) − 1

)Nhence contributes to

the operator Op(�Nξ aN ). Thus, in order to get the asymptotic expansion (4.18),

we have to prove that aN ∈ Amρ,δ(T

1) for every N ∈ Z+. From the proof of Theorem3.4.2 we see that aN is given by

aN (x, y, ξ) =aN−1(x, y, ξ)− aN−1(x, x, ξ)

ei2π(y−x) − 1,

and that it is in C∞(T1 × T1) for every ξ ∈ Z1. Here aN has the same orderas aN−1 does, so that recursively aN ∈ Am

ρ,δ(T1), since a0 = a ∈ Am

ρ,δ(T1). This

completes the proof. �

Remark 4.4.7 (Classical periodic pseudo-differential operators). The operator A ∈Op(Sm(Tn)) is called a classical periodic pseudo-differential operator, if its symbolhas an asymptotic expansion

σA(x, ξ) ∼∞∑

j=0

σj(x, ξ),


where the symbols σj are positively homogeneous of degree m − j: they satisfyσj(x, ξ) = σj(x, ξ/‖ξ‖)‖ξ‖m−j for large ξ. In [142] and [102], it is shown thatany classical periodic pseudo-differential operator can be expressed as a sum ofperiodic integral operators of the type (4.44) – other contributions to periodicintegral operators and classical operators are made in [34], [62], [142], and [102].The research on these operators is of interest, but in the sequel we will ratherconcentrate on questions of the symbolic analysis.

4.5 Extension of toroidal symbols

In the study of periodic pseudo-differential operators some of the applications ofthe calculus of finite differences, for example the discrete Taylor series, can beeliminated. We are going to explain how this can be done by interpolating a sym-bol (x, ξ) �→ σ(x, ξ) in the second argument ξ in a smooth way, so that it becomesdefined on Tn×Rn instead of Tn×Zn. This process will be called an extension ofthe toroidal symbol. By using such extensions one can work with the familiar toolsof classical analysis yielding the same theory as before, and for some practicalexamples this may be more convenient than operating with differences. However,this approach is quite alien to the idea of periodic symbols, as the results can bederived using quite simple difference calculus. In addition, difference operationscan easily be carried out with computers, whereas program realisations of nu-merical differentiation are computationally expensive and troublesome. Moreover,such an extension explores the intricate relation between Tn and Rn and can notbe readily generalised to symbols on other compact Lie groups (thus while veryuseful on Tn yet unfortunately not providing an additional intuition for operatorsin Part IV).

Thus, it is often useful to extend toroidal symbols from Tn×Zn to Tn×Rn,ideally getting symbols in Hormander’s symbol classes. The case of n = 1 and(ρ, δ) = (1, 0) was considered in [141] and [102]. This extension can be done witha suitable convolution that respects the symbol inequalities. In the following, δ0,ξ

is the Kronecker delta at 0 ∈ Zn, i.e., δ0,0 = 1, and δ0,ξ = 0 if ξ = 0. First weprepare the following useful functions θ, φα ∈ S(Rn):

Lemma 4.5.1. There exist functions φα ∈ S(Rn) (for each α ∈ Nn0 ) and a function

θ ∈ S(Rn) such that

Pθ(x) :=∑

k∈Zn

θ(x + k) ≡ 1,

(FRnθ)|Zn(ξ) = δ0,ξ and ∂αξ (FRnθ)(ξ) = �α

ξ φα(ξ)

for all ξ ∈ Zn.

The idea of this lemma may be credited to Yves Meyer [29, p. 4]. It will beused in the interpolation presented in Theorem 4.5.3.

4.5. Extension of toroidal symbols 357

Proof. Let us first consider the one-dimensional case. Let θ = θ1 ∈ C∞(R1) suchthat

supp(θ1) ⊂ (−1, 1), θ1(−x) = θ1(x), θ1(1− y) + θ1(y) = 1

for x ∈ R and for 0 ≤ y ≤ 1; these assumptions for θ1 are enough for us, and ofcourse the choice is not unique. In any case, θ1 ∈ S(R1), so that also FRθ1 ∈ S(R1).If ξ ∈ Z1 then we have

FRθ1(ξ) =∫

R1θ1(x) e−i2πx·ξ dx

=∫ 1

0

(θ1(x− 1) + θ1(x)) e−i2πx·ξ dx

= δ0,ξ.

If a desired φα ∈ S(R1) exists, it must satisfy∫R1

ei2πx·ξ ∂αξ (FRθ1)(ξ) dξ =

∫R1

ei2πx·ξ �α

ξ φα(ξ) dξ

=(1− ei2πx

)α∫

R1ei2πx·ξφα(ξ) dξ

due to the bijectivity of FR : S(R1) → S(R1). Integration by parts leads to theformula

(−i2πx)αθ1(x) = (1− ei2πx)α(F−1R φα)(x).

Thus

(F−1R φα)(x) =

⎧⎪⎨⎪⎩(−i2πx

1−ei2πx

)α

θ1(x), if 0 < |x| < 1,

1, if x = 0,0, if |x| ≥ 1.

The general n-dimensional case is reduced to the one-dimensional case, since map-ping θ = (x �→ θ1(x1)θ1(x2) · · · θ1(xn)) ∈ S(Rn) has the desired properties. �Remark 4.5.2 (Periodic symbols on Rn). The defining symbol inequalities for theclass Sm

ρ,δ(Tn × Rn) of periodic symbols on Rn are

∀α, β ∈ Nn0 ∃cαβ > 0 :

∣∣∂αξ ∂β

xσ(x, ξ)∣∣ ≤ cαβ (1 + ‖ξ‖)m−ρ|α|+δ|β|. (4.20)

To emphasize the difference with toroidal symbols defined on Tn × Zn we callthem Euclidean symbols.

Lemma 4.5.1 provides us the means to interpolate between the discrete pointsof Zn in a manner that is faithful to the symbol (cf. inequalities (4.6) and (4.20)):

Theorem 4.5.3 (Toroidal vs Euclidean symbols). Let 0 < ρ ≤ 1 and 0 ≤ δ ≤ 1.Symbol a ∈ Sm

ρ,δ(Tn×Zn) is a toroidal symbol if and only if there exists a Euclidean

symbol a ∈ Smρ,δ(T

n × Rn) such that a = a|Tn×Zn . Moreover, this extended symbola is unique modulo S−∞(Tn × Rn).


The relation between the corresponding pseudo-differential operators will begiven in Theorem 4.6.12. For the relation between extensions and ellipticity seeTheorem 4.9.15.

Proof. Let us first prove the “if” part. Let a ∈ Smρ,δ(T

n×Rn), and in this part wecan actually allow any ρ and δ, for example 0 ≤ ρ, δ ≤ 1. By the Lagrange MeanValue Theorem, if |α| = 1 then

�αξ ∂β

x a(x, ξ) = �αξ ∂β

xa(x, ξ)

= ∂αξ ∂β

xa(x, ξ)|ξ=η,

where η is on the line from ξ to ξ + α. By the Mean Value Theorem, for a generalmulti-index α ∈ Nn

0 , we also have

�αξ ∂β

x a(x, ξ) = ∂αξ ∂β

xa(x, ξ)|ξ=η

for some η ∈ Q := [ξ1, ξ1+α1]×· · ·×[ξn, ξn+αn]. This can be shown by induction.Indeed, let us write α = ω + γ for some ω = δj . Then we can calculate

�αξ ∂β

x a(x, ξ) = �ωξ

(�γ

ξ ∂βx a

)(x, ξ)

= �ξj

(∂γ

ξ ∂βxa(x, ξ)|ξ=ζ

)= ∂γ

ξ ∂βxa(x, ζ + δj)− ∂γ

ξ ∂βxa(x, ζ)

= ∂αξ ∂β

xa(x, ξ)|ξ=η

for some ζ and η, where we used the induction hypothesis in the first line. There-fore, we can estimate∣∣�α

ξ ∂βx a(x, ξ)

∣∣ =∣∣∂α

ξ ∂βxa(x, ξ)|ξ=η∈Q

∣∣≤ Cαβm 〈η〉m−ρ|α|+δ|β|

≤ C ′αβm 〈ξ〉m−ρ|α|+δ|β|.

Let us now prove the “only if” part. First we show the uniqueness. Leta, b ∈ Sm

ρ,δ(Tn × Rn) and assume that a|Tn×Zn = b|Tn×Zn . Let c = a − b. Then

c ∈ Smρ,δ(T

n×Rn) and it satisfies c|Tn×Zn = 0. If ξ ∈ Rn \Zn, choose η ∈ Zn that isthe nearest point (or one of the nearest points) to ξ. Then we have the first-orderTaylor expansion

c(x, ξ) = c(x, η) +∑

α: |α|=1

rα(x, ξ, ξ − η) (ξ − η)α

=∑

α: |α|=1

rα(x, ξ, ξ − η) (ξ − η)α,

where

rα(x, ξ, θ) =∫ 1

0

(1− t) ∂αξ c(x, ξ + tθ) dt.

4.5. Extension of toroidal symbols 359

Hence we have |c(x, ξ)| ≤ C 〈ξ〉m−ρ. Continuing the argument inductively forc and its derivatives and using that ρ > 0, we obtain the uniqueness moduloS−∞(Tn × Rn).

Let us now show the existence. Let θ ∈ S(Rn) be as in Lemma 4.5.1. Definea : Tn × Rn → C by

a(x, ξ) :=∑

η∈Zn

(FRnθ)(ξ − η) a(x, η). (4.21)

It is easy to see that a = a|Tn×Zn . Furthermore, we have

∣∣∂αξ ∂β

xa(x, ξ)∣∣ =

∣∣∣∣∣∣∑

η∈Zn

∂αξ (FRnθ)(ξ − η) ∂β

x a(x, η)

∣∣∣∣∣∣=

∣∣∣∣∣∣∑

η∈Zn

�α

ξ φα(ξ − η) ∂βx a(x, η)

∣∣∣∣∣∣(3.14)=

∣∣∣∣∣∣∑

η∈Zn

φα(ξ − η) �αη ∂β

x a(x, η) (−1)|α|

∣∣∣∣∣∣≤

∑η∈Zn

|φα(ξ − η)| Cαβm 〈η〉m−ρ|α|+δ|β|

≤ C ′αβm 〈ξ〉m−ρ|α|+δ|β| ∑η∈Zn

|φα(η)| 〈η〉|m−ρ|α|+δ|β||

≤ C ′′αβm 〈ξ〉m−ρ|α|+δ|β|,

where we used the summation by parts formula (3.14). In the last two lines we alsoused that φα ∈ S(Rn) and also a simple fact that for p > 0 we have 〈ξ + η〉p ≤〈ξ〉p〈η〉p and 〈ξ + η〉−p〈η〉−p ≤ 〈ξ〉−p, for all ξ, η ∈ Rn. Thus a∈Sm

ρ,δ(Tn×Rn). �

From now on, we can exploit inequalities (4.20), but it is good to rememberthat all the information was contained already in the original definition of symbolson Tn × Zn. In a sense, the extension is arbitrary (yet unique up to order −∞),as the demands for the function θ ∈ S(Rn) were quite modest in the proof ofLemma 4.5.1.Remark 4.5.4. The extension process can also be modified for amplitudes to geta(x, y, ξ) (continuous ξ ∈ Rn) from a(x, y, ξ) (discrete ξ ∈ Zn).Remark 4.5.5 (Extension respects ellipticity.). Moreover, the extension respectsellipticity, as we will show in Theorem 4.9.15.

Exercise 4.5.6. Work out details of the proof of Remark 4.5.4.

We also observe that the same proof yields the following limited regularityversion of Theorem 4.5.3:


Corollary 4.5.7 (Limited regularity extensions). Let the function a : Tn×Rn → Csatisfy ∣∣∂α

ξ ∂βxa(x, ξ)

∣∣ ≤ cαβ 〈ξ〉m−ρ|α|+δ|β| for all x ∈ Tn, ξ ∈ Rn, (4.22)

for all |α| ≤ N1 and |β| ≤ N2. Then its restriction a := a|Tn×Zn satisfies∣∣�αξ ∂β

x a(x, ξ)∣∣ ≤ cαβ 〈ξ〉m−ρ|α|+δ|β| for all x ∈ Tn, ξ ∈ Zn, (4.23)

and all |α| ≤ N1 and |β| ≤ N2. Conversely, every function a : Tn × Zn → Csatisfying (4.23) for all |α| ≤ N1 and |β| ≤ N2 is a restriction a = a|Tn×Zn ofsome function a : Tn × Rn → C satisfying (4.22) for all |α| ≤ N1 and |β| ≤ N2.

4.6 Periodisation of pseudo-differential operators

In this section we describe the relation between operators with Euclidean andtoroidal quantizations and between operators corresponding to symbols a(x, ξ)and a = a|Tn×Zn , given by the operator of the periodisation of functions.

First we state a property of a pseudo-differential operator a(X, D) to mapthe space S(Rn) into itself, which will be of importance. The following class willbe sufficient for our purposes, and the proof is straightforward.

Proposition 4.6.1. Let a = a(x, ξ) ∈ C∞(Rn × Rn) and assume that there existε > 0 and N ∈ R such that for every α, β there are constants Cαβ and M(α, β)such that the estimate∣∣∣∂α

x ∂βξ a(x, ξ)

∣∣∣ ≤ Cαβ〈x〉N+(1−ε)|β|〈ξ〉M(α,β)

holds for all x, ξ ∈ Rn. Then the pseudo-differential operator a(X, D) with symbola(x, ξ) is continuous from S(Rn) to S(Rn).

Exercise 4.6.2. Prove Proposition 4.6.1.

Before analysing the relation between operators with Euclidean and toroidalquantizations, we will describe the periodisation operator that will be of impor-tance for such analysis.

Theorem 4.6.3 (Periodisation). The periodisation Pf : Rn → C of a functionf ∈ S(Rn) is defined by

Pf(x) :=∑

k∈Zn

f(x + k). (4.24)

Then P : S(Rn)→ C∞(Tn) is surjective and ‖Pf‖L1(Tn) ≤ ‖f‖L1(Rn). Moreover,we have

Pf(x) = F−1Tn ((FRnf)|Zn) (x) (4.25)

and ∑k∈Zn

f(x + k) =∑ξ∈Zn

ei2πx·ξ (FRnf)(ξ). (4.26)

4.6. Periodisation of pseudo-differential operators 361

Taking x = 0 in (4.26), we obtain

Corollary 4.6.4 (Poisson summation formula). For f ∈ S(Rn) we have∑k∈Zn

f(k) =∑ξ∈Zn

f(ξ).

As a consequence of the Poisson summation formula and the Fourier trans-form of Gaussians in Lemma 1.1.23 we obtain the so-called Jacobi identity:

Exercise 4.6.5 (Jacobi identity for Gaussians). For every ε > 0 we have

+∞∑j=−∞

e−πj2/ε =√

ε+∞∑

j=−∞e−πεj2

.

Remark 4.6.6. We note that by Theorem 4.6.3 we may extend the periodisationoperator P to L1(Rn), and also this extension is surjective from L1(Rn) to L1(Tn).This is actually rather trivial compared to the smooth case of Theorem 4.6.3because we can find a preimage f ∈ L1(Rn) of g ∈ L1(Tn) under the periodisationmapping P by for example setting f = g|[0,1]n and f = 0 otherwise.

Exercise 4.6.7. Observe that the periodisation operator P : S(Rn) → C∞(Tn) isdualised to Pt : D′(Tn)→ S ′(Rn) by the formula⟨

Ptu, ϕ⟩

:= 〈u,Pϕ〉 for all ϕ ∈ S(Rn).

Indeed, if ϕ ∈ S(Rn) we have that Pϕ ∈ C∞(Tn), so that this definition makessense for u ∈ D′(Tn). What is the meaning of the operator Pt?

Proof of Theorem 4.6.3. The estimate ‖Pf‖L1(Tn) ≤ ‖f‖L1(Rn) is straightforward.Next, for ξ ∈ Zn, we have

FTn(Pf)(ξ) =∫

Tn

e−i2πx·ξ Pf(x) dx

=∫

Tn

e−i2πx·ξ ∑k∈Zn

f(x + k) dx

=∫

Rn

e−i2πx·ξ f(x) dx

= (FRnf)(ξ).

From this we can see that∑k∈Zn

f(x + k) = Pf(x)

=∑ξ∈Zn

ei2πx·ξ FTn(Pf)(ξ)

=∑ξ∈Zn

ei2πx·ξ (FRnf)(ξ),


proving (4.26). Let us show the surjectivity of P : S(Rn) → C∞(Tn). Let θ ∈S(Rn) be a function defined in Lemma 4.5.1. Then for any g ∈ C∞(Tn) it holdsthat

P(gθ)(x) =∑

k∈Zn

g(x + k) θ(x + k) = g(x)∑

k∈Zn

θ(x + k) = g(x),

where gθ is the product of θ with Zn-periodic function g on Rn. We omit thestraightforward proofs of the other claims. �

We saw in Proposition 3.1.34 that Dirac delta comb δZn can be viewed as asum of Dirac deltas. We can also relate it to the partial sums of Fourier coefficients:

Proposition 4.6.8. Let us define Qj ∈ S ′(Rn) by

〈Qj , ϕ〉 :=∑

k∈Zn: |k|≤j

∫Rn

ei2πk·ξ ϕ(ξ) dξ (4.27)

for ϕ ∈ S(Rn). Then we have the convergence Qj → δZn in S ′(Rn) to the Diracdelta comb.

Proof. Indeed, we have

〈Qj , ϕ〉 =∑

k∈Zn: |k|≤j

∫Rn

ei2πk·ξ ϕ(ξ) dξ

=∑

k∈Zn: |k|≤j

(FRnϕ)(k)

−−−→j→∞

∑k∈Zn

(FRnϕ)(k)

Poisson=∑ξ∈Zn

ϕ(ξ)

= 〈δZn , ϕ〉 ,

for all ϕ ∈ S(Rn). �Remark 4.6.9 (Inflated torus). For N ∈ N we write NTn = (R/NZ)n which wecall the N -inflated torus, or simply an inflated torus if the value of N is not ofimportance. We note that in the case of the N -inflated torus NTn we can use theperiodisation operator PN instead of P, where PN : S(Rn) → C∞(NTn) can bedefined by

(PNf)(x) = F−1NTn

(FRnf | 1

N Zn

)(x), x ∈ NTn. (4.28)

Exercise 4.6.10. Generalise Theorem 4.6.3 to the N -inflated torus using opera-tor PN .

Let us now establish some basic properties of pseudo-differential operatorswith respect to periodisation.


Definition 4.6.11. We will say that a function a : Rn × Rn → C is 1-periodic (wewill always mean that it is periodic with respect to the first variable x ∈ Rn) ifthe function x �→ a(x, ξ) is Zn-periodic for all ξ.

As in Theorem 4.5.3, we use tilde to denote the restriction of a ∈ C∞(Rn ×Rn) to Rn×Zn. If a(x, ξ) is 1-periodic, we can view it as a function on Tn×Zn, andwe write a = a|Tn×Zn . For such a the corresponding operator Op(a) = a(X, D) isdefined by (4.7) in Definition 4.1.9.

Theorem 4.6.12 (Periodisation of operators). Let a = a(x, ξ) ∈ C∞(Rn × Rn) be1-periodic with respect to x for every ξ ∈ Rn. Assume that for every α, β ∈ Nn

0

there are constants Cαβ and M(α, β) such that the estimate∣∣∣∂αx ∂β

ξ a(x, ξ)∣∣∣ ≤ Cαβ〈ξ〉M(α,β)

holds for all x, ξ ∈ Rn. Let a = a|Tn×Zn . Then

P ◦ a(X, D)f = a(X, D) ◦ Pf (4.29)

for all f ∈ S(Rn).

Note that it is not important in this theorem that a is in any of the symbolclasses Sm

ρ,δ(Rn × Rn).

Combining Theorems 4.5.3 and 4.6.12 we get

Corollary 4.6.13 (Equality of operator classes). For 0 ≤ δ ≤ 1 and 0 < ρ ≤ 1 wehave

Op(Smρ,δ(T

n × Rn)) = Op(Smρ,δ(T

n × Zn)),

i.e., classes of 1-periodic pseudo-differential operators with Euclidean (Horman-der’s) symbols in Sm

ρ,δ(Tn × Rn) and toroidal symbols in Sm

ρ,δ(Tn × Zn) coincide.

Remark 4.6.14. Note that by Proposition 4.6.1 both sides of (4.29) are well-definedfunctions in C∞(Tn). Moreover, equality (4.29) can be justified for f in largerclasses of functions. For example, (4.29) remains true pointwise if f ∈ Ck

0 (Rn) is aCk compactly supported function for k sufficiently large. In any case, an equalityon S(Rn) allows an extension to S ′(Rn) by duality.

Proof of Theorem 4.6.12. Let f ∈ S(Rn). Then we have

P(a(X, D)f)(x) =∑

k∈Zn

(a(X, D)f)(x + k)

=∑

k∈Zn

∫Rn

ei2π(x+k)·ξ a(x + k, ξ) (FRnf)(ξ) dξ

=∫

Rn

( ∑k∈Zn

ei2πk·ξ)

ei2πx·ξ a(x, ξ) (FRnf)(ξ) dξ


=∫

Rn

δZn(ξ) ei2πx·ξ a(x, ξ) (FRnf)(ξ) dξ

=∑ξ∈Zn

ei2πx·ξ a(x, ξ) (FRnf)(ξ)

=∑ξ∈Zn

ei2πx·ξ a(x, ξ) FTn(Pf)(ξ) = a(X, D)(Pf)(x),

where δZn is the Dirac δ comb from Definition 3.1.32. As usual, these calculationscan be justified in the sense of distributions (see Remark 4.6.15). �Remark 4.6.15 (Distributional justification). We now give the distributional in-terpretation of the calculations in Theorem 4.6.12. Let us define some useful vari-ants of the Dirac delta comb from Definition 3.1.32: for x ∈ R and j ∈ Z+, letPx,Px

j ∈ S ′(Rn) be such that

〈Px, ϕ〉 :=∑

k∈Zn

ϕ(x + k), 〈Pxj , ϕ〉 :=

∑k∈Zn: |k|≤j

ϕ(x + k),

for ϕ ∈ S(Rn). We can easily observe that Pxj → Px in S ′(Rn). Then we can

calculate:

P(a(X, D)f)(x)= 〈Px, a(X, D)f〉

Pxj →Px

= limj→∞

⟨Px

j , a(X, D)f⟩

= limj→∞

∑k∈Zn: |k|≤j

(a(X, D)f)(x + k)

= limj→∞

∑k∈Zn: |k|≤j

∫Rn

e2πi(x+k)·ξ a(x + k, ξ) (FRnf)(ξ) dξ

= limj→∞

∑k∈Zn: |k|≤j

∫Rn

ei2πk·ξ e2πix·ξ a(x, ξ) (FRnf)(ξ) dξ

Qj from (4.27)= lim

j→∞⟨Qj , ξ �→ e2πix·ξ a(x, ξ) (FRnf)(ξ)

⟩Qj→δZn

=⟨δZn , ξ �→ e2πix·ξ a(x, ξ) (FRnf)(ξ)

⟩(4.25)=

∑ξ∈Zn

e2πix·ξ a(x, ξ) (FRnf)(ξ)

=∑ξ∈Zn

ei2πx·ξ a(x, ξ) FTn(Pf)(ξ)

= a(X, D)(Pf)(x).

As we can see, the distributional justifications are quite natural, in the end.


Let us now formulate a useful corollary of Theorem 4.6.12 that will be ofimportance later, in particular in composing a pseudo-differential operator with aFourier series operator in the proof of Theorem 4.13.11. If in Theorem 4.6.12 wetake function f such that f = g|[0,1]n for some g ∈ C∞(Tn), and f = 0 otherwise,it follows immediately that g = Pf . Adjusting this argument by shifting the cube[0, 1]n if necessary and shrinking the support of g to make f smooth, we obtain

Corollary 4.6.16. Let a = a(x, ξ) be as in Theorem 4.6.12, let g ∈ C∞(Tn), andlet V be an open cube in Rn with side length equal to 1. Assume that the supportof g|V is contained in V . Then we have the equality

a(X, D)g = P ◦ a(X, D)(g|V ),

where g|V : Rn → C is defined as the restriction of g to V and equal to zero outsideof V .

Exercise 4.6.17. Work out the details of the proof of Corollary 4.6.16. Especially,the fact that a is 1-periodic plays an important role.

Since we do not always have periodic symbols on Rn it may be convenientto periodise them.

Definition 4.6.18 (Periodisation of symbols). If a(X, D) is a pseudo-differentialoperator with symbol a(x, ξ), by (Pa)(X, D) we will denote the pseudo-differentialoperator with symbol

(Pa)(x, ξ) :=∑

k∈Zn

a(x + k, ξ).

This procedure makes sense if, for example, a is in L1(Rn) with respect to thevariable x.

In the following proposition we will assume that supports of symbols andfunctions are contained in the cube [−1/2, 1/2]n. We note that this is not restrictiveif these functions are already compactly supported. Indeed, if supports of a(·, ξ)and f are contained in some compact set independent of ξ, we can find someN ∈ N such that they are contained in [−N/2, N/2]n, and then use the analysison the N -inflated torus, with periodisation operator PN instead of P, defined in(4.28).

Proposition 4.6.19 (Operator with periodised symbol). Let a = a(x, ξ) ∈ C∞(Rn×Rn) satisfy supp a ⊂ [−1/2, 1/2]n×Rn and be such that for every α, β ∈ Nn

0 thereare constants Cαβ and M(α, β) such that the estimate∣∣∣∂α

x ∂βξ a(x, ξ)

∣∣∣ ≤ Cαβ〈ξ〉M(α,β)

holds for all x, ξ ∈ Rn. Then we have

a(X, D)f = (Pa)(X, D)f + Rf,


for all f ∈ C∞(Rn) with supp f ⊂ [−1/2, 1/2]n. If moreover a ∈ Smρ,δ(R

n × Rn)with ρ > 0, then the operator R extends to a smoothing pseudo-differential operatorR : D′(Rn)→ S(Rn).

Proof. By the definition we can write

(Pa)(X, D)f(x) =∑

k∈Zn

∫Rn

ei2πx·ξ a(x + k, ξ) FRnf(ξ) dξ,

and let us define Rf := a(X, D)f − (Pa)(X, D)f. The assumption on the supportof a implies that for every x there is only one k ∈ Zn for which a(x + k, ξ) = 0, sothe sum consists of only one term. It follows that Rf(x) = 0 for x ∈ [−1/2, 1/2]n,because for such x the non-zero term corresponds to k = 0. Let now x ∈ Rn \[−1/2, 1/2]n. Since a(x, ξ) = 0 for all ξ ∈ Zn, we have that the sum

Rf(x) = −∑

k∈Zn,k �=0

∫Rn

∫Rn

ei2π(x−y)·ξ a(x + k, ξ) f(y) dy dξ

is just a single term and |x− y| > 0 on supp f , so we can integrate by parts withrespect to ξ any number of times. This implies that R ∈ Ψ−∞(Rn × Rn) becauseρ > 0 and that Rf decays at infinity faster than any power. The proof is completesince the same argument can be applied to the derivatives of Rf as well. �Exercise 4.6.20. Work out the details for the derivatives of Rf .

Proposition 4.6.19 allows us to extend the formula of Theorem 4.6.12 tocompact perturbations of periodic symbols. We will use it later when a(X, D) isa sum of a constant coefficient operator and an operator with compactly (in x)supported symbol.

Corollary 4.6.21 (Periodisation and compactly supported perturbations).Let a(X, D) be an operator with symbol

a(x, ξ) = a1(x, ξ) + a0(x, ξ),

where a1 is as in Theorem 4.6.12, a1 is 1-periodic in x for every ξ ∈ Rn, and a0

is as in Proposition 4.6.19, supported in [−1/2, 1/2]n × Rn. Define

b(x, ξ) := a1(x, ξ) + Pa0(x, ξ), x ∈ Tn, ξ ∈ Zn.

Then we haveP ◦ a(X, D)f = b(X, D) ◦ Pf + P ◦Rf (4.30)

for all f ∈ S(Rn), and operator R extends to R : D′(Rn)→ S(Rn), so that P ◦R :D′(Rn)→ C∞(Tn). Moreover, if a1, a0 ∈ Sm

ρ,δ(Rn×Rn), then b ∈ Sm

ρ,δ(Tn×Zn).

Remark 4.6.22. Recalling Remark 4.6.14, (4.30) can be justified for larger functionclasses, e.g., for f ∈ Ck

0 (Rn) for k sufficiently large (which will be of use in Section4.12).

4.7. Symbolic calculus 367

Proof of Corollary 4.6.21. By Proposition 4.6.19 we can write

a(X, D) = a1(X, D) + (Pa0)(X, D) + R,

with R : D′(Rn)→ S(Rn). Let us define

b(x, ξ) := a1(x, ξ) + (Pa0)(x, ξ),

so that b = b|Tn×Zn . The symbol b is 1-periodic, hence for the operator b(X, D) =a1(X, D) + (Pa0)(X, D) by Theorem 4.6.12 we have

P ◦ b(X, D) = b(X, D) ◦ P= a1(X, D) ◦ P + Pa0(X, D) ◦ P.

Since R : D′(Rn) → S(Rn), we also have P ◦ R : D′(Rn) → C∞(Tn). Finally, ifa1, a0 ∈ Sm

ρ,δ(Rn × Rn), then b ∈ Sm

ρ,δ(Tn × Zn) by Theorem 4.5.3. The proof is

complete. �

4.7 Symbolic calculus

In this section we show that (for suitable ρ, δ) the family of periodic pseudo-differential operators is a ∗-algebra, i.e., it is closed under sums (trivially σj ∈S

mj

ρ,δ (Tn × Zn)⇒ σ1+σ2 ∈ Smax{m1,m2}ρ,δ (Tn × Zn)), products, and taking adjoints;

this algebraic structure is the key property to the applicability of periodic pseudo-differential operators. Furthermore, under these operations the degrees of opera-tors behave as one would expect. In the proofs the symbol analysis techniques areused leaving us with asymptotic expansions, so that there is a point to study peri-odic pseudo-differential operators that are invertible modulo Op(S−∞(Tn × Zn));that is, the elliptic operators, which are discussed in Section 4.9.

Recall that now there are two types of symbols, toroidal and Euclidean, in(4.6) and (4.20), yielding two alternative (toroidal and Euclidean) quantizations foroperators, respectively, see Corollary 4.6.13. As usual, we emphasize this differenceby writing σ ∈ Sm

ρ,δ(Tn × Zn) and σ ∈ Sm

ρ,δ(Tn × Rn), respectively.

Now we will discuss the calculus of pseudo-differential operators with toroidalsymbols. For this, let us fix the notation first and recall discrete versions of deriva-tives from Definition 3.4.1:

D(α)y = D(α1)

y1· · ·D(αn)

yn, (4.31)

where D(0)yj = I and

D(k+1)yj

= D(k)yj

(∂

i2π∂yj− kI

)=

∂

i2π∂yj

(∂

i2π∂yj− I

)· · ·

(∂

i2π∂yj− kI

).


Also, in this section the equivalence of asymptotic sums in Definition 4.1.19 willbe of use. We now observe how the difference operator affects expansions.

Lemma 4.7.1. Let 0 ≤ δ < ρ ≤ 1. Assume that σ ∈ Smρ,δ(T

n × Rn). Then

∞∑γ=0

1γ!�γ

ξ D(γ)x σ(x, ξ) −∞∼

∞∑γ=0

1γ!

∂γξ Dγ

xσ(x, ξ) (4.32)

= exp (∂ξDx) σ(x, ξ),

where exp is used in abbreviating the right-hand side of (4.32).

In the sequel we will drop the infinity sign from −∞∼ and will simply write ∼.

Proof. We apply Theorem 3.3.39 in order to translate differences into derivatives,and use the definition of the Stirling numbers1 of the second kind:∑

α≥0

1α!�α

ξ D(α)x σ(x, ξ)

m−(ρ−δ)N,ρ,δ∼∑|α|<N

1α!

⎡⎣ ∑|γ|<N

α!γ!

{γα

}∂γ

ξ

⎤⎦D(α)x σ(x, ξ)

=∑|γ|<N

1γ!

∂γξ

⎡⎣ ∑|α|<N

{γα

}D(α)

x

⎤⎦σ(x, ξ)

=∑|γ|<N

1γ!

∂γξ Dγ

xσ(x, ξ).

Since we can do this for any N , the proof is complete. �

According to Remark 4.5.4, we can do the same for amplitudes:

Proposition 4.7.2. Let 0 ≤ δ < ρ ≤ 1. If a ∈ Amρ,δ(T

n) is an extended defined onTn × Tn × Rn, then

σOp(a)(x, ξ) ∼∞∑

γ=0

1γ!

∂γξ Dγ

ya(x, y, ξ)|y=x (4.33)

= exp (∂ξDy) a(x, y, ξ)|y=x.

Proof. Replace the symbol σ(x, ξ) and the operators D(γ)x , Dγ

x of (4.32) by a(x, y, ξ)and D

(γ)y , Dγ

y , respectively. Then the expansion (4.18) yields the claim. �

1For Stirling numbers see Definition 3.3.33 and for their properties see Section 3.3.4.


It is time to prove that the adjoints of periodic pseudo-differential operatorsare periodic pseudo-differential operators by constructing an asymptotic expan-sion. As a secondary result we notice that the adjoint operation A �→ A∗ does notchange the order of the periodic pseudo-differential operator. A direct proof of thisby-product, not using the expansion method, is presented in Remark 4.9.10. In Rn

we proved the adjoint case in Theorem 2.5.13 leaving the transpose as Exercise2.5.18, and here we will do the opposite. Thus, we will prove this result for trans-pose operators first, for their definition see Definition 2.5.15 as well as Exercise2.5.16 and Remark 2.5.17.

Theorem 4.7.3 (Transpose operators). Let 0 ≤ δ < ρ ≤ 1. Let A ∈ Ψmρ,δ(T

n × Zn)be a pseudo-differential operator with toroidal symbol σA ∈ Sm

ρ,δ(Tn × Zn). Then

the transpose At of A is in Ψmρ,δ(T

n × Zn) and its symbol σAt ∈ Smρ,δ(T

n × Zn) hasthe following asymptotic expansion:

σAt(x, ξ) ∼∑γ≥0

1γ!�γ

ξ D(γ)x σA(x,−ξ). (4.34)

Remark 4.7.4. We note that the constant (2πi)−|α| which is present in the asymp-totic formula in Exercise 2.5.18 does not appear in (4.34). This is due to thefact that discrete modifications of derivatives in (4.31) do not have homogeneoussymbols and hence the constant remains incorporated in their definition.

Proof of Theorem 4.7.3. Assume that u, v ∈ C∞(Tn). We make use of the integralrepresentation of the duality product and the definition of amplitude operators.In general, if the operator A has the amplitude a(x, y, ξ) = σA(x, ξ), we have∫

Tn

v(x) Atu(x) dx

=∫

Tn

u(y) Av(y) dy

=∫

Tn

u(y)

⎛⎝ ∑ξ∈Zn

∫Tn

ei2π(y−x)·ξ a(y, x, ξ) v(x) dx

⎞⎠ dy

=∫

Tn

v(x)

⎛⎝ ∑ξ∈Zn

∫Tn

ei2π(y−x)·ξ a(y, x, ξ) u(y) dy

⎞⎠ dx.

Thus At = Op(at) with at(x, y, ξ) = a(y, x,−ξ). Especially,

(x, y, ξ) �→ σA(y,−ξ)

is an amplitude of At, so that by (4.18) the asymptotic expansion in the discretecase follows. �


Remark 4.7.5. Notice that in the proof of Theorem 4.7.3, the sum over ξ convergesabsolutely if a ∈ Am

ρ,δ(Tn) with m < −n. Otherwise, this can be understood by a

suitable integration by parts, as usual.

In fact, we also proved the following:

Corollary 4.7.6. Let 0 ≤ δ < ρ ≤ 1. If A is an amplitude operator with toroidalamplitude a ∈ Am

ρ,δ(Tn), then At is also an amplitude operator in Op(Am

ρ,δ(Tn))

with amplitudeat(x, y, ξ) = a(y, x,−ξ).

Theorem 4.7.7 (Adjoint operators). Let 0 ≤ δ < ρ ≤ 1. Let A ∈ Ψmρ,δ(T

n × Zn) bea pseudo-differential operator with toroidal symbol σA ∈ Sm

ρ,δ(Tn × Zn). Then the

adjoint A∗ of A is in Ψmρ,δ(T

n × Zn) and its toroidal symbol σA∗ ∈ Smρ,δ(T

n × Zn)has the following asymptotic expansion:

σA∗(x, ξ) ∼∑α≥0

1α!�α

ξ D(α)x σA(x, ξ). (4.35)

Exercise 4.7.8. Accordingly, show that if A is an amplitude operator with toroidalamplitude in Am

ρ,δ(Tn), then A∗ is also an amplitude operator in Op(Am

ρ,δ(Tn))

with amplitude a∗(x, y, ξ) = a(y, x, ξ) . Hence prove Theorem 4.7.7.

The next lemma paves the way for a proof that a composition of periodicpseudo-differential operators is a periodic pseudo-differential operator:

Lemma 4.7.9 (Product of toroidal symbols). The pointwise product of toroidalsymbols σA ∈ Sm

ρ,δ(Tn × Zn) and σB ∈ Sl

ρ,δ(Tn × Zn) is a toroidal symbol in

Sm+lρ,δ (Tn × Zn).

Proof. This is a simple calculation exploiting both the classical and the discreteLeibniz formulae (Theorem 1.5.10, (iv), and Lemma 3.3.6) the defining inequalityof the symbol of a periodic pseudo-differential operator, and Peetre’s inequality(Proposition 3.3.31):∣∣�α

ξ ∂βx [σB(x, ξ) σA(x, ξ)]

∣∣=

∣∣∣∣∣∣∑γ≤β

∑ω≤α

(β

γ

) (α

ω

) [∂γ

x�ωξ σB(x, ξ)

]∂β−γ

x �α−ωξ σA(x, ξ + ω)

∣∣∣∣∣∣≤ 2|α+β| ∑

γ≤β

∑ω≤α

CBγ,ω 〈ξ〉l−ρ|ω|+δ|γ| CA

β−γ,α−ω 〈ξ + ω〉m−ρ|α−ω|+δ|β−γ|

≤ Cαβ 〈ξ〉m+l−ρ|α|+δ|β|. �

Curiously enough, σAσB is the principal symbol of both AB and BA, andtherefore the commutator of periodic pseudo-differential operators is smootherthan it may seem at the first look – this has nice consequences.


Theorem 4.7.10 (Toroidal composition formula). Let 0 ≤ δ < ρ ≤ 1. The compo-sition BA of two operators B ∈ Op(Sl

ρ,δ(Tn × Zn)) and A ∈ Op(Sm

ρ,δ(Tn × Zn))

is in Op(Sm+lρ,δ (Tn × Zn)), and its toroidal symbol has the following asymptotic

expansion:

σBA(x, ξ) ∼∑γ≥0

1γ!

[�γ

ξ σB(x, ξ)]D(γ)

x σA(x, ξ). (4.36)

Thereby the commutator [A,B] = AB −BA belongs to the operator class

Op(Sm+l−(ρ−δ)ρ,δ (Tn × Zn)).

Remark 4.7.11. Formula (4.36) can be compared with the Euclidean formula thatwe can get by Theorem 4.5.3 by extending toroidal symbols to Tn × Rn:

σBA(x, ξ) ∼∑γ≥0

1γ!

[∂γ

ξ σB(x, ξ)]Dγ

xσA(x, ξ) (4.37)

= exp (∂ξDx) [σB(y, ξ)σA(x, η)]|(y,η)=(x,ξ) .

Remark 4.7.12. In Remark 4.13.9 we will derive another formula for the compo-sition of two pseudo-differential operators where the resulting operator BA is inthe form of an amplitude operator with amplitude c(x, y, ξ) having the asymptoticexpansion:

c(x, z, ξ) ∼∑α≥0

1α!

σB(x, ξ) (−Dz)(α)�αξ σA(z, ξ).

Consequently, applying Theorem 4.4.5 we can obtain another representation forthe symbol σBA.

Proof of Theorem 4.7.10. Of course, by going through the procedure

σBA(x, ξ) = e−i2πx·ξ [BAeξ(x)]

= e−i2πx·ξ ∑η∈Zn

∫Tn

ei2π(x−y)·η σB(x, η) Aeξ(y) dy

=∑

η∈Zn

∫Tn

ei2π(x−y)·(η−ξ) σB(x, η) σA(y, ξ) dy

=∑

η∈Zn

ei2πx·(η−ξ) σB(x, η) σA(η − ξ, ξ)

=∑

η∈Zn

ei2πx·η σB(x, η + ξ) σA(η, ξ),

one gets the exact symbol of the composition of the periodic pseudo-differential op-erators. However, as this representation cannot be used to effectively approximateBA, it is of minor importance: this is why we need an asymptotic expansion.


Now we can use the discrete Taylor series of Theorem 3.3.21 for σB(x, η + ξ)to obtain

σBA(x, ξ) =∑

η∈Zn

ei2πx·η σB(x, η + ξ) σA(η, ξ)

=∑

η∈Zn

ei2πx·η

⎡⎣ ∑|γ|<N

1γ!�γ

ξ σB(x, ξ) η(γ) + RN (x, ξ, η)

⎤⎦ σA(η, ξ)

=∑|γ|<N

1γ!

[�γ

ξ σB(x, ξ)] ∑

η∈Zn

ei2πx·η σA(η, ξ) η(γ)

+∑

η∈Zn

ei2πx·η RN (x, ξ, η) σA(η, ξ)

=∑|γ|<N

1γ!

[�γ

ξ σB(x, ξ)]D(γ)

x σA(x, ξ) + EN (x, ξ),

where EN (x, ξ) =∑

η∈Zn ei2πx·η RN (x, ξ, η) σA(η, ξ). By Lemma 4.7.9, the firstfinite sum here is in Sm+l

ρ,δ (Tn × Zn). We only need to prove that the error termEN (x, ξ) satisfies EN ∈ Sm+l−ρN

ρ,δ (Tn × Zn), or

∣∣�αξ ∂β

xEN (x, ξ)∣∣ =

∣∣∣∣∣∣�αξ ∂β

x

∑η∈Zn

ei2πx·η RN (x, ξ, η) σA(η, ξ)

∣∣∣∣∣∣≤ cαβN 〈ξ〉m+l−ρ(N+|α|)+δ|β|.

In fact, we will show even more. Indeed, by inequality (4.12) of Lemma 4.2.1 wehave, with any r ∈ R,∣∣∣�αA

ξ ∂βAx

[ei2πx·η σA(η, ξ)

]∣∣∣ ≤ cAαA,βA,r 〈η〉|βA|−r〈ξ〉m−ρ|αA|+δr. (4.38)

The error term of the discrete Taylor series (see Theorem 3.3.21) and σB ∈Sl

ρ,δ(Tn × Zn) give∣∣∣�αR

ξ ∂βRx RN (x, ξ, η)

∣∣∣ ≤ cRβR,N 〈η〉N max

|ω|=N, ν∈Q(η)

∣∣∣�ω+αR

ξ ∂βRx σB(x, ξ + ν)

∣∣∣≤ cR

βR,N 〈η〉N maxν∈Q(η)

[CB

βR,ω,αR〈ξ + ν〉l−ρ(N+|αR|)+δ|βR|

].

Multiplying this estimate with (4.38) and taking αA + αR = α and βA + βR = β,by the discrete Leibniz formula (Lemma 3.3.6) we get∣∣�α

ξ ∂βxEN (x, ξ)

∣∣ ≤ C〈ξ〉m−ρ|αA|+δr (4.39)

×∑

η∈Zn

〈η〉N+|βA|−r maxν∈Q(η)

[〈ξ + ν〉l−ρ(N+|αR|)+δ|βR|

].


We split the sum in (4.39) into sums∑|η|≤|ξ|/2 and

∑|η|>|ξ|/2. We can estimate

〈ξ〉m−ρ|αA|+δr∑

|η|≤|ξ|/2



]≤ C〈ξ〉m−ρ|αA|+δr+l−ρ(N+|αR|)+δ|βR|+n max

|η|≤|ξ|/2〈η〉N+|βA|−r.

Taking r = N + |βA| and N large enough we can estimate it by any 〈ξ〉−N ′in view

of δ < ρ. Taking N large enough, the other sum in (4.39) can be estimated as

〈ξ〉m−ρ|αA|+δr∑

|η|>|ξ|/2



]≤ C〈ξ〉m−ρ|αA|+δr

∑|η|>|ξ|/2

〈η〉N+|βA|−r

≤ C〈ξ〉m−ρ|αA|+δr+N+|βA|−r+1

if r is large compared to N . Taking r large enough we can estimate the finalexpression by any 〈ξ〉−N ′

in view of δ < 1. Hence EN ∈ Sm+l−ρNρ,δ (Tn × Zn), so

that by Theorem 4.4.1 formula (4.36) is valid.Now the study of the commutator remains. Exploiting the first terms of the

asymptotic expansions, we see that σBA = σBσA +EBA and σAB = σAσB +EAB ,where EBA, EAB ∈ S

m+l−(ρ−δ)ρ,δ (Tn × Zn). Thus

σBA − σAB = (σBA − σBσA)− (σAB − σAσB)= EBA − EAB

∈ Sm+l−(ρ−δ)ρ,δ (Tn × Zn),

and consequently [B,A] ∈ Op(Sm+l−(ρ−δ)ρ,δ (Tn × Zn)).

The asymptotic expansion (4.37) in Remark 4.7.11 of extended symbols isobtained by applying (4.32) on (4.36). �

Remark 4.7.13. Note that one could use the extended symbols in order to be ableto apply the classical analysis (derivatives and traditional Taylor series) – then onewould use equivalence (4.32) to return to differences. But as it has been expressed,this is not a natural way of handling periodic pseudo-differential operators, as thediscrete Taylor series of Theorem 3.3.21 for σB(x, η + ξ) can be used.

For classical periodic pseudo-differential operators, formula (4.37) has beenproved in another way by Johannes Elschner in [34, p. 129-130]. Without proofsformulae (4.34), (4.36), and (4.37) have been announced in [142].


4.8 Operators on L2(Tn) and Sobolev spaces

In this section we will discuss what conditions on the toroidal symbol a guaran-tee the L2-boundedness of the corresponding pseudo-differential operator Op(a) :C∞(Tn) → D′(Tn). We also discuss Sobolev spaces. In Proposition 4.2.3 weshowed in particular that operators with toroidal symbols in the class S0

0,0(Tn×Zn)

are bounded on all Sobolev spaces. However, here we show that for the bound-edness of an operator on L2(Tn) and on Hs(Tn) it is not necessary that symbolsbelong to the toroidal symbol class S0

0,0(Tn × Zn). In fact, already for the Sobolev

boundedness in Corollary 4.2.4 we showed that if m, s ∈ R and if an integer r ∈ Nis such that r > |s−m|+ n, then for a ∈ C∞(Tn × Zn) satisfying estimates∣∣∂β


for all multi-indices |β| ≤ r, all x ∈ Tn and all ξ ∈ Zn, the operator Op(a) extendsto a bounded linear operator from Hs(Tn) to Hs−m(Tn). In particular, if we takes = m = 0, this would require r = n + 1 derivatives of a with respect to x to bebounded. We now show that this number can be improved:

Theorem 4.8.1 (Boundedness on L2(Tn)). Let k ∈ N and k > n/2. Let a : Tn ×Zn → C be such that∣∣∂β

xa(x, ξ)∣∣ ≤ C for all (x, ξ) ∈ Tn × Zn, (4.40)

and all |β| ≤ k. Then the operator Op(a) extends to a bounded linear operator onL2(Tn).

Remark 4.8.2. We note that compared with several well-known theorems on theL2-boundedness of pseudo-differential operators (see, e.g., Calderon and Vaillan-court [22], Coifman and Meyer [23], Cordes [25]), Theorem 4.8.1 does not requireany regularity with respect to the ξ-variable. In fact, the boundedness of all partialdifferences of all orders ≥ 1 with respect to ξ follows automatically from (4.40) inview of the formulae for difference operators in Proposition 3.3.4.

Proof of Theorem 4.8.1. Let us define

Ayf(x) :=∑ξ∈Zn

∫Tn

ei2π(x−z)·ξ a(y, ξ) f(z) dz,

so that Axf(x) = Op(a)f(x). Then

‖Op(a)f‖2L2(Tn) =∫

Tn

|Axf(x)|2 dx ≤∫

Tn

supy∈Tn

|Ayf(x)|2 dx,

and by an application of the Sobolev embedding theorem we get

supy∈Tn

|Ayf(x)|2 ≤ C∑|α|≤k

∫Tn

|∂αy Ayf(x)|2 dy.

4.8. Operators on L2(Tn) and Sobolev spaces 375

Therefore, using Fubini’s theorem to change the order of integration, we obtain

‖Op(a)f‖2L2(Tn) ≤ C∑|α|≤k

∫Tn

∫Tn

|∂αy Ayf(x)|2 dx dy

≤ C∑|α|≤k

supy∈Tn

∫Tn

|∂αy Ayf(x)|2 dx

= Ck

∑|α|≤k

supy∈Tn

‖∂αy Ayf‖2L2(Tn)

≤ C∑|α|≤k

supy∈Tn

supξ∈Zn

‖∂αy a(y, ξ)‖2L2(Tn)‖f‖2L2(Tn),


From a suitable adaptation of the composition Theorem 4.7.10 we immedi-ately obtain the result in Sobolev spaces:

Corollary 4.8.3 (Boundedness on Hs(Tn)). Let m ∈ R and let a : Tn × Zn → Cbe such that ∣∣∂β

xa(x, ξ)∣∣ ≤ C〈ξ〉m for all (x, ξ) ∈ Tn × Zn, (4.41)

and all multi-indices β. Then the operator Op(a) extends to a bounded linearoperator from Hs(Tn) to Hs−m(Tn), for any s ∈ R.

We refer to Remark 4.2.9 for a discussion of this corollary. In particular, wenote that Corollary 4.2.4 still yields a better result because it gives an estimateon the number of derivatives sufficient for the Sobolev boundedness. We note thatCorollary 4.8.3 can be also obtained by a functional analytic argument withoutcalculus, see Remark 10.8.3.

Now, let us briefly discuss the case of the Lp(Tn)-boundedness. First wediscuss Sobolev spaces over Lp(Tn).

Remark 4.8.4 (Pseudo-differential description of Hs(Tn)). Observe that in thepseudo-differential language the definition of Sobolev spaces in Definition 3.2.2 canbe reformulated. Indeed, since the toroidal symbol of the Laplacian L is σL(x, ξ) =−4π2|ξ|2, it is immediate that f ∈ Hs(Tn) if and only if (1 − (4π2)−1L)s/2f ∈L2(Tn). Of course, constant (4π2)−1 here is not essential.

Definition 4.8.5 (Sobolev spaces Lps(T

n)). Let 1 ≤ p ≤ ∞ and s ∈ R. We say thatf ∈ Lp

s(Tn) if (1− L)s/2f ∈ Lp(Tn).

By Theorem C.4.9 we know that Lp(Tn) is a Banach space, and by a similarproof to that of Theorem 1.5.10, (v), we can conclude that the spaces Lp

s(Tn) are

Banach spaces for s ∈ N.

Exercise 4.8.6. Prove that Lps(T

n) are Banach spaces for all s ∈ R.


In [82], Molahajloo and Wong showed that if k > n/2, k ∈ N, and if thesymbol a(x, ξ) satisfies estimates∣∣∂β

x Δαξ a(x, ξ)

∣∣ ≤ C〈ξ〉−|α| for all (x, ξ) ∈ Tn × Zn,

and all multi-indices α, β such that |α| ≤ k and |β| ≤ k, then the operator a(X, D)is bounded on Lp(Tn), 1 < p < ∞. As a consequence of a suitably adaptedTheorem 4.7.10 we obtain the boundedness on Sobolev space over Lp(Tn):

Corollary 4.8.7 (Boundedness on Sobolev spaces Lps(T

n)). Let m ∈ R, k ∈ N,k > n/2, and let a : Tn × Zn → C be such that∣∣∂β

x Δαξ a(x, ξ)

∣∣ ≤ C〈ξ〉m−|α| for all (x, ξ) ∈ Tn × Zn, (4.42)

and all multi-indices α such that |α| ≤ k and all multi-indices β. Then the operatorOp(a) extends to a bounded linear operator from Lp

s(Tn) to Lp

s−m(Tn), for any1 < p <∞ and any s ∈ R.

Exercise 4.8.8. Work out the details of the proof.

4.9 Elliptic pseudo-differential operators on Tn

Clearly, elliptic pseudo-differential operators on compact manifolds have beenstudied in great depth in view of numerous applications, so we can only touchupon the general theory of this subject. However, here we can analyse them interms of the developed toroidal quantizations. Thus, next we study elliptic oper-ators on Tn, that is operators that can be described as operators “invertible withrespect to smoothness”.

Definition 4.9.1 (Elliptic pseudo-differential operators on Tn). A toroidal symbolσA and the corresponding periodic pseudo-differential operator A = Op(σA) ∈Op(Sm(Tn × Zn)) are called elliptic of order m ∈ R, if σA satisfies

∀(x, ξ) ∈ Tn × Zn : ‖ξ‖ ≥ n0 ⇒ |σA(x, ξ)| ≥ c0 〈ξ〉m (4.43)

for some constants n0, c0 > 0.

Remark 4.9.2. Note that if A is elliptic, it cannot belong to any Op(Sl(Tn × Zn))with l < m. That is to say, elliptic periodic pseudo-differential operators havesharp minimal degrees. In the definition n0 does not play any significant role: Adiffers only by an Op(S−∞(Tn × Zn))-operator from a periodic pseudo-differentialoperator A′ that satisfies inequality (4.43) for all ξ ∈ Zn. Therefore we may assumewithout any further comment that |σA(x, ξ)| ≥ c0 〈ξ〉m holds for all ξ ∈ Zn for anelliptic operator A.Remark 4.9.3 (Elliptic =⇒ Fredholm). Elliptic periodic pseudo-differential oper-ators are Fredholm operators, as it will be shown in Theorem 4.9.17. Thereforethe Fredholm theory is a closely associated subject in further studies but it fallsoutside the scope of this treatise. Nevertheless, later we touch upon it briefly.

4.9. Elliptic pseudo-differential operators on Tn 377

In concordance with the pointwise product of symbols (Lemma 4.7.9), thetoroidal symbol τ of an elliptic operator yields an elliptic toroidal symbol 1/τ :

Lemma 4.9.4. Let σ be a toroidal symbol in Sm(Tn × Zn), and let τ ∈ S�(Tn × Zn)be an elliptic toroidal symbol such that |τ(x, ξ)| ≥ c0 〈ξ〉�. Then σ/τ ∈ Sm−�(Tn×Zn), and 1/τ is elliptic of order −�.

Proof. We begin by observing how derivatives and differences act on σ/τ . Thefirst case is familiar from calculus:

∂xj

σ(x, ξ)τ(x, ξ)

=τ(x, ξ)∂xj

σ(x, ξ)− σ(x, ξ)∂xjτ(x, ξ)

τ(x, ξ)2.

The action of the difference is quite similar:

�ξk

σ(x, ξ)τ(x, ξ)

=τ(x, ξ)�ξk

σ(x, ξ)− σ(x, ξ)�ξkτ(x, ξ)

τ(x, ξ)τ(x, ξ + vk).

Thus we are motivated to define σ0,0 := σ, τ0,0 := τ , and recursively⎧⎪⎪⎪⎨⎪⎪⎪⎩σα,β+vj

(x, ξ) = τα,β(x, ξ)∂xjσα,β(x, ξ)− σα,β(x, ξ)∂xj

τα,β(x, ξ),

τα,β+vj (x, ξ) = τα,β(x, ξ)2,σα+vk,β(x, ξ) = τα,β(x, ξ)�ξk

σα,β(x, ξ)− σα,β(x, ξ)�ξkτα,β(x, ξ),

τα+vk,β(x, ξ) = τα,β(x, ξ) τα,β(x, ξ + vk).

By Lemma 4.7.9 we notice that σα,β and τα,β are symbols of periodic pseudo-differential operators, and it holds that

�αξ ∂β

x

σ(x, ξ)τ(x, ξ)

=σα,β(x, ξ)τα,β(x, ξ)

.

Hence induction on both α and β gives{|σα,β(x, ξ)| ≤ cα,β 〈ξ〉(2

|α+β|−1)�+m−|α|,dα,β 〈ξ〉2

|α+β|� ≤ |τα,β(x, ξ)| ≤ eα,β 〈ξ〉2|α+β|�,

where cα,β , dα,β , eα,β > 0 are appropriate constants. These estimates complete theproof. �Exercise 4.9.5. Let us examine the proof of Lemma 4.9.4 once more. Show that

τα,β(x, ξ) =∏γ≤α

τ(x, ξ + γ)2|β|

.

Moreover, verify the induction in the proof of Lemma 4.9.4.

Now it is time to declare that ellipticity is equivalent to invertibility in thealgebra Op(S∞(Tn × Zn))/Op(S−∞(Tn × Zn)). For a detailed discussion and mo-tivation behind the notion of the parametrix in the Euclidean setting of Rn werefer to Section 2.6.


Theorem 4.9.6 (Elliptic ⇐⇒ ∃ Parametrix). Operator A ∈ Op(Sm(Tn × Zn)) iselliptic if and only if there exists B ∈ Op(S−m(Tn × Zn)) such that BA ∼ I ∼ AB.(We call such B a parametrix of A.)

Proof. The proof of the “if” part is similar to that of Theorem 2.6.6 but wegive another argument to prove the “only if” part. Thus, assume first that A ∈Op(Sm(Tn × Zn)) and B ∈ Op(S−m(Tn × Zn)) satisfy BA = I − T and AB =I − T ′ with T, T ′ ∈ Op(S−∞(Tn × Zn)). Then 1 − σBA = σT ∈ S−∞(Tn × Zn)and consequently by Theorem 4.7.10 we have 1− σBσA ∈ S−1(Tn × Zn), so that|1−σBσA| ≤ C〈ξ〉−1. Hence 1− |σB | · |σA| ≤ C〈ξ〉−1, or equivalently |σB | · |σA| ≥1 − C〈ξ〉−1. If we choose n0 > C, then |σB(x, ξ)| · |σA(x, ξ)| ≥ 1 − C〈n0〉−1 > 0for any ‖ξ‖ ≥ n0. Hence A is elliptic of order m. This yields the “if” part of theproof.

Now assume that A ∈ Op(Sm(Tn × Zn)) is elliptic, and define a periodicpseudo-differential operator B0 by σB0(x, ξ) := 1/σA(x, ξ), so that σB0 is inS−m(Tn × Zn) by Lemma 4.9.4, and σB0σA ∼ 1 ∈ S0(Tn × Zn). Then by (4.36)it is true that σB0A = σB0σA − σT ∼ 1 − σT for some σT ∈ S−1(Tn × Zn);equivalently B0A ∼ I − T . Notice that

N−1∑j=0

T j (I − T ) = I − TN ,

where TN ∈ S−N (Tn × Zn), so that R ∼ ∑∞j=0 T j is a formal Neumann series of

the inverse of I − T in the algebra Op(S∞(Tn × Zn))/Op(S−∞(Tn × Zn)). ThenRB0A ∼ R(I − T ) ∼ I, so that RB0 is a candidate for a parametrix B. Indeed,this technique produces also B′ ∈ Op(S−m(Tn × Zn)) satisfying AB′ ∼ I. Then

B′ ∼ (BA)B′ = B(AB′) ∼ B

completes the proof. �

From the proof above it can be immediately seen that the principal symbolof a parametrix of an elliptic periodic pseudo-differential operator A is given bythe symbol σB0 = 1/σA (provided that σA does not vanish anywhere, which wemay assume).

Remark 4.9.7. Notice that the ellipticity of A is really equivalent to the existenceof a periodic pseudo-differential operator B satisfying BA

−ε∼ I−ε∼ AB for some

ε > 0. In general, there is no restriction in defining “functions” of periodic pseudo-differential operators of negative order by

∑∞j=0 cj T j : the coefficients cj may

even grow arbitrarily fast, as it is only required that T ∈ Op(S−ε(Tn × Zn)) forsome ε > 0. For example, one may define sin and cos of these operators, yieldingsin(T )2 + cos(T )2 ∼ I etc., or determine fractional powers of elliptic operators(discussed in, e.g., [135, p. 42–44]).


Corollary 4.9.8. Assume that A ∈ Op(Sm(Tn × Zn)). Then its adjoint A∗ ∈Op(Sm(Tn × Zn)) and A is elliptic if and only if A∗ is.

Proof. The fact that A∗ ∈ Op(Sm(Tn × Zn)) was stated in Theorem 4.7.3. Now,suppose A is elliptic with parametrix B such that BA = I − T and AB = I − T ′,where T, T ′ ∈ Op(S−∞(Tn × Zn)). Then A∗ has B∗ as its parametrix, becausehere B∗A∗ = I − T ′∗ and A∗B∗ = I − T ∗, where T ∗, T ′∗ ∈ Op(S−∞(Tn × Zn)).

�Exercise 4.9.9. Prove the “transpose” version of Corollary 4.9.8. Namely, showthat A is elliptic if and only if At is.

Remark 4.9.10. For Corollary 4.9.8, we can give a direct proof that

A ∈ Op(Sm(Tn × Zn)) implies A∗ ∈ Op(Sm(Tn × Zn))

via the distributional duality by applying (4.12). The same is true for the transposeoperator At. Since we gave a more detailed proof for the ellipticity of the adjoint inCorollary 4.9.8 and left the transpose case as an exercise, we now do the oppositeand give a direct proof of At ∈ Op(Sm(Tn × Zn)). Indeed, using Theorem 4.1.4,the Fourier coefficients of the symbol of At in the first variable can be found as

σAt(η, ξ) = 〈x �→ σAt(x, ξ), e−η〉=

⟨e−ξA

teξ, e−η

⟩=

⟨Ateξ, e−η−ξ

⟩= 〈eξ, Ae−η−ξ〉= 〈e−η, eη+ξAe−η−ξ〉= 〈e−η, x �→ σA(x,−η − ξ)〉= σA(η,−η − ξ).

Using this and Peetre’s inequality (Proposition 3.3.31) we show that

At ∈ Op(Sm(Tn × Zn)) :

∣∣�αξ ∂β

xσAt(x, ξ)∣∣ =

∣∣∣∣∣∣�αξ ∂β

x

∑η∈Zn

ei2πx·η σA(η,−η − ξ)

∣∣∣∣∣∣(4.12)

≤∑

η∈Zn

cr,α 〈η〉|β| 〈η〉−r 〈ξ + η〉m−|α|

Peetre≤ 2|m−|α|| cr,α,β 〈ξ〉m−|α|

∑η∈Zn

〈η〉|m−|α||+|β|−r

≤ Cα,β 〈ξ〉m−|α|,

if we take r large enough. �


Exercise 4.9.11. Give a similar direct proof (without using asymptotic expansions)of the fact that A ∈ Op(Sm(Tn × Zn)) implies that A∗ ∈ Op(Sm(Tn × Zn)).

Remark 4.9.12 (∗-algebra of periodic pseudo-differential operators). Let us collectthe obtained results concerning the ∗-algebra

Op(S∞(Tn × Zn))/Op(S−∞(Tn × Zn)).

Here the classes are [A] = A + Op(S−∞(Tn)). Now [A] + [B] = [A + B], [A][B] =[AB], and [A]∗ = [A∗]. If A is elliptic with a parametrix C, it makes sense towrite [A]−1 = [C]. The zero and identity elements are [0] = Op(S−∞(Tn × Zn))and [I], respectively. One could then define the action of [A] in [u] := u+C∞(Tn)(u ∈ Hs(Tn)) by [A][u] := [Au]. We do not develop this line of thought any further.

The proof of Theorem 4.9.6 did not use the asymptotic expansion of theproduct of periodic pseudo-differential operators. But applying the compositionformula (4.36), we can obtain explicitly an asymptotic expansion of the parametrix:

Theorem 4.9.13 (Formula for the parametrix). Assume that periodic pseudo-dif-ferential operators A ∈ Op(Sm(Tn × Zn)) and B ∈ Op(S−m(Tn × Zn)) are para-metrices to each other, that is AB ∼ I ∼ BA. Let A ∼ ∑∞

j=0 Aj be an asymp-totic expansion, where Aj ∈ Op(Sm−j(Tn × Zn)). Then an asymptotic expansionB ∼∑∞

k=0 Bk is obtained by σB0 = 1/σA0 , and then recursively

σBN(x, ξ) =

−1σA0(x, ξ)

N−1∑k=0

N−k∑j=0

∑|γ|=N−j−k

1γ!

[�γ

ξ σBk(x, ξ)

]D(γ)

x σAj(x, ξ).

Proof. Now I ∼ BA, so that by (4.36)

1 ∼ σBA(x, ξ)

∼∑γ≥0

1γ!

[�γ

ξ σB(x, ξ)]D(γ)

x σA(x, ξ)

∼∑γ≥0

1γ!

[�γ

ξ

∞∑k=0

σBk(x, ξ)

]D(γ)

x

∞∑j=0

σAj(x, ξ),

where we want to solve σBk. Notice that A0 is elliptic if and only if A is elliptic

(and now A is elliptic by Theorem 4.9.6). Moreover, without a loss of generalitywe may assume that σA0 does not vanish anywhere. Obviously, we can demandthat 1 = σB0(x, ξ) σA0(x, ξ), and that

0 =∑

j+k+|γ|=N

1γ!

[�γ

ξ σBk(x, ξ)

]D(γ)

x σAj(x, ξ).

Then the trivial solution of these equations is the recursion of the theorem. Thereader may verify that σBN

∈ S−m−N (Tn × Zn). Thus B ∼∑∞k=0 Bk. �


Exercise 4.9.14. Show that σBN∈ S−m−N (Tn × Zn) in Theorem 4.9.13.

Of course, when the extended symbols are used, one directly obtains anotherasymptotic expansion for parametrices like in the cases of adjoints and products.

In the theorem above the expansions for A and B may sometimes containonly a finite number of terms. The simplest example demonstrating this is the casewhen A = I = B: the expansions are simply A0 = I = B0.

We now show that the extension of symbols from Theorem 4.5.3 respectsellipticity:

Theorem 4.9.15 (Extensions and ellipticity). Let the toroidal and Euclidean sym-bols a ∈ Sm(Tn × Zn) and a ∈ Sm(Tn × Rn) be such that a = a|Tn×Zn . Then a iselliptic if and only if a is elliptic.

Proof. The “if” part is straightforward because it is simply the restriction of thecondition

∀(x, ξ) ∈ Tn × Rn : ‖ξ‖ ≥ n0 ⇒ |σA(x, ξ)| ≥ c0 〈ξ〉m

to Tn × Zn. Conversely, assume that a is elliptic. For ξ ∈ Rn let η ∈ Zn be (oneof the) closest integers to ξ. We can assume that ξ and η are sufficiently large.Taking the Taylor expansion of a at η we get

a(x, ξ) = a(x, η) +∑|α|=1

(η − ξ)α

∫ t

0

(1− t)∂αξ a(x, ξ + t(ξ − η)) dt.

Now, we have |a(x, η)| = |a(x, η)| ≥ C〈η〉m ≥ C〈ξ〉m since a is elliptic. On theother hand,∣∣∣∣∣∣

∑|α|=1

(η − ξ)α

∫ t

0

(1− t)∂αξ a(x, ξ + t(ξ − η)) dt

∣∣∣∣∣∣ ≤ C〈ξ〉m−1,

implying that a is elliptic. �

Finally, we discuss the Fredholmness of elliptic periodic pseudo-differentialoperators.

Definition 4.9.16 (Fredholm operators and index). Let X and Y be Banach spaces.A continuous linear operator A ∈ L(X, Y ) is a Fredholm operator if its kernelKer(A) = {x ∈ X : Ax = 0} is finite-dimensional and the range (or image)Im(A) = {Ax : x ∈ X} is closed and of finite codimension. Then its index is

Ind(A) := dim Ker(A)− codim Im(A),

or equivalentlyInd(A) := dim Ker(A)− dim Ker(A∗),

where A∗ is the (Banach or Hilbert) adjoint of A. Fredholm operators can begeneralised to locally convex topological vector spaces, see, e.g., [135].


Theorem 4.9.17 (Elliptic =⇒ Fredholm). Let A ∈ Op(Sm(Tn × Zn)) be elliptic.Then it is a Fredholm operator A ∈ L(Hs(Tn),Hs−m(Tn)) for every s ∈ R. More-over, the kernels Ker(A),Ker(A∗) are in C∞(Tn), so that Ker(A), Ker(A∗), andInd(A) are independent of s.

Proof. Let B ∈ Op(S−m(Tn × Zn)) be a parametrix of A such that BA = I −T and AB = I − T ′, T, T ′ ∈ Op(S−∞(Tn × Zn)), and let Kers(A) := {u ∈Hs(Tn)|Au = 0}. Recall that C∞(Tn) =

⋂t∈R Ht(Tn) by Corollary 3.2.12. Now,

T and T ′ are compact (on the basis of Corollary 4.3.3), and

Kers(A) ⊂ Kers(BA) = Kers(I − T ) ⊂ C∞(Tn),A Hs(Tn) ⊃ AB Hs−m(Tn) = (I − T ′)Hs−m(Tn).

By the Fredholm theory (see, e.g., [55, Chapter XIX] or [69, Chapter 4]), I−T andI−T ′ are Fredholm operators. Hence A Hs(Tn) is closed and of finite codimension,and Kers(A) ⊂ C∞(Tn) is finite-dimensional. Similarly Kerm−s(A∗) ⊂ C∞(Tn) isfinite dimensional. Since these kernels are in C∞(Tn), they do not depend on s,and thus the Fredholm index is independent of s. �

Proposition 4.9.18 (All indices are attainable). For every j ∈ Z there are periodicpseudo-differential operators of index j.

Proof. For instance, consider A1 ∈ L(L2(T1)) defined by

A1u(x) := [P+ + ei2πxP−]u(x),

where P+u(x) :=∑

ξ≥0 u(ξ), and P− := I − P+. Clearly A1 is an elliptic peri-odic pseudo-differential operator with one-dimensional kernel, and it maps everyHs(T1) onto itself, being thus a Fredholm operator with Ind(A1) = 1. GenerallyInd(AB) = Ind(A) + Ind(B) and Ind(A∗) = −Ind(A), so that Ind(Aj

1) = j andInd(A∗1

j) = −j for every j ∈ N0. �

4.10 Smoothness properties

In Section 3.2 a close connection of Sobolev spaces and differentiability of distribu-tions was put forward. This section handles these smoothness properties and theirrelation to the continuity properties of periodic pseudo-differential operators.

As it is well known, linear differential operators are local: such an operator Adecreases the support of any distribution u: supp(Au) ⊂ supp(u) (see Definition2.3.3 and exercises thereafter), i.e., for the calculation of Au(x) only the infor-mation of the behavior of u in an arbitrary small neighbourhood of x is needed.Periodic pseudo-differential operators do not satisfy this in general, but there is areasonable version of this property for them, called pseudolocality.

4.10. Smoothness properties 383

Definition 4.10.1 (Pseudolocality and singular support). The singular supportsing supp(u) ⊂ Tn of a distribution u ∈ D′(Tn) is the complement of the maximalopen set where u is C∞-smooth. An operator A is called pseudolocal, if

sing supp(Au) ⊂ sing supp(u)

for every distribution u in the domain of A. This is to say that pseudolocalitymeans locality with respect to smoothness.

These definitions give rise to further smoothness characterisations:

Definition 4.10.2 (Ht-smoothness). Let u ∈ D′(Tn). Let U ⊂ Tn be an openset. We say that u is Ht(Tn)-smooth in U , denoted u ∈ Ht(U), if ψ(x)u(x) ∈Ht(Tn) for every ψ ∈ C∞(Tn) supported in U . The Ht-singular support of u,sing suppt(u), is the set Tn \ U , where U is the maximal open set for whichu ∈ Ht(U). We say that u ∈ D′(Tn) is locally Ht-smooth at x ∈ Tn, denotedu ∈ Ht(x), if u ∈ Ht(U) for some open set U � x.

Exercise 4.10.3. How is C∞(U) related to the spaces Ht(U)?

Exercise 4.10.4. How is sing supp(u) related to⋂

t∈R Ht(x)?

Theorem 4.10.5 (Sobolev pseudolocality). Let 0 ≤ δ < ρ ≤ 1. Let

A ∈ Op(Smρ,δ(T

n × Zn)), and u ∈ Hs(Tn).

Then u ∈ Ht(x) only if Au ∈ Ht−m(x), so that

sing suppt−m(Au) ⊂ sing suppt(u).

Proof. (This approach was proposed by Gennadi Vainikko [143].) Assume thatu ∈ Ht(U), where U is an neighbourhood of x, and let ψ ∈ C∞(Tn) be such thatsupp(ψ) ⊂ U . We notice that ((x, ξ) �→ ψ(x)) ∈ S0(Tn × Zn) and supp(ψk) ⊂ Ufor all k ∈ N0. Let us define A0 := A, u0 := 0, and recursively{

Ak+1 := [ψI, Ak],uk+1 := Ak(ψ · u) + ψ · uk.

By induction on k one proves that ψk · Au = Aku + uk, where uk ∈ Ht−m(Tn)and Ak ∈ Op(Sm−k(ρ−δ)

ρ,δ (Tn × Zn)) (by Theorems 4.2.3 and 4.7.10). Hence we getψk ·Au ∈ Ht−m(U), so that Au ∈ Ht−m(x). �

Exercise 4.10.6. Verify the induction in the proof of Theorem 4.10.5.

Immediately we can state:

Corollary 4.10.7. For 0 ≤ δ < ρ ≤ 1, periodic pseudo-differential operators arepseudolocal.


Remark 4.10.8. We can present an alternative proof for the pseudolocality ofperiodic pseudo-differential operators. Indeed, let KA be the Schwartz kernel ofa periodic pseudo-differential operator A: the approach involves an application ofsmoothness of the kernel outside the diagonal x = y from Theorem 4.3.6. Assumethat u ∈ D′(Tn) such that u|U ∈ C∞(U) for some open set U ⊂ Tn. Let (Vj)∞j=1

be a sequence of open sets such that

Vj ⊂ Vj+1 ⊂ U =∞⋃

j=1

Vj ,

and let ψj ∈ C∞(Tn) be associated functions satisfying supp(ψj) ⊂ U and ψj |Vj=

1. Then

Au(x) = A [ψj · u + (1− ψj) · u] (x)

= A(ψj · u)(x) +∫

Tn

u(y) (1− ψj(y)) KA(x, y) dy.

As a periodic pseudo-differential operator, A maps ψj ·u ∈ C∞(Tn) into C∞(Tn).Now (1 − ψj(y)) KA(x, y) vanishes when x = y ∈ Vj , so that according to The-orem 4.3.6 it is a C∞-smooth kernel when x ∈ Vj . Hence Theorem 4.3.1 impliesthat Au|Vj

∈ C∞(Vj), and thereby Au|U ∈ C∞(U), as U =⋃∞

j=1 Vj . �A general periodic pseudo-differential operator can locally (and even globally)

smoothen a distribution more than its order indicates. This is not the case withelliptic operators, which faithfully transform the information encoded in smooth-ness:

Proposition 4.10.9 (Sobolev hypoellipticity). Let A ∈ Op(Sm(Tn × Zn)) be anelliptic periodic pseudo-differential operator, and let u ∈ D′(Tn). Then

sing suppt−m(Au) = sing suppt(u)

for every t ∈ R, and sing supp(Au) = sing supp(u).

Proof. Let B be a parametrix of A, which exists by Theorem 4.9.6, so that BA =I − T . Because T ∈ Op(S−∞(Tn × Zn)), we get by Theorem 4.10.5,

sing suppt(u) = sing suppt(u− Tu)= sing suppt(BAu)⊂ sing suppt−m(Au)⊂ sing suppt(u),

implying sing suppt−m(Au) = sing suppt(u).Similarly one proves the other equality sing supp(Au) = sing supp(u). Al-

ternatively, one can use the equality sing supp(u) =⋃

t sing suppt(u) to completethe proof. �

4.10. Smoothness properties 385

Definition 4.10.10 (Hypoellipticity). Those periodic pseudo-differential operatorsA which satisfy

sing supp(Au) = sing supp(u)

for every u ∈ D′(Tn) are called hypoelliptic. For further information on theseoperators, we refer to [130] ([135] has a slightly different concept of hypoellipticity).

Recall that by Theorem 3.5.1 a periodic pseudo-differential operator mappingsome Hs(Tn) into Hs−l(Tn) is continuous between these two spaces regardless ofits order. Actually, such continuity properties are more widespread.

Theorem 4.10.11. Let 0 ≤ δ < ρ ≤ 1 and s ∈ R. Let σA ∈ Smρ,δ(T

n × Zn) andassume that A Ht(Tn) ⊂ Ht−l(Tn) for some t, l ∈ R where l < m. Then

∀ε > 0 : A ∈ L(Hs(Tn),Hs−l−ε(Tn)).

Furthermore, if m > l ≥ m− (ρ− δ), we can take ε = 0 above.

Proof. Notice that the requirement l < m is not really restricting, since by Theo-rem 4.2.3 we already know that A ∈ L(Hq(Tn),Hq−m(Tn)) for every q ∈ R. Fixε > 0 and assume for clarity that s < t (the case s > t is totally symmetric).Then, by choosing q < s small enough, the interpolation theorems

L(Ht1 ,Ht2) ∩ L(Hq1 ,Hq2) ⊂ L([Ht1 ,Hq1 ]θ, [Ht2 ,Hq2 ]θ),[Htj ,Hqj ]θ = Hθtj+(1−θ)qj

(here 0 < θ < 1; see [72, Theorems 5.1 and 7.7]) imply that

A ∈ L(Hs(Tn),Hs−l−ε(Tn)).

Now suppose l ≥ m− (ρ− δ). With the aid of the canonical Sobolev space isomor-phisms ϕγ and Theorem 4.7.10, we get ϕs−tAϕt−s−A ∈ Op(Sm−(ρ−δ)

ρ,δ (Tn × Zn)).On the other hand,

ϕs−tAϕt−s Hs(Tn) = ϕs−tA Ht(Tn)⊂ ϕs−t Ht−l(Tn)= Hs−l(Tn).


The interpolation theorems [72, Theorems 5.1 and 7.7] of the preceding proofenhanced with norm estimates are significant also in the proofs of [142, Lemma4.3] and [62, Lemma 4.1], which are important results in the analysis of periodicintegral operators.

Finally, we study the connection of orders and continuity in the elementarycases when a periodic pseudo-differential operator is either a multiplier or a multi-plication. The next theorem, Abel–Dini, dwells in the theory of series. We presentonly the proof of the divergence part, which Niels Henrik Abel solved in the 1820s.


Theorem 4.10.12 (Abel–Dini). Let dj be positive numbers and let DN :=∑N

j=1 dj.Assume that (DN )∞N=1 is divergent. Then

∑∞j=1dj/Dr

j diverges exactly when r ≤ 1.

Proof (Abel part). (The whole proof is in [66, p. 290-291].) We assume that r ≤ 1.Since (DN )∞N=1 diverges, it is true that for every i ∈ Z+ there exists ki ∈ Z+ suchthat Di/Di+ki

≤ 1/2. Hence

i+ki∑j=i+1

dj

Drj

≥i+ki∑

j=i+1

dj

Dj≥ 1

Di+ki

i+ki∑j=i+1

dj = 1− Di

Di+ki

≥ 12.

Due to this,∑∞

j=1 dj/Drj diverges. �

If we say that a sequence converges to infinity, pj →∞, it is meant that forevery C <∞ there exists jC ∈ Z+ such that pj > C if j > jC .

Corollary 4.10.13. If (pj)j∈Z+ is a monotone sequence of positive real numbers con-verging to infinity, then there is a convergent series

∑∞j=1 cj such that

∑∞j=1 pj cj

diverges.

Proof. (A modification of [66, p. 302].) Define{d1 := p1,

dj+1 := pj+1 − pj .

Then, in the notation of the Abel–Dini theorem, DN =∑N

j=1 dj = pN →∞, and∑∞j=1 dj/Dj = 1 +

∑∞j=1(pj+1 − pj)/pj+1 diverges. Let us define

cj := (pj+1 − pj)/(pj+1 pj).

Then∑∞

j=1 cj converges, because 1/pj → 0:

∞∑j=1

cj =∞∑

j=1

(1pj− 1

pj+1

)=

1p1

.

Clearly,∑∞

j=1 pjcj =∑∞

j=1(pj+1 − pj)/pj+1 diverges. �

We apply this to obtain the following result concerning multipliers:

Proposition 4.10.14 (Sobolev unboundedness of multipliers).Assume that σA(x, ξ) = k(ξ), where for every C < ∞ there exists ξ ∈ Zn suchthat |k(ξ)| > C〈ξ〉l. Then A Hs(Tn) ⊂ Hs−l(Tn) for any s ∈ R.

Proof. Now there is a subsequence of (〈ξ〉−2l|k(ξ)|2)ξ∈Zn that converges to ∞ as‖ξ‖ → ∞. Corollary 4.10.13 then provides the existence of a sequence (u(ξ))ξ∈Zn

for which∑

ξ∈Zn〈ξ〉2s |u(ξ)|2 converges, but for which∑

ξ∈Zn〈ξ〉2(s−l)|k(ξ) u(ξ)|2diverges. Thus u ∈ Hs(Tn), and it is mapped to Au ∈ Hs−l(Tn). �

4.11. An application to periodic integral operators 387

Example. Proposition 4.2.3 showed that the order of an operator determines itsboundedness properties on Sobolev spaces. The converse is not true. Indeed, thereis no straightforward way of concluding the order of a symbol from observationsabout between which spaces the mapping acts. A simple demonstration of thiskind of phenomenon is σ(x, ξ) := sin(ln(|ξ|)2) (when |ξ| ≥ 1, ξ ∈ R1; the definitionof σ for |ξ| < 1 is not interesting). This symbol is independent of x, and it isbounded, resulting in that Op(σ) maps Hs(T1) into itself for every s ∈ R. On theother hand, σ defines a periodic pseudo-differential operator of degree ε for anyε > 0, as it is easily verified – however, σ ∈ S0(T1), because

∂ξσ(x, ξ) = 2ln(|ξ|)

ξcos(ln(|ξ|)2), (|ξ| > 1),

which certainly is not in O((1 + |ξ|)−1).

The case of pure multiplications can be more easily and thoroughly handled:

Proposition 4.10.15. Any Sobolev space Hs(Tn) is the intersection of the localSobolev spaces of the same order, i.e., Hs(Tn) =

⋂x∈Tn Hs(x) for every s ∈ R.

Moreover, if ϕ ∈ C∞(Tn) such that ϕ(x) = 0, then ϕ defines an automorphism ofHs(x) by multiplication.

Proof. By Theorem 4.2.3, v ∈ Hs(Tn) implies ψv ∈ Hs(Tn) for any ψ ∈ C∞(Tn).Then assume that v ∈ Hs(x) for every x ∈ Tn, so that there exist neighbourhoodsUx of points x where v ∈ Hs(Ux). Since Tn is compact, there is a finite subcoverU = {Ux(j)}N

j=1. Since there exists a smooth partition of unity subordinate to U(see Corollary A.12.15 for a continuous partition, and then make it smooth, e.g.,by mollification), and U is finite, it is true that v ∈ Hs(Tn) – the first claim isproved.

Let us then show that u �→ ϕu defines an automorphism. As above, ϕu ∈Hs(x). By the continuity of ϕ on Tn there exists a neighbourhood U of x suchthat ϕ(y) = 0 whenever y ∈ U , and furthermore U can be chosen so small thatu ∈ Hs(U). Then take such ψ ∈ C∞(Tn) that ψ|U = 1/ϕ. Since ψϕu|U = u|U ,and the result is obtained. �

4.11 An application to periodic integral operators

As an example of the symbolic analysis techniques, here we study periodic integraloperators. Let A be a linear operator defined on C∞(Tn) by

Au(x) :=∫

Tn

a(x, y) k(x− y) u(y) dy, (4.44)

where a is a C∞-smooth biperiodic function, and k is a 1-periodic distribution.Note that when a is a function of a single variable, A is simply a convolution


operator composed with multiplication: either Au(x) = f(x)∫

Tn k(x− y) u(y) dyif a(x, y) = f(x), or Au(x) =

∫Tn g(y) k(x− y) u(y) dy if a(x, y) = g(y).

We are going to show that whenever A of the type (4.44) is a periodic pseudo-differential operator, it is really something like a convolution operator with mul-tiplication, or on the Fourier side, almost a multiplier:

Theorem 4.11.1. Let ρ > 0. The operator A defined by (4.44) is a periodic pseudo-differential operator of order m if and only if the Fourier coefficients of the distri-bution k satisfy

∀α ∈ Nn0 ∃Cα ∈ R ∀ξ ∈ Zn :

∣∣�αξ k(ξ)

∣∣ ≤ Cα〈ξ〉m−ρ|α|. (4.45)

In this case A ∈ Op(Smρ,0(T

n × Zn)) and the symbol of A has the following asymp-totic expansion:

σA(x, ξ) ∼∑|γ|≥0

1γ!�γ

ξ k(ξ) D(γ)y a(x, y)|y=x.

Proof. An amplitude of A is right in front of our eyes:

Au(x) =∫

Tn

u(y) a(x, y) k(x− y) dy

=∫

Tn

u(y) a(x, y)∑ξ∈Zn

k(ξ) ei2π(x−y)·ξ dy

= Op(a)u(x),

where a(x, y, ξ) = a(x, y)k(ξ). Certainly k satisfies the estimates (4.45) if and onlyif a ∈ Am

ρ,0(Tn). Accordingly, a yields the asymptotic expansion in view of (4.18)

in Theorem 4.4.5. �Remark 4.11.2. By Theorem 4.11.1 it is readily seen that a periodic pseudo-differential operator A of the periodic integral operator form (4.44), that is

Au(x) =∫

Tn

a(x, y) k(x− y) u(y) dy, (4.46)

is elliptic if and only if k(ξ) is an elliptic symbol and a(x, x) = 0 for all x ∈ Tn.Consequently in this case by Theorem 4.9.17 it is a Fredholm operator. The indexis invariant under compact perturbations (see [55, Corollary 19.1.8], or [135, p.99]), so that we can add to A any periodic pseudo-differential operator of strictlylower degree and still get an operator with the same index.

Exercise 4.11.3. Let A in (4.46) be elliptic. Show that index Ind(A) = 0.

Theorem 4.11.1 implies that the principal symbol of the periodic integraloperator in (4.44) viewed as a periodic pseudo-differential operator is a(x, x)k(ξ).By combining Propositions 4.10.14 and 4.10.15 with this observation, we obtainanother application to periodic integral operators:

4.12. Toroidal wave front sets 389

Proposition 4.11.4. If a periodic pseudo-differential operator A is of the periodicintegral operator form (4.44)

Au(x) =∫

Tn

u(y) a(x, y) k(x− y) dy,

where a(x, x) = 0 for all x ∈ Tn and

∀C ∈ R ∃ξ ∈ Zn : |k(ξ)| > C〈ξ〉l,

then A Hs(Tn) ⊂ Hs−l(Tn) for all s ∈ R.

Remark 4.11.5. In [142] and [102], it is shown that any classical periodic pseudo-differential operator can be expressed as a sum of periodic integral operators of thetype (4.44), see Remark 4.4.7. Other contributions to periodic integral operatorsand classical operators are made, e.g., in [34], [62], [142], and [102].

4.12 Toroidal wave front sets

Here we shall briefly study microlocal analysis not on the cotangent bundle of thetorus but on Tn×Zn, which is better suited for the Fourier series representations.Let us define mappings

πRn : Rn \ {0} → Sn−1, πRn(ξ) := ξ/‖ξ‖,πTn×Rn : Tn × (Rn \ {0})→ Tn × Sn−1, πTn×Rn(x, ξ) := (x, ξ/‖ξ‖).

We set πZn = πRn |Zn : Zn\{0} → Sn−1.

Definition 4.12.1 (Discrete cones). We say that K ⊂ Rn \ {0} is a cone in Rn ifξ ∈ K and λ > 0 imply λξ ∈ K. We say that Γ ⊂ Zn \ {0} is a discrete cone ifΓ = Zn ∩K for some cone K in Rn; moreover, if this K is open then Γ is calledan open discrete cone. The set S := πRn(Zn\{0}) is the set of points with rationaldirections on the unit sphere.

Proposition 4.12.2. Γ ⊂ Zn \ {0} is a discrete cone if and only if Γ = Zn ∩π−1

Rn (πRn(Γ)).

Proof. We must show that if K is a cone in Rn then

Zn ∩K = π−1Zn πZn(Zn ∩K).

The inclusion “⊂” is obvious. Let us show the inclusion “⊃”. Let ξ ∈ π−1Zn πZn(Zn∩

K). Then ξ ∈ Zn so we need to show that ξ ∈ K. It follows that πZn(ξ) ∈πZn(Zn ∩K) = S ∩ πRn(K), which implies ξ ∈ π−1

Zn (S ∩ πRn(K)) ⊂ K, completingthe proof. �


Definition 4.12.3 (Toroidal wave front sets). Let u ∈ D′(Tn). The toroidal wavefront set WFT (u) ⊂ Tn × (Zn \ {0}) is defined as follows: we say that (x0, ξ0) ∈Tn× (Zn \ {0}) does not belong to WFT (u) if and only if there exist χ ∈ C∞(Tn)and an open discrete cone Γ ⊂ Zn \ {0} such that χ(x0) = 0, ξ0 ∈ Γ and

∀N > 0 ∃CN <∞ ∀ξ ∈ Γ : |FTn(χu)(ξ)| ≤ CN 〈ξ〉−N ;

in such a case we say that FTn(χu) decays rapidly in Γ.We say that a pseudo-differential operator A ∈ Ψm(Tn×Zn) = OpSm(Tn×

Zn) is elliptic at the point (x0, ξ0) ∈ Tn × (Zn \ {0}) if its toroidal symbol σA :Tn × Zn → C satisfies

|σA(x0, ξ)| ≥ C 〈ξ〉m

for some constant C > 0 as ‖ξ‖ → ∞, where ξ ∈ Γ and Γ ⊂ Zn \ {0} is an opendiscrete cone containing ξ0. Should ξ �→ σA(x0, ξ) be rapidly decaying in an opendiscrete cone containing ξ0 then A is said to be smoothing at (x0, ξ0). The toroidalcharacteristic set of A ∈ Ψm(Tn × Zn) is

charT (A) := {(x0, ξ0) ∈ Tn × (Zn \ {0}) : A is not elliptic at (x0, ξ0)},and the toroidal wave front set of A is

WFT (A) := {(x0, ξ0) ∈ Tn × (Zn \ {0}) : A is not smoothing at (x0, ξ0)}.Proposition 4.12.4. We have

WFT (A) ∪ charT (A) = Tn × (Zn \ {0}).Proof. The statement follows because if (x, ξ) ∈ charT (A), it means that A iselliptic at (x, ξ), and hence not smoothing. �Exercise 4.12.5. Show that WFT (A) = ∅ if and only if A is smoothing, i.e., mapsD′(Tn) to C∞(Tn) (equivalently, the Schwartz kernel is smooth by Theorem 4.3.6).

Proposition 4.12.6. Let A,B ∈ OpSm(Tn × Zn). Then

WFT (AB) ⊂WFT (A) ∩WFT (B).

Proof. By Theorem 4.7.10 applied to pseudo-differential operators A and B wenotice that the toroidal symbol of AB ∈ OpS2m(Tn × Zn) has an asymptoticexpansion

σAB(x, ξ) ∼∑α≥0

1α!

(�α

ξ σA(x, ξ))

D(α)x σB(x, ξ)

∼∑α≥0

1α!

(∂α

ξ σA(x, ξ))

∂αx σB(x, ξ),

where in the latter expansion we have used smooth extensions of toroidal symbols.This expansion says that AB is smoothing at (x0, ξ0) if A or B is smoothing at(x0, ξ0). �

4.12. Toroidal wave front sets 391

The notion of the toroidal wave front set is compatible with the action ofpseudo-differential operators:

Proposition 4.12.7 (Transformation of toroidal wave fronts). Let u ∈ D′(Tn) andA ∈ OpSm

ρ,δ(Tn × Zn), where 0 ≤ ρ ≤ 1, 0 ≤ δ < 1. Then

WFT (Au) ⊂WFT (u).

Especially, if ϕ ∈ C∞(Tn) does not vanish, then WFT (ϕu) = WFT (u).

Proof. Let FTnu decay rapidly in an open discrete cone Γ ⊂ Zn. Let us estimate

FTn(Au)(η) =∑ξ∈Zn

σA(η − ξ, ξ) FTnu(ξ),

where σA(η, ξ) =∫

Tn e−i2πx·η σA(x, ξ) dx. Integration by parts yields

|σA(η, ξ)| ≤ CM 〈η〉−M 〈ξ〉m+δM ,

because σA ∈ Smρ,δ(T

n × Zn). Due to the rapid decay of FTnu on Γ, we get∑ξ∈Γ

|σA(η − ξ, ξ)| |FTnu(ξ)| ≤ CM,N

∑ξ∈Γ

〈η − ξ〉−M 〈ξ〉m+δM 〈ξ〉−N

≤ 2MCM,N

∑ξ∈Γ

〈η〉−M 〈ξ〉m+(1+δ)M−N

≤ CM 〈η〉−M ,

where we used Peetre’s inequality and chose N large enough. Next, take an opendiscrete cone Γ1 ⊂ Γ such that η ∈ Γ1 and that 〈ω− ξ〉 ≥ C1 max{〈ω〉, 〈ξ〉} for allω ∈ Γ1 and ξ ∈ Zn \Γ (where C1 is a constant). Then 〈ω− ξ〉 ≥ C1〈ω〉1/k〈ξ〉1−1/k

for all k ∈ N. Notice that |FTnu(ξ)| ≤ CN 〈ξ〉N for some positive N . Thereby∑ξ∈Zn\Γ

|σA(η − ξ, ξ)| |FTnu(ξ)|

≤ C∑

ξ∈Zn\Γ〈η − ξ〉−M 〈ξ〉m+δM 〈ξ〉N

≤ C ′∑

ξ∈Zn\Γ〈η〉−M/k 〈ξ〉m+(δ−(k−1)/k)M+N

≤ CM 〈η〉−M/k,

where we chose (k − 1)/k > δ and then M large enough. Thus FTn(Au) decaysrapidly in Γ1. �

We will not pursue the complete analysis of toroidal wave front sets muchfurther because most of their properties can be obtained from the known propertiesof the usual wave front sets and the following relation, where WF(u) stands forthe usual Hormander’s wave front set of a distribution u.


Theorem 4.12.8 (Characterisation of toroidal wave front sets). Let u ∈ D′(Tn).Then

WFT (u) = (Tn × Zn) ∩WF(u).

Proof. Without loss of generality, let u ∈ Ck(Tn) for some large k, and let χ ∈C∞0 (Rn) such that supp(χ) ⊂ (0, 1)n. If FRn(χu) decays rapidly in an open coneK ⊂ Rn then FTn(P(χu)) = FRn(χu)|Zn decays rapidly in the open discrete coneZn ∩K. Hence WFT (u) ⊂ (Tn × Zn) ∩WF(u).

Next, we need to show that

(Tn × Zn) \WFT (u) ⊂ (Tn × Zn) \WF(u).

Let (x0, ξ0) ∈ (Tn × Zn) \WFT (u) (where ξ0 = 0). We must show that (x0, ξ0) ∈WF(u). There exist χ ∈ C∞(Tn) (we may assume that supp(χ) ⊂ (0, 1)n as above)and an open cone K ⊂ Rn such that χ(x0) = 0, ξ0 ∈ Zn∩K and that FTn(P(χu))decays rapidly in Zn ∩K.

Let K1 ⊂ Rn be an open cone such that ξ0 ∈ K1 ⊂ K and that the closureK1 ⊂ K ∪ {0}. Take any function w ∈ C∞(Sn−1) such that

w(ω) =

{1, if ω ∈ Sn−1 ∩K1,

0, if ω ∈ Sn−1 \K.

Let a ∈ C∞(Rn × Rn) be independent of x and such that a(x, ξ) = w(ξ/‖ξ‖)whenever ‖ξ‖ ≥ 1. Then a ∈ S0(Rn × Rn). Let a = a|Tn×Zn , so that a ∈ S0(Tn ×Zn) by Theorem 4.5.3.

By Corollary 4.6.21, we have

P(χ a(X, D)f) = P(χ) a(X, D)(Pf) + P(Rf)

for all Schwartz test functions f , for a smoothing operator R : D′(Rn) → S(Rn).By Remark 4.6.22 we also have

P(χ a(X, D)(χu)) = P(χ) a(X, D)(P(χu)) + P(R(χu)),

where the right-hand side belongs to C∞(Tn), since its Fourier coefficients decayrapidly on the whole Zn. Therefore also P(χ a(X, D)(χu)) belongs to C∞(Tn).

Thus χ a(X, D)(χu) ∈ C∞0 (Rn). Let ξ ∈ K1 such that ‖ξ‖ ≥ 1. Then wehave

FRn(a(X, D)(χu))(ξ) = w(ξ/‖ξ‖) FRn(χu)(ξ) = FRn(χu)(ξ).

Thus FRn(χu) decays rapidly on K1. Therefore (x0, ξ0) does not belong to WF(u).�

Exercise 4.12.9. Show that for every u ∈ D′(Tn) we have

WFT (u) =⋂

A∈Ψ0,Au∈C∞charT (A).

4.13. Fourier series operators 393

4.13 Fourier series operators

In this section we consider analogues of Fourier integral operators on the torus Tn.We will call such operators Fourier series operators and study their compositionformulae with pseudo-differential operators on the torus.

Definition 4.13.1 (Fourier series operators). Fourier series operators (FSO) areoperators of the form

Tu(x) :=∑ξ∈Zn

∫Tn

e2πi(φ(x,ξ)−y·ξ) a(x, y, ξ) u(y) dy, (4.47)

where a ∈ C∞(Tn×Tn×Zn) is a toroidal amplitude and φ is a real-valued phasefunction such that conditions of the following Remark 4.13.2 are satisfied.

Remark 4.13.2 (Phase functions). We note that if u ∈ C∞(Tn), for the functionTu to be well defined on the torus we need that the integral (4.47) is 1-periodicin x. Therefore, by identifying functions on Tn with 1-periodic functions on Rn,we will require that the phase function φ : Rn ×Zn → R is such that the functionx �→ e2πiφ(x,ξ) is 1-periodic for all ξ ∈ Zn. Note that here it is not necessary thatthe function x �→ φ(x, ξ) itself is 1-periodic.Remark 4.13.3. Assume that the function φ : Rn × Zn → R is in Ck with respectto x for all ξ ∈ Zn. Assume also that the function x �→ e2πiφ(x,ξ) is 1-periodic for allξ ∈ Zn. Differentiating it with respect to x we get that the functions x �→ ∂α

x φ(x, ξ)are 1-periodic for all ξ ∈ Zn and all α ∈ Nn

0 with 1 ≤ |α| ≤ k.Remark 4.13.4. The operator T : C∞(Tn) → D′(Tn) in (4.47) can be justified inthe usual way for oscillatory integrals. If we have more information on the symbolwe have better properties, for example:

Proposition 4.13.5. Let φ ∈ C∞(Tn×Zn) be such that the function x �→ e2πiφ(x,ξ)

is 1-periodic for all ξ ∈ Zn, and such that for some � ∈ R we have

|∂αx φ(x, ξ)| ≤ Cα〈ξ〉�

for all multi-indices α, all x ∈ Tn and ξ ∈ Zn. Let a ∈ C∞(Tn×Tn×Zn) be suchthat there is m, δ1 ∈ R and δ2 < 1 such that for all multi-indices α, β we have

|∂αx ∂β

y a(x, y, ξ)| ≤ Cαβ〈ξ〉m+δ1|α|+δ2|β|

for all x, y ∈ Tn and ξ ∈ Zn. Then the operator T in (4.47) is a well-definedcontinuous linear operator from C∞(Tn) to C∞(Tn).

Proof. Let u ∈ C∞(Tn) and let Ly be the Laplacian with respect to y. Expression(4.47) can be justified by integration by parts with the operator Ly = 1−(4π2)−1Ly

which satisfies 〈ξ〉−2Ly e2πiy·ξ = e2πiy·ξ. Consequently, we interpret (4.47) as

Tu(x) =∑ξ∈Zn

〈ξ〉−2N∫

Tn

e2πi(φ(x,ξ)−y·ξ) LNy [a(x, y, ξ) u(y)] dy, (4.48)


so that both y-integral and ξ-sum converge absolutely if N is large enough in viewof δ2 < 1. Consequently, Tu is 1-periodic by our assumptions and by Remark4.13.2, and (4.48) can be differentiated any number of times with respect to x toyield a function Tu ∈ C∞(Tn) by Remark 4.13.3. Continuity of T on C∞(Tn) fol-lows from Lebesgue’s dominated convergence theorem on Tn × Zn (see TheoremsC.3.22 and 1.1.4). �

Remark 4.13.6. Thus, we will always interpret (4.47) as (4.48). Composition for-mulae of this section can be compared with those obtained in [94, 96] globallyon Rn under minimal assumptions on phases and amplitudes. However, on thetorus, the assumptions on the regularity or boundedness of higher-order deriva-tives of phases and amplitudes are redundant due to the fact that ξ ∈ Zn takesonly discrete values.

We recall the notation for the toroidal version of Taylor polynomials and thecorresponding derivatives introduced in (3.15) and (4.31), which will be used inthe formulation of the following theorems. However, we need the following:

Definition 4.13.7 (Warning: operators (−Dy)(α)). Before we define operators(−Dy)(α) below we warn the reader that one should not formally plug in theminus sign in the definition of the previously defined operators (Dy)(α) in Defini-tion 3.4.1! Please compare these operators with those in (4.31) and observe howthe sign changes. With this warning in place, we define

(−Dy)(α) = (−Dy1)(α1) · · · (−Dyn

)(αn), (4.49)

where −D(0)yj = I and

(−Dyj)(k+1) = (−Dyj

)(k)

(− ∂

2πi∂yj− kI

)= − ∂

2πi∂yj

(− ∂

2πi∂yj− I

)· · ·

(− ∂

2πi∂yj− kI

).

We now study composition formulae of Fourier series operators with pseudo-differential operators.

Theorem 4.13.8 (Composition FSO◦ΨDO). Let φ : Rn × Zn → R be such thatfunction x �→ e2πiφ(x,ξ) is 1-periodic for all ξ ∈ Zn. Let T : C∞(Tn)→ D′(Tn) bedefined by

Tu(x) :=∑ξ∈Zn

∫Tn

e2πi(φ(x,ξ)−y·ξ) a(x, y, ξ) u(y) dy, (4.50)

where the toroidal amplitude a ∈ C∞(Tn × Tn × Zn) satisfies∣∣∂αx ∂β

y a(x, y, ξ)∣∣ ≤ Cαβm 〈ξ〉m


for all x, y ∈ Tn, ξ ∈ Zn and α, β ∈ Nn0 . Let p ∈ S�(Tn × Zn) be a toroidal

symbol and P = Op(p) the corresponding pseudo-differential operator. Then thecomposition TP has the form

TPu(x) =∑ξ∈Zn

∫Tn

e2πi(φ(x,ξ)−z·ξ) c(x, z, ξ) u(z) dz,

wherec(x, z, ξ) =

∑η∈Zn

∫Tn

e2πi(y−z)·(η−ξ) a(x, y, ξ) p(y, η) dy

satisfies ∣∣∂αx ∂β

z c(x, z, ξ)∣∣ ≤ Cαβmt 〈ξ〉m+�

for every x, z ∈ Tn, ξ ∈ Zn and α, β ∈ Nn0 . Moreover, we have an asymptotic

expansion

c(x, z, ξ) ∼∑α≥0

1α!

(−Dz)(α)[a(x, z, ξ) �α

ξ p(z, ξ)].

Furthermore, if 0 ≤ δ < ρ ≤ 1, p ∈ S�ρ,δ(T

n × Zn) and a ∈ Amρ,δ(T

n), thenc ∈ Am+�

ρ,δ (Tn).

Remark 4.13.9. We note that if T in (4.50) is a pseudo-differential operator withphase φ(x, ξ) = x · ξ and amplitude a(x, y, ξ) = a(x, ξ) independent of y, thenthe asymptotic expansion formula for the composition of two pseudo-differentialoperators T ◦ P becomes

c(x, z, ξ) ∼∑α≥0

1α!

a(x, ξ) (−Dz)(α)�αξ p(z, ξ).

This is another representation for the composition compared to Theorem 4.7.10,with an amplitude realisation of the pseudo-differential operator T ◦P , see Remark4.7.12.

Proof of Theorem 4.13.8. Let us calculate the composition TP :

TPu(x) =∑ξ∈Zn

∫Tn

e2πi(φ(x,ξ)−y·ξ) a(x, y, ξ) Pu(y) dy

=∑ξ∈Zn

∫Tn

e2πi(φ(x,ξ)−y·ξ) a(x, y, ξ)

×∑

η∈Zn

∫Tn

e2πi(y−z)·η p(y, η) u(z) dz dy

=∑ξ∈Zn

∫Tn



wherec(x, z, ξ) =

∑η∈Zn

∫Tn

e2πi(y−z)·(η−ξ) a(x, y, ξ) p(y, η) dy.

Denote θ := η − ξ, so that by the discrete Taylor expansion (Theorem 3.3.21), weformally get

c(x, z, ξ) ∼∑θ∈Zn

∫Tn

e2πi(y−z)·θ a(x, y, ξ)∑α≥0

1α!

θ(α) �αξ p(y, ξ) dy

=∑α≥0

1α!

∑θ∈Zn

∫Tn

θ(α) e2πi(y−z)·θ a(x, y, ξ) �αξ p(y, ξ) dy

=∑α≥0

1α!

(−Dy)(α)(a(x, y, ξ) �α

ξ p(y, ξ))|y=z.

Now we have to justify the asymptotic expansion. First we take a discrete Taylorexpansion and using Theorem 3.3.21 again, we obtain

p(y, ξ + θ) =∑|ω|<M

1ω!

θ(ω) �ωξ p(y, ξ) + rM (y, ξ, θ).

Let Q(θ) = {ν ∈ Zn : |νj | ≤ |θj | for all j = 1, . . . , n} as in Theorem 3.3.21. Thenby Peetre’s inequality (Proposition 3.3.31) we have∣∣�α

ξ ∂βy rM (y, ξ, θ)

∣∣ ≤ C 〈θ〉M max|ω|=Mν∈Q(θ)

∣∣∣�α+ωξ ∂β

y p(y, ξ + ν)∣∣∣

≤ C 〈θ〉M maxν∈Q(θ)

〈ξ + ν〉�−|α|−M

≤ C 〈θ〉M maxν∈Q(θ)

〈ν〉|�−|α|−M | 〈ξ〉�−|α|−M

≤ C 〈θ〉2M+|�|+|α| 〈ξ〉�−|α|−M .

The corresponding remainder term in the asymptotic expansion of c(x, z, ξ) is

RM (x, z, ξ) =∑θ∈Zn

∫Tn

e2πi(y−z)·θ a(x, y, ξ) rM (y, ξ, θ) dy

=∑θ∈Zn

∫Tn

e2πi(y−z)·θ 〈θ〉−2N

×(I − (4π2)−1Ly)N [a(x, y, ξ) rM (y, ξ, θ)] dy,

where we integrated by parts exploiting that

(I − (4π2)−1Ly) e2πi(y−z)·θ = 〈θ〉2 e2πi(y−z)·θ,


where Ly is the Laplacian with respect to y. Thus we get the estimate∣∣�αξ ∂β

x∂γz RM (x, z, ξ)

∣∣ ≤ C∑θ∈Zn

〈θ〉|γ|−2N+2M+|�|+|α| 〈ξ〉m+�−M ,

and we take N ∈ N so large that this sum over θ converges. Hence∣∣�αξ ∂β

x∂γz RM (x, z, ξ)

∣∣ ≤ C 〈ξ〉m+�−M .

This completes the proof of the first part of the theorem. Finally, we assumethat a ∈ Am

ρ,δ(Tn). Then also the terms in the asymptotic expansion and the

remainder RM have corresponding decay properties in the ξ-differences, leadingto the amplitude c ∈ Am+�

ρ,δ (Tn). This completes the proof. �Exercise 4.13.10. Work out all the details of the proof in the (ρ, δ)-case.

We now formulate the theorem about compositions of operators in the op-posite order.

Theorem 4.13.11 (Composition ΨDO◦FSO). Let φ : Rn × Zn → R be 1-periodicfor all ξ ∈ Zn. Let T : C∞(Tn)→ D′(Tn) be such that

Tu(x) :=∑ξ∈Zn

∫Tn

e2πi(φ(x,ξ)−y·ξ) a(x, y, ξ) u(y) dy,

where a ∈ C∞(Tn × Tn × Zn) satisfies∣∣∂αx ∂β

y a(x, y, ξ)∣∣ ≤ Cαβm 〈ξ〉m

for all x, y ∈ Tn, ξ ∈ Zn and α, β ∈ Nn0 . Assume that for some C > 0 we have

C−1 〈ξ〉 ≤ 〈∇xφ(x, ξ)〉 ≤ C 〈ξ〉 (4.51)

for all x ∈ Tn, ξ ∈ Zn, and that

|∂αx φ(x, ξ)| ≤ Cα 〈ξ〉,

∣∣∣∂αx�β

ξ φ(x, ξ)∣∣∣ ≤ Cαβ (4.52)

for all x ∈ Tn, ξ ∈ Zn and α, β ∈ Nn0 with |β| = 1. Let p ∈ S�(Tn × Zn) be a

toroidal symbol let p(x, ξ) denote an extension of p(x, ξ) to a symbol in S�(Tn×Rn)as given in Theorem 4.5.3. Let P = Op(p) be the corresponding pseudo-differentialoperator. Then

PTu(x) =∑ξ∈Zn

∫Tn


where we have ∣∣∂αx ∂β

z c(x, z, ξ)∣∣ ≤ Cαβ 〈ξ〉m+�


all every x, z ∈ Tn, ξ ∈ Zn and α, β ∈ Nn0 . Moreover, we have the asymptotic

expansion

c(x, z, ξ) ∼∑α≥0

(2πi)−|α|

α!∂α

η p(x, η)|η=∇xφ(x,ξ) ∂αy

[e2πiΨ(x,y,ξ)a(y, z, ξ)

]|y=x,

(4.53)

whereΨ(x, y, ξ) := φ(y, ξ)− φ(x, ξ) + (x− y) · ∇xφ(x, ξ).

Remark 4.13.12. Let us make some remarks about quantities appearing in theasymptotic extension formula (4.53). It is geometrically reasonable to evaluatethe symbol p(x, ξ) at the real Hamiltonian flow generated by the phase func-tion φ of the Fourier series operator T . This is the main complication comparedwith pseudo-differential operators for which we have Proposition 4.12.7. How-ever, although a priori the symbol p is defined only on Tn × Zn, we can stillextend it to a symbol p(x, ξ) on Tn × Rn by Theorem 4.5.3, so that the restric-tion ∂α

η p(x, η)|η=∇xφ(x,ξ) makes sense. We also note that the function Ψ(x, y, ξ)can not be in general considered as a function on Tn × Tn × Zn because it maynot be periodic in x and y. However, we can still observe that the derivatives∂α

y

[e2πiΨ(x,y,ξ)a(y, z, ξ)

]|y=x are periodic in x and z, so all terms in the right-

hand side of (4.53) are well-defined functions on Tn × Tn × Zn. In any case, for astandard theory of Fourier integral operators on Rn we refer the reader to [56].Remark 4.13.13. In Theorem 4.13.11, we note that if φ ∈ S1

ρ,δ(Rn × Rn), p ∈

S�ρ,δ(T

n×Zn), a ∈ Amρ,δ(T

n), and 0 ≤ δ < ρ ≤ 1, then we also have c ∈ Am+�ρ,δ (Tn).

Exercise 4.13.14. Prove this remark.

Proof of Theorem 4.13.11. To simplify the notation, let us drop writing tilde onp, and denote both symbols p and p by the same letter p. There should be noconfusion since they coincide on Tn × Zn. Let P = Op(p). We can write

PTu(x) =∑

η∈Zn

∫Tn

e2πi(x−y)·η p(x, η) Tu(y) dy

=∑

η∈Zn

∫Tn

e2πi(x−y)·η p(x, η)

×∑ξ∈Zn

∫Tn

e2πi(φ(y,ξ)−z·ξ) a(y, z, ξ) u(z) dz dy

=∑ξ∈Zn

∫Tn


where

c(x, z, ξ) =∑

η∈Zn

p(x, η)∫

Tn

e2πi(φ(y,ξ)−φ(x,ξ)+(x−y)·η) a(y, z, ξ) dy.


Let us fix some x ∈ Rn, with corresponding equivalence class [x] ∈ Tn which westill denote by x. Let V ⊂ Rn be an open cube with side length equal to 1 centredat x. Let χ = χ(x, y) ∈ C∞(Tn × Tn) be such that 0 ≤ χ ≤ 1, χ(x, y) = 1 for‖x− y‖ < κ for some sufficiently small κ > 0, and such that suppχ(x, ·)∩ V ⊂ V .The last condition means that χ(x, ·)|V ∈ C∞0 (V ) is supported away from theboundaries of the cube V . Let

c(x, z, ξ) = cI(x, z, ξ) + cII(x, z, ξ),

where

cI(x, z, ξ) =∑

η∈Zn

∫Tn

e2πi(φ(y,ξ)−φ(x,ξ)+(x−y)·η) (1− χ(x, y)) a(y, z, ξ) p(x, η) dy,

and

cII(x, z, ξ) =∑

η∈Zn

∫Tn

e2πi(φ(y,ξ)−φ(x,ξ)+(x−y)·η) χ(x, y) a(y, z, ξ) p(x, η) dy.

1. Estimate on the support of 1− χ. By making a decomposition into cones(sectors) centred at x viewed as a point in Rn, it follows that we can assumewithout loss of generality that the support of 1 − χ is contained in a set whereC < |xj − yj |, for some 1 ≤ j ≤ n. In turn, because of the assumption on thesupport of χ(x, ·)|V it follows that C < |xj −yj | < 1−C, for some C > 0. Now weare going to apply the summation by parts formula (3.14) to estimate cI(x, z, ξ).First we notice that it follows that

�ηje2πi(x−y)·η = e2πi(x−y)·(η+ej) − e2πi(x−y)·η

= e2πi(x−y)·η(

e2πi(xj−yj) − 1)

= 0

on supp(1− χ). Hence by the summation by parts formula (3.14) we get that∑η∈Zn

e2πi(x−y)·η p(x, η) =(

e2πi(xj−yj) − 1)−N1 ∑

η∈Zn

e2πi(x−y)·η �ηj

N1p(x, η),

where the sum on the right-hand side converges absolutely for large enough N1.On the other hand, we can integrate by parts with the operator

tLy =1− (4π2)−1Ly

〈∇yφ(y, ξ)〉2 − (2π)−1i Lyφ(y, ξ),

where Ly is the Laplace operator with respect to y, and for which we haveLN2

y e2πiφ(y,ξ) = e2πiφ(y,ξ). Note that in view of our assumption (4.51) on φ,we have

|〈∇yφ(y, ξ)〉2 − (2π)−1i Lyφ(y, ξ)| ≥ |〈∇yφ(y, ξ)〉|2 ≥ C1〈ξ〉2.


Therefore,

cI(x, z, ξ) =∑

η∈Zn

∫Tn

e2πi(φ(y,ξ)−φ(x,ξ)+x·η)LN2y

{e−2πiy·η

×(

e2πi(xj−yj) − 1)−N1

�ηj

N1p(x, η) (1− χ(x, y)) a(y, z, ξ)

}dy.

From the properties of amplitudes, we get

|cI(x, z, ξ)| ≤ C∑

η∈Zn

∫V ∩{2π−c>|xj−yj |>c}

〈ξ〉m−2N2〈η〉2N2+�−N1 dy

≤ C〈ξ〉−N

for all N , if we choose large enough N2 and then large enough N1. We can easilysee that similar estimates work for the derivatives of cI , completing the proof onthe support of 1− χ.

2. Estimate on the support of χ. Extending p ∈ S�(Tn × Zn) to a symbol inp ∈ S�(Tn × Rn) as in Theorem 4.5.3, we will make its usual Taylor expansion atη = ∇xφ(x, ξ), so that we have

p(x, η) =∑|α|<M

(η −∇xφ(x, ξ))α

α!∂α

ξ p(x,∇xφ(x, ξ))

+∑|α|=M

Cα (η −∇xφ(x, ξ))α rα(x, ξ, η −∇xφ(x, ξ)),

rα(x, ξ, η −∇xφ(x, ξ)) =∫ 1

0

(1− t)M1 ∂αξ p(x, tη + (1− t)∇xφ(x, ξ)) dt.

Then

cII(x, z, ξ) =∑|α|<M

1α!

cα(x, z, ξ) +∑|α|=M

CαRα(x, z, ξ),

where

Rα(x, z, ξ) =∑

η∈Zn

∫Tn

e2πi(φ(y,ξ)−φ(x,ξ)+(x−y)·η)(η −∇xφ(x, ξ))α

× χ(x, y) rα(x, ξ, η −∇xφ(x, ξ)) a(y, z, ξ) dy,

cα(x, z, ξ) =∑

η∈Zn

∫Tn

e2πi(φ(y,ξ)−φ(x,ξ)+(x−y)·η)(η −∇xφ(x, ξ))α

× χ(x, y) a(y, z, ξ) ∂αξ p(x,∇xφ(x, ξ)) dy.


Now using Corollary 4.6.16 we can calculate

cα(x, z, ξ) = ∂αξ p(x,∇xφ(x, ξ)) [Dy −∇xφ(x, ξ)]α

×{

e2πi(φ(y,ξ)−φ(x,ξ)) χ(x, y) a(y, z, ξ)}|y=x

= ∂αξ p(x,∇xφ(x, ξ))

∫Rn

∫V

e2πi(x−y)·η [η −∇xφ(x, ξ)]α

× e2πi(φ(y,ξ)−φ(x,ξ)) χ(x, y) a(y, z, ξ) dy dη

= ∂αξ p(x,∇xφ(x, ξ))

×Dαy

[e2πi(φ(y,ξ)−φ(x,ξ)+(x−y)·∇xφ(x,ξ))χ(x, y) a(y, z, ξ)

]|y=x,

where we wrote the derivative [Dy −∇xφ(x, ξ)]α as a pseudo-differential operatorwith symbol [η −∇xφ(x, ξ)]α, x, ξ ∈ Rn, and changed the variables η �→ η +∇xφ(x, ξ). Since χ is identically equal to one for y near x, we obtain the asymptoticformula (4.53), once the remainders Rα are analysed.

3. Estimates for the remainder. Let us first write the remainder in the form

Rα(x, z, ξ) =∑

η∈Zn

∫Tn

e2πi(x−y)·η rα(x, ξ, η −∇xφ(x, ξ))

× (η −∇xφ(x, ξ))α χ(x, y) g(y) dy,

(4.54)

withg(y) = e2πi(φ(y,ξ)−φ(x,ξ)) χ(x, y) a(y, z, ξ),

which is a 1-periodic function of y. Now, we can use Corollary 4.6.16 to concludethat Rα(x, z, ξ) in (4.54) is equal to the periodisation with respect to x in the formRα(x, z, ξ) = PxSα(x, z, ξ), where

Sα(x, z, ξ) = rα(x, ξ, Dy −∇xφ(x, ξ)) (Dy −∇xφ(x, ξ))αg(y)|y=x

=∫

Rn

∫V

e2πi(x−y)·η rα(x, ξ, η −∇xφ(x, ξ))

× (η −∇xφ(x, ξ))α χ(x, y) g(y) dy dη

=∫

Rn

∫V

e2πi(x−y)·θ e2πiΨ(x,y,ξ) θα χ(x, y) a(y, z, ξ) rα(x, ξ, θ) dy dθ,

where we changed the variables by θ = η −∇xφ(x, ξ) and where

Ψ(x, y, ξ) := φ(y, ξ)− φ(x, ξ) + (x− y) · ∇xφ(x, ξ)

and

rα(x, ξ, θ) =∫ 1

0

(1− t)M1 ∂αξ p(x,∇xφ(x, ξ) + tθ) dt.

Since the periodisation Px does not change the orders in z and ξ it is enough toderive the required estimates for Sα(x, z, ξ).


Let ρ ∈ C∞0 (Rn) be such that ρ(θ) = 1 for ‖θ‖ < ε/2, and ρ(θ) = 0 for‖θ‖ > ε, for some small ε > 0 to be chosen later. We decompose

Sα(x, z, ξ) = SIα(x, z, ξ) + SII

α (x, z, ξ),

where

SIα(x, z, ξ) =

∫Rn

∫v

e2πi(x−y)·θ ρ

(θ

〈ξ〉

)× rα(x, ξ, θ) Dα

y

[e2πiΨ(x,y,ξ) χ(x, y) a(y, z, ξ)

]dy dθ,

SIIα (x, z, ξ) =

∫Rn

∫V

e2πi(x−y)·θ(

1− ρ

(θ

〈ξ〉

))× rα(x, ξ, θ) Dα

y


]dy dθ.

3.1. Estimate for ‖θ‖ ≤ ε〈ξ〉. For sufficiently small ε > 0, for any 0 ≤ t ≤ 1,〈∇xφ(x, ξ) + tθ〉 and 〈ξ〉 are equivalent. Indeed, if we use the inequalities

〈z〉 ≤ 1 + ‖z‖ ≤√

2〈z〉,

we get

〈∇xφ(x, ξ) + tθ〉 ≤ (C2

√2 + ε)〈ξ〉

√2〈∇xφ(x, ξ) + tθ〉, ≥ 1 + ‖∇xφ‖ − ‖θ‖ ≥ 〈∇xφ〉 − ‖θ‖ ≥ (C1 − ε)〈ξ〉,

so we will take ε < C1. This equivalence means that for ‖θ‖ ≤ ε〈ξ〉, the functionrα(x, ξ, θ) is dominated by 〈ξ〉�−|α| since p ∈ S�(Tn × Rn). We will need twoauxiliary estimates. The first estimate∣∣∣∣∂γ

θ

(ρ

(θ

〈ξ〉

)rα(x, ξ, θ)

)∣∣∣∣ ≤ C∑δ≤γ

∣∣∣∣∂δθρ

(θ

〈ξ〉

)∂γ−δ

θ rα(x, ξ, θ)∣∣∣∣

≤ C∑δ≤γ

〈ξ〉−|δ|〈ξ〉�−|α|−|γ−δ|

≤ C〈ξ〉�−|α|−|γ|

(4.55)

follows from the properties of rα. Before we state the second estimate, let us anal-yse the structure of ∂α

y e2πiΨ(x,y,ξ). It has at most |α| powers of terms ∇yφ(y,ξ)−∇xφ(x,ξ), possibly also multiplied by at most |α| higher-order derivatives ∂δ

yφ(y,ξ).The product of the terms of the form ∇yφ(y, ξ)−∇xφ(x, ξ) can be estimated byC(‖y−x‖〈ξ〉)|α|. The terms containing no difference ∇yφ(y, ξ)−∇xφ(x, ξ) are theproducts of at most |α|/2 terms of the type ∂δ

yφ(y, ξ), and the product of all suchterms can be estimated by C〈ξ〉|α|/2. Altogether, we obtain the estimate

|∂αy e2πiΨ(x,y,ξ)| ≤ Cα (1 + 〈ξ〉‖y − x‖)|α| 〈ξ〉|α|/2

.


The second auxiliary estimate now is∣∣∣Dαy


]∣∣∣ ≤ Cα (1 + 〈ξ〉‖y − x‖)|α| 〈ξ〉|α|2 +m

. (4.56)

Now we are ready to prove a necessary estimate for SIα(x, z, ξ). Let

Lθ =(1− (4π2)−1〈ξ〉2Lθ)

1 + 〈ξ〉2‖x− y‖2, LN

θ e2πi(x−y)·θ = e2πi(x−y)·θ,

where Lθ is the Laplace operator with respect to θ. Integration by parts with Lθ

yields

SIα(x, z, ξ)

=∫

Rn

∫V

e2πi(x−y)·θ

(1 + 〈ξ〉2‖x− y‖2)N(1− (4π2)−1〈ξ〉2Lθ)N

×{

ρ

(θ

〈ξ〉

)rα(x, ξ, θ) Dα

y


]}dy dθ

=∫

Rn

∫V

e2πi(x−y)·θ

(1 + 〈ξ〉2‖x− y‖2)N

∑|γ|≤2N

Cγ〈ξ〉|γ|

×{

Dαy


]∂γ

θ

(ρ

(θ

〈ξ〉

)rα(x, ξ, θ)

)}dy dθ.

Using estimates (4.55), (4.56) and the fact that the measure of the support offunction θ �→ ρ(θ/〈ξ〉) is estimated by (ε〈ξ〉)n, we obtain the estimate

|SIα(x, z, ξ)| ≤ C

∑|γ|≤2N

〈ξ〉n+|γ|+ |α|2 +m〈ξ〉�−|α|−|γ|

∫V

(1 + 〈ξ〉‖y − x‖)|α|(1 + 〈ξ〉2‖x− y‖2)N

dy

≤ C〈ξ〉�+m+n− |α|2 ,

if we choose N large enough, e.g., N ≥M = |α|.Each derivative of SI

α(x, z, ξ) with respect to x or ξ gives an extra power of θunder the integral. Integrating by parts, this amounts to taking more y-derivatives,giving a higher power of 〈ξ〉. However, this is not a problem if for the estimatefor a given number of derivatives of the remainder SI

α(x, z, ξ), we choose M = |α|sufficiently large.

3.2. Estimate for ‖θ‖ > ε〈ξ〉. Let us define

ω(x, y, ξ, θ) := (x− y) · θ + Ψ(x, y, ξ)= (x− y) · (∇xφ(x, ξ) + θ) + φ(y, ξ)− φ(x, ξ).


From (4.51) and (4.52) we have

‖∇yω‖ = ‖ − θ +∇yφ−∇xφ‖ ≤ 2C2(‖θ‖+ 〈ξ〉),‖∇yω‖ ≥ ‖θ‖ − ‖∇yφ−∇xφ‖

≥ 12‖θ‖+

( ε

2− C0‖x− y‖

)〈ξ〉

≥ C(‖θ‖+ 〈ξ〉),

(4.57)

if we choose κ < ε2C0

, since ‖x − y‖ < κ in the support of χ in V (recall that wewere free to choose κ > 0). Let us write

σγ1(x, y, ξ) := e−2πiΨ(x,y,ξ) Dγ1y e2πiΨ(x,y,ξ).

For any ν we have an estimate

|∂νy σγ1(x, y, ξ)| ≤ C〈ξ〉|γ1|, (4.58)

because of our assumption (4.52) that |∂νy φ(y, ξ)| ≤ Cν〈ξ〉. For M = |α| > � we

also observe that

|rα(x, ξ, θ)| ≤ Cα, |∂νy a(y, z, ξ)| ≤ Cβ〈ξ〉m. (4.59)

Let us take tLy = i‖∇yω‖−2∑n

j=1(∂yjω)∂yj

. It can be shown by induction thatthe operator LN

y has the form

LNy =

1‖∇yω‖4N

∑|ν|≤N

Pν,N∂νy , Pν,N =

∑|μ|=2N

cνμδj(∇yω)μ∂δ1

y ω · · · ∂δNy ω,

where |μ| = 2N, |δj | ≥ 1,∑N

j=1 |δj | + |ν| = 2N. It follows from (4.52) and (4.57)

that |Pν,N | ≤ C(‖θ‖ + 〈ξ〉)3N , since for all δj we have |∂δjy ω| ≤ C(‖θ‖ + 〈ξ〉). By

the Leibniz formula we have

SIIα (x, z, ξ)

=∫

Rn

∫V

e2πi(x−y)·θ(

1− ρ

(θ

〈ξ〉

))rα(x, ξ, θ)

×Dαy


]dy dθ

=∫

Rn

∫V

e2πiω(x,y,ξ,θ)

(1− ρ

(θ

〈ξ〉

))rα(x, ξ, θ)

×∑

γ1+γ2+γ3=α

σγ1(x, y, ξ) Dγ2y χ(x, y) Dγ3

y a(y, z, ξ) dy dθ

=∫

Rn

∫V

e2πiω(x,y,ξ,θ)‖∇yω‖−4N∑|ν|≤N

Pν,N (x, y, ξ, θ)(

1− ρ

(θ

〈ξ〉

))× rα(x, ξ, θ)

∑γ1+γ2+γ3=α

∂νy

(σγ1(x, y, ξ) Dγ2

y χ(x, y) Dγ3y a(y, z, ξ)

)dy dθ.

4.14. Boundedness of Fourier series operators on L2(Tn) 405

It follows now from (4.58) and (4.59) that

|SIIα (x, z, ξ)| ≤ C

∫‖θ‖>ε〈ξ〉/2

(‖θ‖+ 〈ξ〉)−N 〈ξ〉|α|〈ξ〉m dθ

≤ C〈ξ〉m+|α|+n−N,

which yields the desired estimate if we take large enough N . For the derivativesof SII

α (x, z, ξ), similar to Part 3.1 for SIα, we can get extra powers of θ, which

can be taken care of by choosing large N . The proof of Theorem 4.13.11 is nowcomplete. �Remark 4.13.15. Note that we could also use the following asymptotic expansionfor c based on the discrete Taylor expansion from Theorem 3.3.21:

c(x, z, ξ) ∼∑θ∈Zn

∑α≥0

1α!

θ(α) [�αωp(x, ω)]ω=∇xφ(x,ξ)

×∫

Tn

e2πi(Ψ(x,y,ξ)+(x−y)·θ) a(y, z, ξ) dy

=∑α≥0

1α!

[�αωp(x, ω)]ω=∇xφ(x,ξ)

×∑θ∈Zn

∫Tn

θ(α) e2πi(x−y)·θ e2πiΨ(x,y,ξ) a(y, z, ξ) dy

=∑α≥0

1α!

[�αωp(x, ω)]ω=∇xφ(x,ξ) D(α)

y

[e2πiΨ(x,y,ξ) a(y, z, ξ)

]y=x

.

Exercise 4.13.16. Justify this expansion to obtain yet another composition formula.

4.14 Boundedness of Fourier series operators on L2(Tn)

In Theorem 4.8.1 we proved the boundedness of operators on L2(Tn) in termsof estimates on their symbols. In particular, in applications it is important toknow how many derivatives (or differences in the present toroidal approach) of thesymbol must be estimated for the boundedness of the operator. In this section wepresent the L2(Tn)-boundedness theorem for Fourier series operators also payingattention to the number of required derivatives for the amplitude.

However, first we need an auxiliary result which is of great importance on itsown. The following statement is a modification of the well-known Cotlar’s lemmataking into account the fact that operators in our application Theorem 4.14.2,especially the Fourier transform on the torus, act on functions on different Hilbertspaces. The proof below follows [118, p. 280] but there is a difference in how weestimate operator norms because we cannot immediately replace the operator Sby S∗S in the estimates since they act on functions on different spaces.


Theorem 4.14.1 (Cotlar’s lemma in Hilbert spaces). Let H,G be Hilbert spaces.Assume that a family of bounded linear operators {Sj : H → G}j∈Zr and positiveconstants {γ(j)}j∈Zr satisfy

‖S∗l Sk‖H→H ≤ [γ(l − k)]2 , ‖SlS∗k‖G→G ≤ [γ(l − k)]2 ,

andA =

∑j∈Zr

γ(j) <∞.

Then the operatorS =

∑j∈Zr

Sj

satisfies‖S‖H→G ≤ A.

Proof. First let us assume that there are only finitely many (say N) non-zerooperators Sj . We want to establish an estimate uniformly in N and then pass tothe limit. We observe that we have the estimate ‖S‖ ≤ ‖S∗S‖ for operator normsbecause we can estimate

||S||2H→G = sup‖f‖H≤1

(Sf, Sf)G = sup‖f‖H≤1

(S∗Sf, f)H

≤ ‖S∗S‖H→H.

For any k ∈ N and B ∈ L(H) we have ‖B‖2k

= ‖(B∗B)2k−1‖, which follows

inductively from ‖B‖2 = ‖B∗B‖. Thus if m = 2k and B = S∗S then ‖S∗S‖m =‖(S∗S)m‖, so we can conclude

‖S‖2mH→G ≤ ‖S∗S‖

mH→H = ‖(S∗S)m‖H→H

=

∥∥∥∥∥∥∑

i1,...,i2m

S∗i1Si2 · · ·S∗i2m−1Si2m

∥∥∥∥∥∥H→H

. (4.60)

Now, we can group products in the sum in different ways. Grouping the terms inthe last product as (S∗i1Si2)(S

∗i3

Si4) · · · (S∗i2m−1Si2m

), we can estimate∥∥∥S∗i1Si2 · · ·S∗i2m−1Si2m

∥∥∥H→H

≤ γ(i1 − i2)2 γ(i3 − i4)2 · · · γ(i2m−1 − i2m)2. (4.61)

Alternatively, grouping them as S∗i1(Si2S∗i3

) · · · (Si2m−2S∗i2m−1

)Si2m , we can esti-mate∥∥∥S∗i1Si2 · · ·S∗i2m−1

Si2m

∥∥∥H→H

≤ A2 γ(i2 − i3)2 γ(i4 − i5)2 · · · γ(i2m−2 − i2m−1)2.

(4.62)


Taking the geometric mean of (4.61) and (4.62) and using it in (4.60), we get theestimate

‖S‖2mH→G ≤

∑i1,...,i2m

A γ(i1 − i2) γ(i2 − i3) · · · γ(i2m−1 − i2m).

Now, taking the sum first with respect to i1 and using that∑

i1γ(i1−i2) ≤ A, then

taking the sum with respect to i2, etc., we can estimate ‖S‖2mH→G ≤ A2m

∑i2m

1.Now, if there are only N non-zero Si’s, we obtain the estimate

‖S‖H→G ≤ A N1

2m

which proves the statement if we let m→∞. Since this conclusion is uniform overN , the proof is complete. �

We recall that in the analysis in this chapter we wrote 2π in the exponentialto assure that functions e2πix·ξ are 1-periodic. In this section, the only functionthat occurs in the exponential is φ(x, k) and so we do not need to keep writing 2πin the exponential.

Theorem 4.14.2 (Fourier series operators on L2(Tn)). Let T : C∞(Tn)→ D′(Tn)be defined by

Tu(x) =∑

k∈Zn

eiφ(x,k) a(x, k) (FTnu)(k),

where φ : Rn×Zn → R and a : Tn×Zn → C. Assume that the function x �→ eiφ(x,ξ)

is 1-periodic for every ξ ∈ Zn, and that for all |α| ≤ 2n + 1 and |β| = 1 we have

|∂αx a(x, k)| ≤ C and

∣∣∣∂αx�β

kφ(x, k)∣∣∣ ≤ C (4.63)

for all x ∈ Tn and k ∈ Zn. Assume also that

|∇xφ(x, k)−∇xφ(x, l)| ≥ C|k − l| (4.64)

for all x ∈ Tn and k, l ∈ Zn. Then T extends to a bounded linear operator fromL2(Tn) to L2(Tn).

Remark 4.14.3. Note that condition (4.64) is a discrete version of the usual lo-cal graph condition for Fourier integral operators, necessary for the local L2-boundedness. We also note that conditions on the boundedness of the higher-order differences of phase and amplitude would follow automatically from con-dition (4.63). Therefore, this theorem relaxes assumptions on the behaviour withrespect to the dual variable, compared, for example, with the corresponding globalresult for Fourier integral operators in [95] in Rn.


Proof of Theorem 4.14.2. Since for u : Tn → C we have ‖u‖L2(Tn) = ‖FTnu‖�2(Zn),it is enough to prove that the operator

Sw(x) =∑

k∈Zn

eiφ(x,k) a(x, k) w(k)

is bounded from �2(Zn) to L2(Tn). Let us define

Slw(x) := eiφ(x,l) a(x, l) w(l),

so that S =∑

l∈Zn Sl. From the identity

(w,S∗v)�2(Zn) = (Sw, v)L2(Tn) =∫

Tn

∑k∈Zn

eiφ(x,k) a(x, k) w(k) v(x) dx

we find that the adjoint S∗ to S is given by

(S∗v)(k) =∫

Tn

e−iφ(x,k) a(x, k) v(x) dx

and so we also have

(S∗l v)(m) = δlm

∫Tn

e−iφ(x,m) a(x, m) v(x) dx = δlm(S∗v)(l).

It follows that

SkS∗l v(x) = eiφ(x,k) a(x, k) (S∗l v)(k)

= δlk

∫Tn

eiφ(x,k) a(x, k) e−iφ(y,k) a(y, k) v(y) dy

=∫

Tn

Kkl(x, y) v(y) dy,

whereKkl(x, y) = δkl ei[φ(x,k)−φ(y,k)] a(x, k) a(y, k).

From (4.63) and compactness of the torus it follows that the kernel Kkl is boundedand that ‖SkS∗l v‖L2(Tn) ≤ Cδkl‖v‖L2(Tn). In particular, we can trivially concludethat for any N ≥ 0 we have the estimate

‖SkS∗l ‖L2(Tn)→L2(Tn) ≤CN

1 + |k − l|N . (4.65)

On the other hand, we have

(S∗l Skw)(m) = δlm

∫Tn

e−iφ(x,l) a(x, l) (Skw)(x) dx

= δlm

∫Tn

ei[φ(x,k)−φ(x,l)] a(x, k) a(x, l) w(k) dx

=∑

μ∈Zn

Klk(m,μ) w(μ),


whereKlk(m,μ) = δlmδkμ

∫Tn

ei[φ(x,k)−φ(x,l)] a(x, k) a(x, l) dx.

Now, if k = l, integrating by parts (2n + 1)-times with operator

1i∇xφ(x, k)−∇xφ(x, l)‖∇xφ(x, k)−∇xφ(x, l)‖2 · ∇x

and using the periodicity of a and ∇xφ (so there are no boundary terms), we getthe estimate

|Klk(m,μ)| ≤ C δlm δkμ

1 + |k − l|2n+1, (4.66)

where we also used that by the discrete Taylor expansion (Theorem 3.3.21) thesecond condition in (4.63) implies that

|∇xφ(x, k)−∇xφ(x, l)| ≤ C|k − l| for all x ∈ Tn, k, l ∈ Zn.

Estimate (4.66) implies

supm

∑μ

|Klk(m,μ)| = |Klk(l, k)| ≤ C

1 + |k − l|2n+1,

and similarly for supμ

∑m, so that we have

‖S∗l Sk‖�2(Zn)→�2(Zn) ≤C

1 + |k − l|2n+1. (4.67)

These estimates for norms ‖SkS∗l ‖L2(Tn)→L2(Tn) and ‖S∗l Sk‖�2(Zn)→�2(Zn) in (4.65)and (4.67), respectively, imply the theorem by a modification of Cotlar’s lemmagiven in Proposition 4.14.1, which we use with H = �2(Zn) and G = L2(Tn). �

Using Theorems 4.13.8, 4.13.11, and 4.14.2, we obtain the result on theboundedness of Fourier series operators on Sobolev spaces:

Corollary 4.14.4 (Fourier series operators on Sobolev spaces). Let T : C∞(Tn)→D′(Tn) be defined by

Tu(x) =∑

k∈Zn

eiφ(x,k) a(x, k) u(k),

where φ : Tn × Zn → R and a : Tn × Zn → C. Assume that for all α and |β| = 1we have

|∂αx a(x, k)| ≤ Cα〈k〉m,

as well as|∂α

x φ(x, k)| ≤ Cα〈k〉 and∣∣∣∂α

x�βkφ(x, k)

∣∣∣ ≤ Cαβ


for all x ∈ Tn and k ∈ Zn. Assume that for some C > 0 we have

C−1 〈k〉 ≤ 〈∇xφ(x, k)〉 ≤ C 〈k〉

for all x ∈ Tn, k ∈ Zn, and that

|∇xφ(x, k)−∇xφ(x, l)| ≥ C|k − l|

for all x ∈ Tn and k, l ∈ Zn. Then T extends to a bounded linear operator fromHs(Tn) to Hs−m(Tn) for all s ∈ R.

Exercise 4.14.5. Work out all the details of the proof.

4.15 An application to hyperbolic equations

In this section we briefly discuss how the toroidal analysis can be applied to con-struct global parametrices for hyperbolic equations on the torus and how to embedcertain problems on Rn into the torus. The finite propagation speed of singular-ities for solutions to hyperbolic equations allows one to cut-off the equation andthe Cauchy data for large x for the local analysis of singularities of solutions forbounded times. Then the problem can be embedded into Tn, or into the inflatedtorus NTn (Remark 4.6.9), in order to apply the periodic analysis developed here.One of the advantages of this procedure is that since phases and amplitudes noware only evaluated at ξ ∈ Zn one can apply this also for problems with low regular-ity in ξ, in particular to problems for weakly hyperbolic equations or systems withvariable multiplicities. For example, if the principal part has constant coefficientsthen the loss of regularity occurs only in ξ so techniques developed in this chaptercan be applied.

Let a(X, D) be a pseudo-differential operator with symbol a satisfying a =a(x, ξ) ∈ Sm(Rn×Rn) (with some properties to be specified). There is no differencein the subsequent argument if a = a(t, x, ξ) also depends on t. For a functionu = u(t, x) of t ∈ R and x ∈ Rn we write

a(X, D)u(t, x) =∫

Rn

e2πix·ξ a(x, ξ) (FRnu)(t, ξ)dξ

=∫

Rn

∫Rn

e2πi(x−y)·ξ a(x, ξ) u(t, y) dy dξ.

Let u(t, ·) ∈ L1(Rn) (0 < t < t0) be a solution to the hyperbolic problem{i ∂∂tu(t, x) = a(X, D)u(t, x),

u(0, x) = f(x),(4.68)

where f ∈ L1(Rn) is compactly supported.

4.15. An application to hyperbolic equations 411

Assume now that a(X, D) = a1(X, D)+a0(X, D) where a1(x, ξ) is 1-periodicand a0(x, ξ) is compactly supported in x (assume even that supp a0(·, ξ) ⊂ [0, 1]n).A simple example is a constant coefficient symbol a1(x, ξ) = a1(ξ). Let us alsoassume that supp f ⊂ [0, 1]n.

We will now describe a way to periodise problem (4.68). According to Propo-sition 4.6.19, we can replace (4.68) by{

i ∂∂tu(t, x) = (a1(x,D) + (Pa0)(X, D))u(t, x) + Ru(t, x),

u(x, 0) = f(x),

where the symbol a1 + Pa0 is periodic and R is a smoothing operator. To studysingularities of (4.68), it is sufficient to analyse the Cauchy problem{

i ∂∂tv(t, x) = (a1(x, D) + (Pa0)(X, D))v(t, x),

v(x, 0) = f(x)

since by Duhamel’s formula we have WF(u− v) = ∅. This problem can be trans-ferred to the torus. Let w(t, x) = Pv(·, t)(x). By Theorem 4.6.12 it solves theCauchy problem on the torus Tn, with operator P from (4.24) in Theorem 4.6.3:{

i ∂∂tw(t, x) = (a1(x,D) + Pa0(X, D))w(t, x),

w(x, 0) = Pf(x).

Now, if a ∈ S1(Rn × Rn) is of the first order, the calculus constructed in previoussections yields the solution in the form

w(t, x) ≡ Ttf(x) =∑

k∈Zn

e2πiφ(t,x,k) b(t, x, k) FTn(Pf)(k),

where φ(t, x, ξ) and b(t, x, ξ) satisfy discrete analogues of the eikonal and trans-port equations. Here we note that FTn(Pf)(k) = (FRnf)(k). We also note thatthe phase φ(t, x, k) is defined for discrete values of k ∈ Zn, so there is no issueof regularity, making this representation potentially applicable to low regularityproblems and weakly hyperbolic equations.Example. If the symbol a1(x, ξ) = a1(ξ) has constant coefficients and belongs toS1(Rn × Rn), and a0 belongs to S0(Rn × Rn), we can find that the phase is givenby φ(t, x, k) = x · k + ta1(k). In particular, ∇xφ(x, k) = k. Applying a(X, D) tow(t, x) = Ttf(x) and using the composition formula from Theorem 4.13.11 weobtain

a(X, D)Ttf(x) =∑

k∈Zn

∫Rn

e2πi((x−z)·k+ta1(k)) c(t, x, k) f(z) dz,

where

c(t, x, k) ∼∑α≥0

(2πi)−|α|

α!∂α

ξ a(x, ξ)∣∣ξ=k

∂αx b(t, x, k), (4.69)


since the function Ψ in Theorem 4.13.11 vanishes. From this we can find amplitudeb from the discrete version of the transport equations, details of which we omithere. Finally, we note that we can also have an asymptotic expansion for theamplitude b in (4.69) in terms of the discrete differences�α

ξ and the corresponding

derivatives ∂(α)x instead of derivatives ∂α

ξ and ∂αx , respectively, if we use Remark

4.13.15 instead of Theorem 4.13.11.

Exercise 4.15.1. Work out the details for the arguments above.

Remark 4.15.2 (Schrodinger equation). Let u(t, x), t ∈ R, x ∈ Tn, be the solutionto a constant coefficients Schrodinger equation on the torus, i.e., u satisfies

i∂tu + Lu = 0, u|t=0 = f,

where L is the Laplace operator. This equation can be solved by taking the Fouriertransform, and thus the Fourier series representation of the solution is

u(t, x) = eitLf(x) =∑ξ∈Zn

ei2π(x·ξ−2πt|ξ|2)f(ξ).

This representation shows, in particular, that the solution is periodic in time. In[16, 17, 18], employing this representation, Bourgain used, for example, the equal-ity ‖u‖4L4(T×Tn) =

∥∥u2∥∥2

L2(T×Tn)leading to the corresponding Strichartz estimates

and global well-posedness results for nonlinear equations. We can note that sincethe torus is compact, the usual dispersive estimates fail even locally in time. Wewill not pursue this topic further, and refer to the aforementioned papers for thedetails.

Chapter 5

Commutator Characterisation ofPseudo-differential Operators

On a smooth closed manifold the pseudo-differential operators can be characterisedby taking commutators with vector fields, i.e., first-order partial derivatives. Thisapproach is due to Beals ([12], 1977), Dunau ([32], 1977), and Coifman and Meyer([23], 1978); perhaps the first ones to consider these kind of commutator prop-erties were Calderon and his school [21]. For other contributions, see also [26],[133] and [80].

In this chapter we present a Sobolev space version of these characterisations.This will be one of the steps in developing global quantizations of operators onLie groups in Part IV. Indeed, a commutator characterisation in Sobolev spacesas opposed to only L2 will have an advantage of allowing us to control the ordersof operators.

In particular, the commutators provide us a new, quite simple way of provingthe equivalence of local and global definitions of pseudo-differential operators on atorus, and we derive related commutator characterisations for operators of generalorder on the scale of Sobolev spaces.

The structure of the treatment is the following. First, we review necessarypseudo-differential calculus on Rn, obtaining a commutator characterisation of lo-cal pseudo-differential operators (Theorem 5.1.4). After that, the correspondingglobal characterisation is given on closed manifolds (Theorem 5.3.1). Lastly, weapply this to the global symbolic analysis of periodic pseudo-differential opera-tors on Tn (Theorem 5.4.1). Section 5.2 is devoted to a brief introduction to thenecessary concepts of pseudo-differential operators on manifolds.

5.1 Euclidean commutator characterisation

In this section we discuss the case of the Euclidean space Rn. We will concentrateon the localisation of pseudo-differential operators which is just a local way to look

414 Chapter 5. Commutator Characterisation of Pseudo-differential Operators

at pseudo-differential operators from Chapter 2 where we dealt with global analysison Rn. The commutator characterisation of local pseudo-differential operators onRn provided by Theorem 5.1.4 is needed in the next section for the commutatorcharacterisation result on closed manifolds.

Definition 5.1.1 (Order of an operator on the Sobolev scale). A linear operatorA : S(Rn)→ S(Rn) is said to be of order m ∈ R on the Sobolev scale (Hs(Rn))s∈R,if it has bounded extensions As,s−m ∈ L(Hs(Rn),Hs−m(Rn)) for every s ∈ R. Inthis case, the extension is unique in the sense that the operator A has the extensionAS′ ∈ L(S ′(Rn)) satisfying AS′ |Hs(Rn) = As,s−m. Thereby any of the operatorsAs,s−m or AS′ is also denoted by A.

By Theorem 2.6.11 a pseudo-differential operator of order m in the classΨm(Rn × Rn) is also of order m on the Sobolev scale.

Definition 5.1.2 (Local pseudo-differential operators). A linear operator

A : C∞0 (Rn)→ D′(Rn)

is called a local pseudo-differential operator of order m∈R on Rn, A∈OpSmloc(R

n),if φAψ ∈ OpSm(Rn) for every φ, ψ ∈ C∞0 (Rn). Naturally, here

((φAψ)u)(x) = φ(x)A(ψu)(x).

In addition to the symbol inequalities (2.3) in Definition 2.1.1, there is an-other appealing way of characterising pseudo-differential operators, namely viacommutators. This characterisation dates back to [12] by Beals, to [32] by Dunau,and to [23] by Coifman and Meyer. We present a related result, Theorem 5.1.4,about local pseudo-differential operators. First we introduce the following nota-tion:

Definition 5.1.3 (Notation). Let us define the commutators

Lj(A) := [∂xj, A] and Rk(A) := [A,Mxk

],

where Mxkis the multiplication operator (Mxk

f)(x) = xkf(x).Set Rα = Rα1

1 · · ·Rαnn and accordingly Lβ = Lβ1

1 · · ·Lβnn for multi-indices

α, β, with convention L0j = I = R0

k. Finally, for a partial differential operator Con Rn, let deg(C) denote its order. By Theorem 2.6.11, deg(C) is also the orderof C on the Sobolev scale.

The following theorem characterises local pseudo-differential operators on Rn

in terms of the orders of their commutators on the Sobolev scale:

Theorem 5.1.4 (Commutator characterisation on Rn). Let m ∈ R and let A be alinear operator defined on C∞0 (Rn). Then the following conditions are equivalent:

(i) A ∈ OpSmloc(R

n).

5.1. Euclidean commutator characterisation 415

(ii) For any φ, ψ ∈ C∞0 (Rn), for any s ∈ R and for any sequence C = (Cj)∞j=0 ⊂OpS1

loc(Rn) of partial differential operators of first order, it holds that{

B0 = φAψ ∈ L(Hs(Rn),Hs−m(Rn)),

Bk+1 = [Bk, Ck] ∈ L(Hs(Rn),Hs−m+dC,k(Rn)),

where dC,k =∑k

j=0(1− deg(Cj)).(iii) For any φ, ψ ∈ C∞0 (Rn), for any s ∈ R and for every α, β ∈ N0, it holds that

RαLβ(φAψ) ∈ L(Hs(Rn),Hs−(m−|α|)(Rn)).

Remark 5.1.5. At first sight, condition (ii) in Theorem 5.1.4 may seem awkward,at least when compared to condition (iii). However, this result will be neededin the pseudo-differential analysis on manifolds, and it is crucial in the proof ofTheorem 5.3.1. Also notice the similarities in the formulations of Theorems 5.1.4and 5.3.1, and in the proofs of Theorems 5.1.4 and 5.4.1.

Proof of Theorem 5.1.4. First, let A ∈ OpSmloc(R

n), and fix φ, ψ ∈ C∞0 (Rn). ThenB0 = φAψ ∈ OpSm(Rn). Let χ ∈ C∞0 (Rn) be such that χ(x) = 1 in a neigh-bourhood of the compact set supp(φ)∪ supp(ψ) ⊂ Rn, so that Bk+1 = [Bk, Ck] =[Bk, χCk]. Notice that χCk ∈ OpSdeg(Ck)(Rn). Hence by induction and by thecomposition Theorem 2.5.1 it follows that Bk+1 ∈ OpSm−dC,k(Rn). This provesthe implication (i)⇒ (ii) by Theorem 2.6.11 with p = 2.

It is really trivial that (ii) implies (iii).Finally, let us show that (iii) implies (i). Assume (iii), and fix φ, ψ ∈ C∞0 (Rn);

we have to prove that φAψ ∈ OpSm(Rn). Let χ ∈ C∞0 (Rn) be such that χ(x) =1 in a neighbourhood of the compact set supp(φ) ∪ supp(ψ) ⊂ Rn. We denoteeξ(x) = e2πix·ξ. Evidently, φAψ is of order m, and

∂αξ ∂β

xσφAψ(x, ξ) = σRαLβ(φAψ)(x, ξ)

= e−2πix·ξ(RαLβ(φAψ)eξ)(x)= e−2πix·ξ(RαLβ(φAψ)(χeξ))(x).

If 2s > n = dim(Rn), s ∈ N, then by the Cauchy-Schwartz inequality for u ∈Hs(Rn) we have:

|u(x)| ≤∫

Rn

|u(ξ)| dξ

≤[∫

Rn

(1 + |ξ|)−2s dξ

]1/2 [∫Rn

(1 + |ξ|)2s|u(ξ)|2 dξ

]1/2

= Cs‖u‖Hs(Rn) ≤ Cs

⎛⎝ ∑|γ|≤s

‖∂γxu‖2H0(Rn)

⎞⎠1/2

.


Applied to the symbol ∂αξ ∂β

xσφAψ this implies

|∂αξ ∂β

xσφAψ(x, ξ)| ≤ C

⎛⎝ ∑|γ|≤s

‖∂αξ ∂β+γ

x σφAψ(·, ξ)‖2H0(Rn)

⎞⎠1/2

= C

⎛⎝ ∑|γ|≤s

‖e−ξ

(RαLβ+γ(φAψ)

)(χeξ)‖2H0(Rn)

⎞⎠1/2

≤ C

⎛⎝ ∑|γ|≤s

‖e−ξ‖2L(H0)‖RαLβ+γ(φAψ)‖2L(Hm−|α|,H0)‖χeξ‖2Hm−|α|

⎞⎠1/2

.

By a version of Peetre’s inequality in (3.25) we have

∀s ∈ R ∀η, ξ ∈ Rn : (1 + |η + ξ|)s ≤ 2|s|(1 + |η|)|s|(1 + |ξ|)s

(where |ξ| = ‖ξ‖ in the notation of the torus chapters is just the Euclidean normof the vector ξ), so that we obtain

‖χeξ‖Hm−|α|(Rn) =(∫

Rn

(1 + |η|)2(m−|α|) |χeξ(η)|2 dη

)1/2

=(∫

Rn

(1 + |η + ξ|)2(m−|α|) |χ(η)|2 dη

)1/2

≤ 2|m−|α||‖χ‖H|m−|α||(Rn)(1 + |ξ|)m−|α|.

Hence|∂α

ξ ∂βxσφAψ(x, ξ)| ≤ Cαβ,φ,ψ(1 + |ξ|)m−|α|,

and consequently A ∈ OpSmloc(R

n). Thus (i) is obtained from (iii). �

5.2 Pseudo-differential operators on manifolds

Here we briefly provide a background on pseudo-differential operators on man-ifolds. The differential geometry needed in the study is quite simple, sufficientgeneral reference being any text book in the field, e.g., [54].

Definition 5.2.1 (Atlases on topological spaces). Let X be a topological space. Anatlas on X is a collection of pairs {(Uα, κα)}α, where all sets Uα ⊂ X are open inX,

⋃α Uα = X, and for every α the mapping κα : Uα → Rn is a homeomorphism

of Uα onto an open subset of Rn; such n is called the dimension of the chart(Uα, κα), and pairs (Uα, κα) are called charts of the atlas. For every two charts(Uα, κα) and (Uβ , κβ) with Uα ∩ Uβ = ∅, the functions

καβ := κα ◦ κ−1β : κβ(Uα ∩ Uβ)→ κα(Uα ∩ Uβ)

5.2. Pseudo-differential operators on manifolds 417

are called transition maps of the atlas. We note that each transition map καβ is ahomeomorphism between open subsets of Euclidean spaces, so that the dimensionn is the same for such charts. We will say that a point x ∈ X belongs to a chart(U, κ) if x ∈ U .

Definition 5.2.2 (Manifolds). Let X be a Hausdorff topological space such that itstopology has a countable base1. Then X equipped with an atlas A = {(Uα, κα)}α

of charts of the same dimension n is called a locally Euclidean topological space.Since n is the same for all charts, we can set dimX := n to be the dimension of X.A locally Euclidean topological space with atlas A is called a (smooth) manifold,or a C∞ manifold, if all the transition maps of the atlas A are smooth. A manifoldM is called compact if X is compact.

Example. Simple examples of n-dimensional manifolds include Euclidean spacesRn, spheres Sn, tori Tn.Remark 5.2.3. We assume that X has a countable topological base and that it isHausdorff to ensure that there are not too many open sets and that the topologyof compact manifolds is especially nice, respectively. We also note that given twoatlases we can look at transition maps in the atlas which is then union. Thus, if theunion of two atlases is again an atlas we will call these atlases equivalent. This leadsto a notion of equivalent atlases and thus a manifold is rather an equivalence classM = (X, [A]), if we do not want to worry about which atlas to fix. However, wewill avoid such technicalities because of the limited differential geometry requiredfor our purposes. In the sequel we will often omit writing the atlas at all becauseon the manifolds that we are dealing with the choice of an atlas will be more orless canonical. However, an important property for us is that if (U, κ) is a chartand V ⊂ U is open, then (V, κ) is also a chart (in an equivalent atlas, hence achart in M). We also note that Hausdorff follows from the existence of an atlas,which also implies the existence of a locally countable topological base. Instead ofthe first countability one may directly assume the existence of a countable atlas.

Definition 5.2.4 (Smooth mappings). Let f : M → N be a mapping betweenmanifolds M = (X,A) and N = (Y,B). Let x ∈ X, let (U, κ) ∈ A be a chart inM containing x, and let (V, ψ) ∈ B be a chart in N containing f(x). By shrinkingthe set U if necessary we may assume that f(U) ⊂ V. We will say that f is smoothat x ∈ X if the mapping

ψ ◦ f ◦ κ−1 : κ(U)→ ψ(V ) (5.1)

is smooth. As usual, f is smooth if it is smooth at all points. The space C∞(M)is the set of smooth complex-valued functions on M , and C∞0 (U) is the set ofsmooth functions with compact supports in an open set U ⊂M .

If k ∈ N and if all the mappings (5.1) are in Ck(κ(U)) for all charts, then wewill say that f ∈ Ck(M).

1For a topological base see Definition A.8.16


Exercise 5.2.5. Check that the definition of “f is smooth at x” does not dependon a particular choice of charts (U, κ) and (V, ψ).

Remark 5.2.6 (Whitney’s embedding theorem). We will deal only with smoothmanifolds. It is a fundamental fact that every compact manifold admits a smoothembedding as a submanifold of RN for sufficiently large N . An interesting questionis how small can N be. In 1936, in [150], for general (also non-compact) manifoldsWhitney showed that one can take N = 2n + 1 for this to be true, later alsoimproving it to N = 2n. We will not pursue this topic here and can refer to [54]for further details, but we will revisit it in a simpler context of Lie groups inCorollary 8.0.4 as well as use it in Section 10.6.Remark 5.2.7 (Orientable manifolds). The natural n-form on Rn is given by thevolume element Ω = dx1∧· · ·∧ dxn which is non-degenerate. For every open U ⊂Rn the restriction ΩU := Ω|U defines a volume element on U . A diffeomorphismF : U → V ⊂ Rn is called orientation preserving if F ∗ΩV = fΩU for somef ∈ C∞(U) such that f > 0 everywhere. A manifold M is called orientable if ithas an atlas such that all the transition maps are orientation preserving. One canshow that orientable manifolds have a non-degenerate volume element, i.e., it ispossible to define a smooth n-form on M which is not zero at any point.

Definition 5.2.8 (Localisation of operators). If A : C∞(M)→ C∞(M) and φ, ψ ∈C∞(M), we define the operator φAψ : C∞(M) → C∞(M) by ((φAψ)u)(x) =φ(x) ·A(ψ · u)(x).

Definition 5.2.9 (κ-transfers). If (U, κ) is a chart on M , the κ-transfer Aκ :C∞(κ(U))→ C∞(κ(U)) of an operator A : C∞(U)→ C∞(U) is defined by

Aκu := A(u ◦ κ) ◦ κ−1.

Similarly, the κ-transfer of a function φ is φκ = φ ◦ κ−1.

Exercise 5.2.10. Prove that the transfer of a commutator is the commutator oftransfers:

[A,B]κ = [Aκ, Bκ]. (5.2)

Pseudo-differential operators on the manifold M in the Hormander sense aredefined as follows:

Definition 5.2.11 (Pseudo-differential operators on manifolds). A linear operatorA : C∞(M) → C∞(M) is a pseudo-differential operator of order m ∈ R onM , if for every chart (U, κ) and for any φ, ψ ∈ C∞0 (U), the operator (φAψ)κ

is a pseudo-differential operator of order m on Rn. Since the class of pseudo-differential operators of order m on Rn is diffeomorphism invariant, it followsthat the corresponding class on M is well defined. We denote the set of pseudo-differential operators of order m on M by Ψm(M).

Exercise 5.2.12. Check that the class of pseudo-differential operators of order mon Rn is diffeomorphism invariant (and see Section 2.5.2).

5.2. Pseudo-differential operators on manifolds 419

Definition 5.2.13 (Diff(M)). Let Diff(M) be the ∗-algebra

Diff(M) =∞⋃

k=0

Diffk(M),

where Diffk(M) is the set of at most kth order partial differential operators onM with smooth coefficients. Here, Diff0(M) ∼= C∞(M), and Diff1(M) \Diff0(M)corresponds to the non-trivial smooth vector fields on M , i.e., the non-trivialsmooth sections of the tangent bundle TM .

Definition 5.2.14 (Closed manifolds). A compact manifold without boundary iscalled closed.

Throughout this section and further in this chapter, M will be a closed smoothorientable manifold. Then we can equip it with the volume element from Remark5.2.7. One can think of it as a suitable pullback of the Euclidean volume n-form(the Lebesgue measure) in local charts.Remark 5.2.15 (Spaces D(M) and D′(M)). A differential operator D ∈ Diff(M)defines a seminorm pD on C∞(M) by pD(u) = supx∈M |(Du)(x)|. The seminormfamily {pD : C∞(M) → R | D ∈ Diff(M)} induces a Frechet space structure onC∞(M). This test function space is denoted by D(M), and the distributions byD′(M) = L(D(M), C). In particular, similar to Remark 1.3.7 in Rn, for u ∈ Lp(M)and ϕ ∈ C∞(M), the duality

〈u, ϕ〉 :=∫

M

u(x) ϕ(x) dx

gives a canonical way to identify u ∈ Lp(M) with a distribution in D′(M). Heredx stands for a volume element on M .

Definition 5.2.16 (Sobolev space Hs(M)). The Sobolev space Hs(M) (s ∈ R) isthe set of those distributions u ∈ D′(M) such that (φu)κ ∈ Hs(Rn) for every chart(U, κ) on M and for every φ ∈ C∞0 (U). Let U = {(Uj , κj)} be a cover of M withcharts. Due to the compactness of M , we can require the cover to be finite. Fixa smooth partition of unity {(Uj , φj)} with respect to the cover U . We equip theSobolev space Hs(M) with the norm

‖u‖Hs(M),{(Uj ,κj ,φj)} :=

⎛⎝∑j

‖(φju)κj‖2Hs(Rn)

⎞⎠1/2

.

Exercise 5.2.17. Show that any other choice of Uj , κj , φj would have resulted inan equivalent norm. Prove that Hs(M) is a Hilbert space.

As a consequence of Corollary 1.5.15, as well as Propositions 1.5.18 and1.5.19, we get:


Corollary 5.2.18 (Density). Let M be a closed manifold. The space C∞(M) issequentially dense in Lp(M) for all 1 ≤ p < ∞. Also, C∞(M) is sequentiallydense in Hs(M) for every s ∈ R.

Remark 5.2.19. The last statement is true for any s ∈ R but requires more of themanifold theory than developed here. Such statements can be easily found in theliterature, and in the case of Rn see an even more general statement in Theorem1.3.31. However, the reader is encouraged to provide the details for the proof ofthe density for all s ∈ R.

Definition 5.2.20 (Order of an operator on the Sobolev scale). A linear operatorA on C∞(M) is said to be of order m ∈ R on M , if it extends boundedly be-tween Hs(M) and Hs−m(M) for every s ∈ R. Thereby the operator A has alsothe continuous extension AD′ : D′(M) → D′(M). As is in the case of Rn in Def-inition 5.1.1, any of these extensions coincide in their mutual domains, so that itis meaningful to denote any one of them by A.

Exercise 5.2.21. Prove that

C∞(M) =⋂s∈R

Hs(M) and D′(M) =⋃s∈R

Hs(M).

Remark 5.2.22 (All operators are properly supported). We recall the notion ofproperly supported operators from Definition 2.5.20. Since the support of theintegral kernel is closed, we immediately see that all pseudo-differential operatorson a closed manifold are properly supported.

We briefly address the Lp issue on compact manifolds, we formulate

Theorem 5.2.23 (Boundedness on Lp(M)). Let M be a compact manifold and letA ∈ Ψ0(M). Then A is bounded from Lp(M) to Lp(M) for any 1 < p < ∞, andits operator norm is bounded by

‖A‖L(Lp(M)) ≤ C max|β|≤n+1

|α|≤[n/2]+1

∣∣∂βx∂α

ξ a(x, ξ)∣∣|ξ||α|,

where ∂βx∂α

ξ a(x, ξ) is defined on one of some finite number of selected coordinatesystems covering M .

The proof of this theorem can be carried out by reducing the problem tothe corresponding Lp-boundedness statement of pseudo-differential operators inΨ0(Rn × Rn) with compactly supported amplitudes which would follow from The-orem 2.6.22. However, the advantage of this theorem is that one also obtains abound on the number of necessary derivatives (as well as a corresponding resultfor Theorem 2.6.22) if one reduces the problem to the Lp-multipliers problem. Werefer to [130, p. 267] for further details.

We refer to Section 13.1 for a further discussion of these concepts.

5.3. Commutator characterisation on closed manifolds 421

5.3 Commutator characterisation on closed manifolds

The main result of this section is Theorem 5.3.1 about the commutator character-isation (cf. Theorems 5.1.4 and 5.4.1), which was stated by Coifman and Meyer[23] in the case of 0-order operators on L2(M) (see also [32] for a kindred trea-tise). This will be applied in the final part of this chapter concerning periodicpseudo-differential operators (Theorem 5.4.1) and in Part IV (Theorem 10.7.7).Theorem 5.3.1 shows that pseudo-differential operators on closed manifolds canbe characterised by the orders of their commutators on the Sobolev scale.

Let M be a closed manifold. Naturally, an operator D ∈ Diffk(M) fromDefinition 5.2.13 is of order deg(D) = k. Observe that the algebra Diff(M) hasthe “almost-commuting property”:

[Diffj(M),Diffk(M)] ⊂ Diffj+k−1(M),

which follows by the Leibniz formula. Actually, more general pseudo-differentialoperators are also characterised by the “almost-commuting” with differential op-erators:

Theorem 5.3.1 (Commutator characterisation on closed manifolds). Let m ∈ Rand let A : C∞(M)→ C∞(M) be a linear operator. Then the following conditionsare equivalent:

(i) A ∈ Ψm(M).

(ii) For any s ∈ R and for any sequence D = (Dj)∞j=0 ⊂ Diff1(M), it holds that{A0 = A ∈ L(Hs(M),Hs−m(M)),Ak+1 = [Ak, Dk] ∈ L(Hs(M),Hs−m+dD,k(M)),

where dD,k =∑k

j=0(1− deg(Dj)).

We need the following auxiliary result:

Lemma 5.3.2. Let M be a closed smooth manifold. Then there exists a smoothpartition of unity with respect to a cover U on M such that U ∪ V is a chartneighbourhood whenever U, V ∈ U .

Proof. Let V be a cover of M with chart neighbourhoods. Since M is a compactmetrisable space by the Whitney embedding theorem (Remark 5.2.6), the cover Vhas the Lebesgue number λ > 0 – i.e., if S ⊂M has a small diameter, diam(S) < λ,then there exists V ∈ V such that S ⊂ V , see Lemma A.13.12. Let W be a coverof M with chart neighbourhoods of diameter less than λ/2, and choose a finitesubcover U ⊂ W. Now there exists a smooth partition of unity on M with respectto U , and if U, V ∈ U intersect, then diam(U ∪ V ) < λ. On the other hand, ifU ∩ V = ∅, then U ∪ V is clearly a chart neighbourhood. �


Proof of Theorem 5.3.1. ((i)⇒(ii)) Assume that A ∈ Ψm(M). Lemma 5.3.2 pro-vides a smooth partition of unity {(Uj , φj)}N

j=1 such that Ui∪Uj is always a chartneighbourhood, so that the study can be localised:

A =N∑

i=1

N∑j=1

φiAφj .

Let (Ui ∪ Uj , κij) be a chart. Now φi, φj ∈ C∞0 (Ui ∪ Uj), so that the κij-transfer(φiAφj)κij is a pseudo-differential operator of order m on Rn, hence belongingto L(Hs(Rn),Hs−m(Rn)) by Theorem 2.6.11 . Thereby φiAφj = ((φiAφj)κij )κ−1

ij

belongs to L(Hs(M),Hs−m(M)), and consequently A ∈ L(Hs(M),Hs−m(M)).Thus we have the result Ψm(M) ⊂ L(Hs(M),Hs−m(M)).

In order to get (ii), also inclusions

[Ψm(M),Diff1(M)] ⊂ Ψm(M) and [Ψm(M),Diff0(M)] ⊂ Ψm−1(M)

must be proven. Let A ∈ Ψm(M) and D ∈ Diff1(M), and fix an arbitrary chart(U, κ) and arbitrary functions φ, ψ ∈ C∞0 (U). By a direct calculation,

φ[A,D]ψ = [φAψ,D]− φA[ψ, D]− [φ,D]Aψ,

so that(φ[A,D]ψ)κ = [(φAψ)κ, Dκ]− (φA[ψ, D])κ − ([φ,D]Aψ)κ

by (5.2). Because A ∈ Ψm(M), Theorem 5.1.4 implies that the operators on theright-hand side of the previous equality are pseudo-differential operators of orderm−(1−deg(D)) on Rn. Therefore [A,D] ∈ Ψm−(1−deg(D))(M), proving implication(i)⇒(ii).

((ii)⇒(i)) Let A : C∞(M) → C∞(M) satisfy condition (ii), and fix a chart(U, κ) on M and φ, ψ ∈ C∞0 (U). To get (i), we have to prove that (φAψ)κ ∈OpSm(Rn), which by Theorem 5.1.4 follows, if we can prove the following variantof condition (ii):

(ii)′ For any s ∈ R and for any sequence C = (Cj)∞j=0 ⊂ OpS1loc(R

n) of partialdifferential operators, it holds that{

B0 = (φAψ)κ ∈ L(Hs(Rn),Hs−m(Rn)),Bk+1 = [Bk, Ck] ∈ L(Hs(Rn),Hs−m+dC,k(Rn)),

where dC,k =∑k

j=0(1− deg(Cj)).

Indeed, B0 = (φAψ)κ ∈ L(Hs(Rn),Hs−m(Rn)) by (ii). Let χ ∈ C∞0 (κ(U)) suchthat χ(x) = 1 in a neighbourhood of the compact set supp(φκ) ∪ supp(ψκ) ⊂ Rn.Then define D = (Dj)∞j=0 ⊂ Diff1(M) so that Dj |M\U = 0, and Dj |U = (χCj)κ−1 .

5.4. Toroidal commutator characterisation 423

Then dD,k ≥ dC,k, and due to condition (ii), we get

Bk+1 = [Bk, Ck] = [Bk, χCk] = [(Bk)κ−1 , Dk]κ∈ L(Hs(Rn),Hs−m+dD,k(Rn))⊂ L(Hs(Rn),Hs−m+dC,k(Rn)),

verifying (ii)′. Hence A ∈ Ψm(M). �

Remark 5.3.3 (Ψ(M) is a ∗-algebra). The pseudo-differential operators on M forma ∗-algebra

Ψ(M) =⋃

m∈R

Ψm(M),

where Ψm(M) ⊂ L(Hs(M),Hs−m(M)). It is true that Diffk(M) ⊂ Ψk(M),and Ψ(M) has properties analogous to those of the algebra Diff(M). Especially,[Ψm1(M),Ψm2(M)] ⊂ Ψm1+m2−1(M).

Exercise 5.3.4 (Paracompact manifolds). Generalise the result in Lemma 5.3.2 tosmooth paracompact manifolds. Recall that a Hausdorff topological space is calledparacompact if every open cover admits an open locally finite subcover.

5.4 Toroidal commutator characterisation

On the torus Tn = Rn/Zn one has a well-defined global symbol analysis of periodicoperators from the class Ψ(Tn × Zn), as developed in Chapter 4. In this section, asone application of the commutator characterisation Theorem 5.3.1, we provide aproof of the equality of operator classes Ψ(Tn × Zn) = Ψ(Tn). For the equality ofoperator classes Ψ(Tn × Zn) = Ψ(Tn × Rn) see Corollary 4.6.13 that was obtainedusing the extension and periodisation techniques. However, a similar applicationof Theorem 5.3.1 will be important on Lie groups (Theorem 10.7.7) where theseother techniques are not readily available.

For 1 ≤ j, k ≤ n, let us define the operators Lj and Rk acting on periodicpseudo-differential operators by

Lj(A) := [Dxj , A] and Rk(A) := [A, ei2πxkI].

Moreover, for α, β ∈ Nn0 , let

Lβ(A) = Lβ11 · · ·Lβn

n (A),Rα(A) = Rα1

1 · · ·Rαnn (A)

(here the letters L and R refer to “left” and “right”). By the composition Theorem4.7.10, if A ∈ Op(Sm(Tn × Zn)) then Lj(A) ∈ Op(Sm(Tn × Zn)) and Rk(A) :=


Op(Sm−1(Tn × Zn)). Let us explain how these commutators arise, and why theyare so interesting. First,

DxjσA(x, ξ) = Dxj

(e−i2πx·ξAeξ(x)

)= e−i2πx·ξ (

DxjA− ξjA

)eξ(x)

= e−i2πx·ξ (Dxj A−ADxj

)eξ(x)

= e−i2πx·ξ [Dxj

, A]eξ(x)

= σLj(A)(x, ξ).

Thus the partial derivative with respect to xj of the symbol σA leads to the symbolof the commutator [Dxj

, A]. As regards to the difference, the situation is almostsimilar (where vk stands for the standard kth unit basis vector of Zn):

�ξkσA(x, ξ) = σA(x, ξ + vk)− σA(x, ξ)

= e−i2πx·(ξ+vk)Aeξ+vk(x)− e−i2πx·ξAeξ(x)

= e−i2πxke−i2πx·ξ (A ◦

(ei2πxkI

)−

(ei2πxkI

)◦A

)eξ(x)

= e−i2πxke−i2πx·ξ [A, ei2πxkI

]eξ(x)

= e−i2πxkσRk(A)(x, ξ).

The minor asymmetry in �ξkσA(x, ξ) = e−i2πxkσRk(A)(x, ξ) caused by e−i2πxk is

due to the nature of differences. In [12, p. 46-49] the pseudo-differential operatorsof certain degree have been characterised using analogues of these commutatorsrepresenting the differentiations of a symbol. As before, the approach on Tn issomewhat simpler:

Theorem 5.4.1 (Commutator characterisation on Tn). Let A be a linear operatoron C∞(Tn). Then we have A ∈ Op(Sm(Tn × Zn)) if and only if

∀α, β ∈ Nn0 : LβRα(A) ∈ L(Hm−|α|(Tn),H0(Tn)). (5.3)

Thus the classes of periodic pseudo-differential operators and pseudo-differentialoperators on Tn coincide. More precisely, for any m ∈ R it holds that

OpSm(Tn × Zn) = Ψm(Tn). (5.4)

Proof of Theorem 5.4.1 for the T1 case. The “only if”-part is trivial by Propo-sition 4.2.3, since Theorem 4.7.10 implies that L1(B) ∈ Op(Sl(T1 × Z1)) andR1(B) ∈ Op(Sl−1(T1 × Z1)) for any B ∈ Op(Sl(T1 × Z1)).

For the “if”-part we have to estimate �αξ ∂β

xσA(x, ξ). Let us define opera-tor R′1 by R′1(A) = e−i2πxR1(A). Because u(x) �→ e−i2πxu(x) is a homeomor-phism from Hs(T1) to Hs(T1) for every s ∈ R, it is true that Lα

1 R′α1 (A) ∈L(Hm−α(T1),H0(T1)). Notice that

|u(x)| ≤∑ξ∈Z1

|u(ξ)| ≤

⎡⎣ ∑ξ∈Z1

〈ξ〉−2

⎤⎦1/2 ⎡⎣ ∑ξ∈Z1

|ξu(ξ)|2⎤⎦1/2

= C‖∂xu‖H0(T1).

5.4. Toroidal commutator characterisation 425

Using this we get∣∣�αξ ∂β

xσA(x, ξ)∣∣ ≤ C‖�α

ξ ∂β+1x σA(x, ξ)‖H0(T1)

= C‖e−ξLβ+11 R′α1 (A)eξ‖H0(T1)

≤ C‖e−ξI‖L(H0,H0) ‖Lβ+11 R′α1 (A)‖L(Hm−α,H0) ‖eξ‖Hm−α

= C‖Lβ+11 R′α1 (A)‖L(Hm−α(T1),H0(T1)) 〈ξ〉m−α

≤ Cαβ 〈ξ〉m−α.

This completes the proof of the one-dimensional case. �

General proof of Theorem 5.4.1. Let us first prove the inclusion OpSm(Tn×Zn)⊂Ψm(Tn). We know by Proposition 4.2.3 that

OpSm(Tn × Zn) ⊂ L(Hs(Tn),Hs−m(Tn)).

Therefore by Theorem 5.3.1, in view of Proposition 4.2.3 it suffices to verify that

[OpSm(Tn × Zn),Diff1(Tn)] ⊂ OpSm(Tn × Zn) (5.5)

and that[Op Sm(Tn × Zn),Diff0(Tn)] ⊂ OpSm−1(Tn × Zn). (5.6)

This is true due to the asymptotic expansion of the composition of two periodicpseudo-differential operators (see Theorem 4.7.10). However, we present a briefindependent and instructive proof of the inclusion (5.5). Let A ∈ OpSm(Tn) andlet X ∈ Diff1(Tn), Xx = φ(x)∂xk

(1 ≤ k ≤ n). Now

σ[A,X](x, ξ) = i2πξk

∑η

[σA(x, ξ + η)− σA(x, ξ)]φ(η)ei2πx·η

−φ(x)(∂xkσA)(x, ξ).

Notice that

σA(x, ξ + η)− σA(x, ξ)

=n∑

j=1

ηj−(sgn(ηj)+1)/2∑ωj=(sgn(ηj)−1)/2

sgn(ηj)�ξjσA(x, ξ + η1δ1 + · · ·+ ηj−1δj−1 + ωjδj),

where

sgn(ηj) =

⎧⎨⎩ 1, ηj > 0,0, ηj = 0,−1, ηj < 0,

and there are at most∑

j |ηj | <√

n(1 + ‖η‖) non-zero terms in the sum. Hence,applying the ordinary Leibniz formula with respect to x, the discrete Leibnizformula with respect to ξ (Lemma 3.3.6), the inequality of Peetre (Proposition


3.3.31) and Lemma 4.2.1, we get

|�αξ ∂β

xσ[A,X](x, ξ)|≤ Cαβ,φ,r(1 + ‖ξ‖)

∑η

(1 + ‖ξ‖)m−(|α|+1)√

n(1 + ‖η‖)|m−(|α|+1)|+|β|+1−r

+Cαβ,φ(1 + ‖ξ‖)m−|α|.

By choosing r > |m− (|α|+ 1)|+ |β|+ 2, the series above converges, so that

|�αξ ∂β

xσ[A,X](x, ξ)| ≤ Cαβ(1 + ‖ξ‖)m−|α|.

Hence [A,X] ∈ OpSm(Tn × Zn). Similarly, but with less effort, one proves (5.6).Thus A ∈ Ψm(Tn) by Theorem 5.3.1, and hence also (5.3) by Theorem 5.3.1.

Now assume that A ∈ Ψm(Tn). We have to prove that σA satisfies inequalitiesdefining the toroidal symbol class Sm

1,0(Tn×Zn) in (4.6) from Definition 4.1.7. We

also note that A ∈ Ψm(Tn) implies (5.3) by Theorem 5.3.1. Let us define thetransformation Rk by Rk(A) := e−i2πxkRk(A), and set Rα := Rα1

1 · · · Rαnn , so that

�αξ ∂β

xσA(x, ξ) = σRαLβ(A)(x, ξ).

By Theorem 5.3.1, we have RαLβ(A) ∈ L(Hm−|α|(Tn),H0(Tn)). Notice that

|u(x)| ≤∑

ξ

|u(ξ)| ≤

⎡⎣∑ξ

(1 + ‖ξ‖)−2s

⎤⎦1/2 ⎡⎣∑ξ

(1 + ‖ξ‖)2s|u(ξ)|2⎤⎦1/2

= Cs‖u‖Hs(Tn) ≤ Cs

⎛⎝ ∑|γ|≤s

‖∂γxu‖2H0(Tn)

⎞⎠1/2

,

where s ∈ N satisfies 2s > n = dim(Tn). Using this we get

∣∣�αξ ∂β

xσA(x, ξ)∣∣ ≤ C

⎛⎝ ∑|γ|≤s

‖�αξ ∂β+γ

x σA(·, ξ)‖2H0(Tn)

⎞⎠1/2

= C

⎛⎝ ∑|γ|≤s

‖e−ξRαLβ+γ(A)eξ‖2H0(Tn)

⎞⎠1/2

≤ C

⎛⎝ ∑|γ|≤s

‖e−ξI‖2L(H0(Tn))

×‖RαLβ+γ(A)‖2L(Hm−|α|(Tn),H0(Tn))‖eξ‖2Hm−|α|(Tn)

)1/2

≤ Cαβ (1 + ‖ξ‖)m−|α|,


Part III

Representation Theoryof Compact Groups

We might call the traditional topology and measure theory by the name “commuta-tive geometry”, referring to the commutative function algebras; “non-commutativegeometry” would then refer to the study of non-commutative algebras. Althoughthe function algebras considered in the sequel are still commutative, the non-commutativity of the corresponding groups is the characteristic feature of Parts IIIand IV.

Here we present the necessary material on compact groups and their rep-resentations. The presentation gradually increases the availability of topologicaland differentiable structures, thus tracing the development from general compactgroups to linear Lie groups. Moreover, we present additional material on the Hopfalgebras joining together the material of this part to the analysis of algebras fromChapter D. Nevertheless, we tried to make the exposition self-contained, providingreferences to Part I when necessary. If the reader wants to gain more profoundknowledge of Lie groups, Lie algebras and their representation, there are manyexcellent monographs available on different aspects of these theories at differentlevels, for example [9, 19, 20, 31, 36, 37, 38, 47, 48, 49, 50, 51, 58, 61, 64, 65, 73,74, 88, 123, 127, 147, 148, 149, 154], to mention a few.

Chapter 6

Groups

6.1 Introduction

Loosely speaking, groups encode symmetries of (geometric) objects: if we considera space X with some specific structure (e.g., a Riemannian manifold), a symmetryof X is a bijection f : X → X preserving the natural involved structure (e.g., theRiemannian metric) – here, the compositions and inversions of symmetries yieldnew symmetries. In a handful of assumptions, the concept of groups captures theessential properties of wide classes of symmetries, and provides powerful tools forrelated analysis.

Perhaps the first non-trivial group that mankind encountered was the set Zof integers; with the usual addition (x, y) �→ x + y and “inversion” x �→ −x this isa basic example of a group. Intuitively, a group is a set G that has two mappingsG×G→ G and G→ G generalising the properties of the integers in a simple andnatural way.

We start by defining the groups, and we study the mappings preserving suchstructures, i.e., group homomorphisms. Of special interest are representations,that is those group homomorphisms that have values in groups of invertible linearoperators on vector spaces. Representation theory is a key ingredient in the theoryof groups.

In this framework we study analysis on compact groups, foremost measuretheory and Fourier transform. Remarkably, for a compact group G there exists aunique translation-invariant linear functional on C(G) corresponding to a proba-bility measure. We shall construct this so-called Haar measure, closely related tothe Lebesgue measure of a Euclidean space. We shall also introduce Fourier seriesof functions on a group.

Groups having a smooth manifold structure (with smooth group operations)are called Lie groups, and their representation theory is especially interesting. Left-invariant first-order partial differential operators on such a group can be identifiedwith left-invariant vector fields on the group, and the corresponding set called theLie algebra is studied.

430 Chapter 6. Groups

Finally, we introduce Hopf algebras and study the Gelfand theory related tothem.Remark 6.1.1 (–morphisms). If X, Y are spaces with the same kind of algebraicstructure, the set Hom(X, Y ) of homomorphisms consists of mappings f : X → Yrespecting the structure. Bijective homomorphisms are called isomorphisms. Ho-momorphisms f : X → X are called endomorphisms of X, and their set is de-noted by End(X) := Hom(X, X). Isomorphism-endomorphisms are called auto-morphisms, and their set is Aut(X) ⊂ End(X). If there exist the zero-elements0X , 0Y in respective algebraic structures X, Y , the null space or the kernel off ∈ Hom(X, Y ) is

Ker(f) := {x ∈ X : f(x) = 0Y } .

Sometimes algebraic structures might have, say, topology, and then the homomor-phisms are typically required to be continuous. Hence, for instance, a homomor-phism f : X → Y between Banach spaces X, Y is usually assumed to be continu-ous and linear, denoted by f ∈ L(X, Y ), unless otherwise mentioned; for short, letL(X) := L(X, X). The assumptions in theorems etc. will still be explicitly stated.

Conventions

N is the set of positive integers,Z+ = N,N0 = N ∪ {0},Z is the set of integers,Q the set of rational numbers,R the set of real numbers,C the set of complex numbers, andK ∈ {R, C}.

6.2 Groups without topology

We start with groups without complications, without assuming supplementaryproperties. This choice helps in understanding the purely algebraic ideas, andonly later we will mingle groups with other structures, e.g., topology.

Definition 6.2.1 (Groups). A group consists of a set G having an element e = eG ∈G and endowed with mappings

((x, y) �→ xy) : G×G→ G,

(x �→ x−1) : G→ G

satisfying

x(yz) = (xy)z,

ex = x = xe,

x x−1 = e = x−1x,

6.2. Groups without topology 431

for all x, y, z ∈ G. We may freely write xyz := x(yz) = (xy)z; the element e ∈ G iscalled the neutral element, and x−1 is the inverse of x ∈ G. If the group operationsare implicitly known, we may say that G is a group. If xy = yx for all x, y ∈ Gthen G is called commutative (or Abelian).

Example. Let us give some examples of groups:

1. (Symmetric group). Let G = {f : X → X | f bijection}, where X = ∅; this isa group with operations (f, g) �→ f ◦ g, f �→ f−1. This group G of bijectionson X is called the symmetric group of X, and it is non-commutative whenever|X| ≥ 3, where |X| is the number of elements of X. The neutral element isidX = (x �→ x) : X → X.

2. The sets Z, Q, R and C are commutative groups with operations (x, y) �→x + y, x �→ −x. The neutral element is 0 in each case.

3. Any vector space is a commutative group with operations (x, y) �→ x + y,x �→ −x; the neutral element is 0.

4. (Automorphism group Aut(V )). Let V be a vector space. The set Aut(V )of invertible linear operators V → V forms a group with operations (A,B) �→AB, A �→ A−1; this group is non-commutative when dim(V ) ≥ 2. The neutralelement is I = (v �→ v) : V → V .

5. Sets Q× := Q \ {0}, R× := R \ {0}, C× := C \ {0} (more generally, invertibleelements of a unital ring) form multiplicative groups with operations (x, y) �→xy (ordinary multiplication) and x �→ x−1 (as usual). The neutral element is1 in each case.

6. (Affine group Aff(V )). The set

Aff(V ) = {Aa = (v �→ Av + a) : V → V | A ∈ Aut(V ), a ∈ V }

of affine mappings forms a group with operations defined to be (Aa, Bb) �→(AB)Ab+a, Aa �→ (A−1)A−1a; this group is non-commutative when dim(V ) ≥1. The neutral element is I0.

7. (Product group). If G and H are groups then G × H has a natural groupstructure:

((g1, h1), (g2, h2)) �→ (g1h1, g2h2), (g, h) �→ (g−1, h−1).

The neutral element is eG×H := (eG, eH).

Exercise 6.2.2. Let G be a group and x, y ∈ G. Prove:

(a) (x−1)−1 = x.(b) If xy = e then y = x−1.(c) (xy)−1 = y−1x−1.

Definition 6.2.3 (Finite groups). If a group has finitely many elements it is saidto be finite.


Example. The symmetry group of a set consisting of n elements is called thepermutation group of n elements. Such group is a finite group and has n! = 1·2 · · ·nelements.

Definition 6.2.4 (Notation). Let G be a group, x ∈ A, A,B ⊂ G and n ∈ Z+. Wewrite

xA := {xa | a ∈ A} ,

Ax := {ax | a ∈ A} ,

AB := {ab | a ∈ A, b ∈ B} ,

A0 := {e} ,

A−1 :={a−1 | a ∈ A

},

An+1 := AnA,

A−n := (An)−1.

Definition 6.2.5 (Subgroups H < G, and normal subgroups H � G). A set H ⊂ Gis a subgroup of a group G, denoted by H < G, if

e ∈ H, xy ∈ H and x−1 ∈ H

for all x, y ∈ H (hence H is a group with the inherited operations). A subgroupH < G is called normal in G if

xH = Hx

for all x ∈ G; then we write H � G.

Remark 6.2.6. With the inherited operations, a subgroup is a group. Normal sub-groups are the well-behaving ones, as exemplified later in Proposition 6.2.16 andTheorem 6.2.20. In some books normal subgroups of G are called normal divisorsof G.

Exercise 6.2.7. Let H < G. Show that if H ⊂ x−1Hx for every x ∈ G, thenH � G.

Exercise 6.2.8. Let H < G. Show that H � G if and only if H = x−1Hx for everyx ∈ G.

Example. Let us collect some instances and facts about subgroups:

1. (Trivial subgroups). We always have normal trivial subgroups {e} � G andG � G. Subgroups of a commutative group are always normal.

2. (Centre of a group). The centre Z(G) � G, where

Z(G) := {z ∈ G | ∀x ∈ G : xz = zx}.

Thus, the centre is the collection of elements that commute with all elementsof the group.


3. If F < H and G < H then F ∩G < H.

4. If F < H and G � H then FG < H.

5. {Ia | a ∈ V } � Aff(V ).

6. The following two examples will be of crucial importance later so we formulatethem as Remarks 6.2.9 and 6.2.10.

Remark 6.2.9 (Groups GL(n, R), O(n), SO(n)). We have

SO(n) < O(n) < GL(n, R) ∼= Aut(Rn),

where the groups consist of real n×n-matrices: GL(n, R) is the real general lineargroup consisting of invertible real matrices (i.e., determinant non-zero); O(n) isthe orthogonal group, where the matrix columns (or rows) form an orthonormalbasis for Rn (so that AT = A−1 for A ∈ O(n), det(A) ∈ {−1, 1}); SO(n) is thespecial orthogonal group, the group of rotation matrices of Rn around the origin,so that

SO(n) = {A ∈ O(n) : det(A) = 1}.Remark 6.2.10 (Groups GL(n, C), U(n), SU(n)). We have

SU(n) < U(n) < GL(n, C) ∼= Aut(Cn),

where the groups consist of complex n × n-matrices: GL(n, C) is the complexgeneral linear group consisting of invertible complex matrices (i.e., determinantnon-zero); U(n) is the unitary group, where the matrix columns (or rows) form anorthonormal basis for Cn (so that A∗ = A−1 for A ∈ U(n), |det(A)| = 1); SU(n)is the special unitary group,

SU(n) = {A ∈ U(n) : det(A) = 1}.

Remark 6.2.11. The mapping

(z �→(z)) : C→ C1×1

identifies complex numbers with complex (1 × 1)-matrices. Thereby the complexunit circle group {z ∈ C : |z| = 1} is identified with the group U(1).

Definition 6.2.12 (Right quotient G/H). Let H < G. Then

x ∼ y ⇐⇒ xH = yH

defines an equivalence relation on G, as can be easily verified. The (right) quotientof G by H is the set

G/H = {xH | x ∈ G} .

Notice that xH = yH if and only if x−1y ∈ H.


Similarly, we can define

Definition 6.2.13 (Left quotient H\G). Let H < G. Then

x ∼ y ⇐⇒ Hx = Hy

defines an equivalence relation on G. The (left) quotient of G by H is the set

H\G = {Hx | x ∈ G} .

Notice that Hx = Hy if and only if x−1y ∈ H.

Remark 6.2.14 (Right for now). We will deal mostly with the right quotient G/Hin Part III. However, we note that in Part IV we will actually need more the leftquotient H\G. It should be a simple exercise for the reader to translate all theresults from “right” to “left”. Indeed, simply replacing the side from which thesubgroup acts from right to left, and changing all the words from “right” to “left”should do the job since the situation is completely symmetric. The reason for ourchange is that once we choose to identify the Lie algebras with the left-invariantvector fields in Part IV it leads to a more natural analysis of pseudo-differentialoperators on left quotients. However, because our intuition about division may bebetter suited to the notation G/H we chose to explain the basic ideas for the rightquotients, keeping in mind that the situation with the left quotients is completelysymmetric.Remark 6.2.15. It is often useful to identify the points xH ∈ G/H with the setsxH ⊂ G. Also, for A ⊂ G we naturally identify the sets

AH = {ah : a ∈ A, h ∈ H} ⊂ G and{aH : a ∈ A} = {{ah : h ∈ H} : a ∈ A} ⊂ G/H.

This provides a nice way to treat the quotient G/H.

Proposition 6.2.16 (When is G/H a group?). Let H � G be normal. Then thequotient G/H can be endowed with the group structure

(xH, yH) �→ xyH, xH �→ x−1H.

Proof. The operations are well-defined mappings (G/H) × (G/H) → G/H andG/H → G/H, respectively, since

xHyHH�G= xyHH

HH=H= xyH,

and

(xH)−1 = H−1x−1 H−1=H= Hx−1 H�G= x−1H.

The group axioms follow, since by simple calculations we have

(xH)(yH)(zH) = xyzH,

(xH)(eH) = xH = (eH)(xH),

(x−1H)(xH) = H = (xH)(x−1H).

Notice that eG/H = eGH = H. �


Definition 6.2.17 (Torus Tn as a quotient group). The quotient Tn := Rn/Zn iscalled the (flat) n-dimensional torus.

Definition 6.2.18 (Homomorphisms and isomorphisms). Let G, H be groups. Amapping φ : G → H is called a homomorphism (or a group homomorphism),denoted by φ ∈ Hom(G, H), if

φ(xy) = φ(x)φ(y)

for all x, y ∈ G. The kernel of φ ∈ Hom(G, H) is

Ker(φ) := {x ∈ G | φ(x) = eH} .

A bijective homomorphism φ ∈ Hom(G, H) is called an isomorphism, denoted byφ : G ∼= H.

Remark 6.2.19. Group homomorphisms are the natural mappings between groups,preserving the group operations. Notice especially that for a group homomorphismφ : G→ H it holds that

φ(eG) = eH and φ(x−1) = φ(x)−1

for all x ∈ G.

Example. Examples of homomorphisms:

1. (x �→ eH) ∈ Hom(G, H).

2. For y ∈ G, (x �→ y−1xy) ∈ Hom(G, G).

3. If H � G then x �→ xH is a surjective homomorphism G→ G/H.

4. For x ∈ G, (n �→ xn) ∈ Hom(Z, G).

5. If φ ∈ Hom(F,G) and ψ ∈ Hom(G, H) then ψ ◦ φ ∈ Hom(F,H).

6. T1 ∼= U(1) ∼= SO(2).

Theorem 6.2.20. Let φ ∈ Hom(G, H) and K = Ker(φ). Then:

1. φ(G) < H.2. K � G.3. ψ(xK) := φ(x) defines a group isomorphism ψ : G/K → φ(G).

Thus we have the commutative diagram

Gφ−−−−→ H

x�→xK

⏐⏐( F⏐⏐y �→y

G/Kψ:G/K∼=φ(G)−−−−−−−−−→ φ(G).


Proof. Let x, y ∈ G. Now φ(G) is a subgroup of H, because

eH = φ(eG) ∈ φ(G),φ(x)φ(y) = φ(xy) ∈ φ(G),

φ(x−1)φ(x) = φ(x−1x) = φ(eG)= eH

= · · · = φ(x)φ(x−1);

notice that φ(x)−1 = φ(x−1). If a, b ∈ Ker(φ) then

φ(eG) = eH ,

φ(ab) = φ(a)φ(b) = eHeH = eH ,

φ(a−1) = φ(a)−1 = e−1H = eH ,

so that K = Ker(φ) < G. If moreover x ∈ G then

φ(x−1Kx) = φ(x−1) φ(K) φ(x) = φ(x)−1 {eH} φ(x) = {eH} ,

meaning x−1Kx ⊂ K. Thus K � G by Exercise 6.2.8. By Proposition 6.2.16, G/Kis a group (with the natural operations). Since φ(xa) = φ(x) for every a ∈ K,ψ = (xK �→ φ(x)) : G/K → φ(G) is a well-defined surjection. Furthermore,

ψ(xyK) = φ(xy) = φ(x)φ(y) = ψ(xK)ψ(yK),

thus ψ ∈ Hom(G/K, φ(G)). Finally,

ψ(xK) = ψ(yK) ⇐⇒ φ(x) = φ(y) ⇐⇒ x−1y ∈ K ⇐⇒ xK = yK,

so that ψ is injective. �Exercise 6.2.21 (Universality of the permutation groups). Let G be a finite group.Show that there is a set X with finitely many elements such that G is isomorphicto a subgroup of the symmetric group of X.

6.3 Group actions and representations

Spaces can be studied by examining their symmetry groups. On the other hand, itis fruitful to investigate groups when they are acting as symmetries of some nicelystructured spaces. Next we study actions of groups on sets. Especially interestinggroup actions are the linear actions on vector spaces, providing the machinery oflinear algebra – this is the fundamental idea in the representation theory of groups.

Definition 6.3.1 (Transitive actions). An action of a group G on a set M = ∅ is amapping

((x, p) �→ x · p) : G×M →M,

6.3. Group actions and representations 437

for which {x · (y · p) = (xy) · p,

e · p = p

for all x, y ∈ G and p ∈M ; the action is transitive if

∀p, q ∈M ∃x ∈ G : x · q = p.

If M is a vector space and the mapping p �→ x · p is linear for each x ∈ G, theaction is called linear.

Remark 6.3.2. To be precise, our action G×M →M in Definition 6.3.1 should becalled a left action, to make a difference to the right actions M ×G→M , whichare defined in the obvious way. When G acts on M , it is useful to think of G as a(sub)group of symmetries of M . Transitivity means that M is highly symmetric:there are enough symmetries to move any point to any other point.Example. Let us present some examples of actions:

1. On a vector space V , the group Aut(V ) acts linearly by (A, v) �→ Av.2. If φ ∈ Hom(G, H) then G acts on H by (x, y) �→ φ(x)y. Especially, G acts

on G transitively by (x, y) �→ xy.3. The rotation group SO(n) acts transitively on the sphere Sn−1 := {x =

(xj)nj=1 ∈ Rn | x2

1 + · · ·+ x2n = 1} by (A, x) �→ Ax.

4. If H < G and ((x, p) �→ x · p) : G×M →M is an action then the restriction((x, p) �→ x · p) : H ×M →M is an action.

Definition 6.3.3 (Isotropy subgroup). Let ((x, p) �→ x · p) : G ×M → M be anaction. The isotropy subgroup of q ∈M is

Gq := {x ∈ G | x · q = q} .

That is, Gq ⊂ G contains those symmetries that fix the point q ∈M .

Theorem 6.3.4. Let ((x, p) �→ x · p) : G × M → M be a transitive action. Letq ∈M . Then the isotropy subgroup Gq is a subgroup for which

fq := (xGq �→ x · q) : G/Gq →M

is a bijection.

Remark 6.3.5. If Gq � G then G/Gq is a group; otherwise the quotient is just aset. Notice also that the choice of q ∈M here is essentially irrelevant.Example. Let G = SO(3), M = S2, and q ∈ S2 be the north pole (i.e., q =(0, 0, 1) ∈ R3). Then Gq < SO(3) consists of the rotations around the vertical axis(passing through the north and south poles). Since SO(3) acts transitively on S2,we get a bijection SO(3)/Gq → S2. The reader may think how A ∈ SO(3) movesthe north pole q ∈ S2 to Aq ∈ S2.


Proof of Theorem 6.3.4. Let a, b ∈ Gq. Then

e · q = q,

(ab) · q = a · (b · q) = a · q = q,

a−1 · q = a−1 · (a · q) = (a−1a) · q = e · q = q,

so that Gq < G. Let x, y ∈ G. Since

(xa) · q = x · (a · q) = x · q,

f = (xGq �→ x · q) : G/Gq →M is a well-defined mapping. If x · q = y · q then

(x−1y) · q = x−1 · (y · q) = x−1 · (x · q) = (x−1x) · q = e · q = q,

i.e., x−1y ∈ Gq, that is xGq = yGq; hence f is injective. Take p ∈ M . By transi-tivity, there exists x ∈ G such that x · q = p. Thereby f(xGq) = x · q = p, i.e., fis surjective. �Remark 6.3.6. If an action ((x, p) �→ x · p) : G ×M → M is not transitive, it isoften reasonable to study only the orbit of q ∈M , defined by

G · q := {x · q | x ∈ G} .

Now((x, p) �→ x · p) : G× (G · q)→ (G · q)

is transitive, and (x · q �→ xGq) : G · q → G/Gq is a bijection. Notice that eitherG · p = G · q or (G · p) ∩ (G · q) = ∅; thus the action of G cuts M into a disjointunion of “slices” (orbits).

Definition 6.3.7 (Unitary groups). Let (v, w) �→ 〈v, w〉H be the inner product of acomplex vector space H. Recall that the adjoint A∗ ∈ Aut(H) of A ∈ Aut(H) isdefined by

〈A∗v, w〉H := 〈v,Aw〉H.

The unitary group of H is

U(H) := {A ∈ Aut(H) | ∀v, w ∈ H : 〈Av,Aw〉H = 〈v, w〉H} ,

i.e., U(H) contains the unitary linear bijections H → H. Clearly A∗ = A−1 forA ∈ U(H). The unitary matrix group for Cn is

U(n) :={A = (aij)n

i,j=1 ∈ GL(n, C) | A∗ = A−1}

,

see Remark 6.2.10; here A∗ = (aji)ni,j=1 = A−1, i.e.,

n∑k=1

akiakj = δij .


Definition 6.3.8 (Representations). A representation of a group G on a vectorspace V is any φ ∈ Hom(G, Aut(V )); the dimension of φ is dim(φ) := dim(V ).A representation ψ ∈ Hom(G,U(H)) is called a unitary representation, and ψ ∈Hom(G, U(n)) is called a unitary matrix representation.

Remark 6.3.9. The main idea here is that we can study a group G by using linearalgebraic tools via representations φ ∈ Hom(G, Aut(V )).Remark 6.3.10. There is a bijective correspondence between the representationsof G on V and linear actions of G on V . Indeed, ifφ ∈ Hom(G, Aut(V )) then

((x, v) �→ φ(x)v) : G× V → V

is an action of G on V . Conversely, if ((x, v) �→ x · v) : G × V → V is a linearaction then

(x �→ (v �→ x · v)) ∈ Hom(G, Aut(V ))

is a representation of G on V .Example. Let us give some examples of representations:

1. If G < Aut(V ) then (A �→ A) ∈ Hom(G, Aut(V )).2. If G < U(H) then (A �→ A) ∈ Hom(G,U(H)).3. There is always the trivial representation (x �→ I) ∈ Hom(G, Aut(V )).4. (Representations πL and πR). Let F(G) = CG, i.e., the complex vector space

of functions f : G → C. Let us define left and right regular representationsπL, πR ∈ Hom(G, Aut(F(G))) by

(πL(y)f)(x) := f(y−1x),(πR(y)f)(x) := f(xy)

for all x, y ∈ G.5. Let us identify the complex (1 × 1)-matrices with the complex numbers by

the mapping ((z) �→ z) : C1×1 → C. Then U(1) is identified with the unitcircle {z ∈ C : |z| = 1}, and (x �→ eix·ξ) ∈ Hom(Rn,U(1)) for all ξ ∈ Rn.

6. Analogously, (x �→ ei2πx·ξ) ∈ Hom(Rn/Zn,U(1)) for all ξ ∈ Zn.7. Let φ ∈ Hom(G, Aut(V )) and ψ ∈ Hom(G, Aut(W )), where V,W are vector

spaces over the same field. Then

φ⊕ ψ = (x �→ φ(x)⊕ ψ(x)) ∈ Hom(G, Aut(V ⊕W )),

φ⊗ ψ|G = (x �→ φ(x)⊗ ψ(x)) ∈ Hom(G, Aut(V ⊗W )),

where V ⊕W is the direct sum and V ⊗W is the tensor product space.8. If φ = (x �→ (φ(x)ij)n

i,j=1) ∈ Hom(G, GL(n, C)) then the conjugate φ = (x �→(φ(x)ij)n

i,j=1) ∈ Hom(G, GL(n, C)).


Definition 6.3.11 (Invariant subspaces and irreducible representations). Let V bea vector space and A ∈ End(V ). A subspace W ⊂ V is called A-invariant if

AW ⊂W,

where AW = {Aw : w ∈ W}. Let φ ∈ Hom(G, Aut(V )). A subspace W ⊂ V iscalled φ-invariant if W is φ(x)-invariant for all x ∈ G (abbreviated φ(G)W ⊂W );moreover, φ is irreducible if the only φ-invariant subspaces are the trivial subspaces{0} and V .

Remark 6.3.12 (Restricted representations). If W ⊂ V is φ-invariant for φ ∈Hom(G, Aut(V )), we may define the restricted representation

φ|W ∈ Hom(G, Aut(W ))

by φ|W (x)w := φ(x)w. If φ is unitary then its restriction is also unitary.

Lemma 6.3.13. Let φ ∈ Hom(G,U(H)). Let W ⊂ H be a φ-invariant subspace.Then its orthocomplement

W⊥ = {v ∈ H | ∀w ∈W : 〈v, w〉H = 0}

is also φ-invariant.

Proof. If x ∈ G, v ∈W⊥ and w ∈W then

〈φ(x)v, w〉H = 〈v, φ(x)∗w〉H = 〈v, φ(x)−1w〉H = 〈v, φ(x−1)w〉H = 0,

meaning that φ(x)v ∈W⊥. �Definition 6.3.14 (Direct sums). Let V be an inner product space and let {Vj}j∈J

be some family of its mutually orthogonal subspaces (i.e., 〈vi, vj〉V = 0 if vi ∈ Vi,vj ∈ Vj and i = j). The (algebraic) direct sum of {Vj}j∈J is the subspace

W =⊕j∈J

Vj := span⋃j∈J

Vj .

If Aj ∈ End(Vj) then let us define

A =⊕j∈J

Aj ∈ End(W )

by Av := Ajv for all j ∈ J and v ∈ Vj . If φj ∈ Hom(G, Aut(Vj)) then we define

φ =⊕j∈J

φj ∈ Hom(G, Aut(W ))

by φ|Vj= φj for all j ∈ J , i.e., φ(x) :=

⊕j∈J φj(x) for all x ∈ G.


Remark 6.3.15. In a sense, irreducible representations are the building blocks ofrepresentations. Given a representation of a group, a fundamental task is to find itsinvariant subspaces, and describe the representation as a direct sum of irreduciblerepresentations. To reach this goal, we often have to assume some extra conditions,e.g., of algebraic or topological nature.

Theorem 6.3.16 (Reducing finite-dimensional representations).Let φ ∈ Hom(G,U(H)) be finite-dimensional. Then φ is a direct sum of irreducibleunitary representations.

Proof (by induction). The claim is true for dim(H) = 1, since then the only sub-spaces of H are the trivial ones. Suppose the claim is true for representations ofdimension n or less. Suppose dim(H) = n+1. If φ is irreducible, there is nothing toprove. Hence assume that there exists a non-trivial φ-invariant subspace W ⊂ H.Then also the orthocomplement W⊥ is φ-invariant by Lemma 6.3.13. Due to theφ-invariance of the subspaces W and W⊥, we may define restricted representa-tions φ|W ∈ Hom(G,U(W )) and φ|W⊥ ∈ Hom(G,U(W⊥)). Hence H = W ⊕W⊥

and φ = φ|W ⊕ φ|W⊥ . Moreover, dim(W ) ≤ n and dim(W⊥) ≤ n; the proofis complete, since unitary representations up to dimension n are direct sums ofirreducible unitary representations by the induction hypothesis. �Remark 6.3.17. By Theorem 6.3.16, finite-dimensional unitary representations canbe decomposed nicely. More precisely, if φ ∈ Hom(G,U(H)) is finite-dimensionalthen

H =k⊕

j=1

Wj , φ =k⊕

j=1

φ|Wj ,

where each φ|Wj ∈ Hom(G,U(Wj)) is irreducible.

Definition 6.3.18 (Equivalent representations). A linear mapping A : V → Wis an intertwining operator between representations φ ∈ Hom(G, Aut(V )) andψ ∈ Hom(G, Aut(W)), denoted by A ∈ Hom(φ, ψ), if

Aφ(x) = ψ(x)A

for all x ∈ G, i.e., if the diagram

Vφ(x)−−−−→ V

A

⏐⏐( ⏐⏐(A

Wψ(x)−−−−→ W

commutes for every x ∈ G. If A ∈ Hom(φ, ψ) is invertible then φ and ψ are saidto be equivalent, denoted by φ ∼ ψ.

Remark 6.3.19. Always 0 ∈ Hom(φ, ψ), and Hom(φ, ψ) is a vector space. Moreover,if A ∈ Hom(φ, ψ) and B ∈ Hom(ψ, ξ) then BA ∈ Hom(φ, ξ).


Exercise 6.3.20. Let G be a finite group and let F(G) be the vector space offunctions f : G→ C. Let ∫

G

f dμG :=1|G|

∑x∈G

f(x),

when f ∈ F(G). Let us endow F(G) with the inner product

〈f, g〉L2(μG) :=∫

G

f g dμG.

Define πL, πR : G→ Aut(F(G)) by

(πL(y) f)(x) := f(y−1x),(πR(y) f)(x) := f(xy).

Show that πL and πR are equivalent unitary representations.

Exercise 6.3.21. Let G be non-commutative and |G| = 6. Endow F(G) with theinner product given in Exercise 6.3.20. Find the πL-invariant subspaces and giveorthogonal bases for them.

Exercise 6.3.22 (Torus Tn). Let us endow the n-dimensional torus Tn := Rn/Zn

with the quotient group structure and with the Lebesgue measure. Let πL, πR :Tn → L(L2(Tn)) be defined by

(πL(y) f)(x) := f(x− y),(πR(y) f)(x) := f(x + y)

for almost every x ∈ Tn. Show that πL and πR are equivalent reducible unitaryrepresentations. Describe the minimal πL- and πR-invariant subspaces containingthe function x �→ ei2πx·ξ, where ξ ∈ Zn.

Remark 6.3.23. One of the main results in the representation theory of groups isSchur’s Lemma 6.3.25, according to which the intertwining space Hom(φ, φ) maybe rather trivial. The most of the work for such a result is carried out in the proofof the following Proposition 6.3.24:

Proposition 6.3.24. Let φ ∈ Hom(G, Aut(Vφ)) and ψ ∈ Hom(G, Aut(Vψ)) be irre-ducible. If A ∈ Hom(φ, ψ) then either A = 0 or A : Vφ → Vψ is invertible.

Proof. The image AVφ ⊂ Vψ of A is ψ-invariant, because

ψ(G) AVφ = A φ(G)Vφ = AVφ,

so that either AVφ = {0} or AVφ = Vψ, as ψ is irreducible. Hence either A = 0 orA is a surjection.


The kernel Ker(A) = {v ∈ Vφ | Av = 0} is φ-invariant, since

A φ(G) Ker(A) = ψ(G) A Ker(A) = ψ(G) {0} = {0} ,

so that either Ker(A) = {0} or Ker(A) = Vφ, as φ is irreducible. Hence either Ais injective or A = 0.

Thus either A = 0 or A is bijective. �

Corollary 6.3.25 (Schur’s Lemma (finite-dimensional (1905))).Let φ ∈ Hom(G, Aut(V )) be irreducible and finite-dimensional. Then Hom(φ, φ) =CI = {λI | λ ∈ C}.

Proof. Let A ∈ Hom(φ, φ). The finite-dimensional linear operator A has an eigen-value λ ∈ C, i.e., λI − A : V → V is not invertible. On the other hand, λI − A ∈Hom(φ, φ), so that λI −A = 0 by Proposition 6.3.24. �

Corollary 6.3.26 (Representations of commutative groups). Let G be a commu-tative group. Irreducible finite-dimensional representations of G are one-dimen-sional.

Proof. Let φ ∈ Hom(G, Aut(V )) be irreducible, dim(φ) <∞. Due to the commu-tativity of G,

φ(x)φ(y) = φ(xy) = φ(yx) = φ(y)φ(x)

for all x, y ∈ G, so that φ(G) ⊂ Hom(φ, φ). By Schur’s Lemma 6.3.25, Hom(φ, φ) =CI. Hence if v ∈ V then

φ(G)span{v} = span{v},

i.e., span{v} is φ-invariant. Therefore either v = 0 or span{v} = V . �

Corollary 6.3.27. Let φ ∈ Hom(G,U(Hφ)) and ψ ∈ Hom(G,U(Hψ)) be finite-dimensional. Then φ ∼ ψ if and only if there exists an isometric isomorphismB ∈ Hom(φ, ψ).

Remark 6.3.28 (Isometries). An isometry f : M → N between metric spaces(M,dM ) and (N, dN ) satisfies dN (f(x), f(y)) = dM (x, y) for all x, y ∈M .

Proof of Corollary 6.3.27. The “if”-part is trivial. Assume that φ ∼ ψ. Recall thatthere are direct sum decompositions

φ =m⊕

j=1

φj , ψ =n⊕

k=1

ψk,

where φj , ψk are irreducible unitary representations on Hφj,Hψk

, respectively.Now n = m, since φ ∼ ψ. Moreover, we may arrange the indices so that φj ∼ ψj

for each j. Choose invertible Aj ∈ Hom(φj , ψj). Then A∗j is invertible, and A∗j ∈


Hom(ψj , φj): if x ∈ G, v ∈ Hφjand w ∈ Hψj

then

〈A∗jψj(x)w, v〉Hφ= 〈w,ψj(x)∗Ajv〉Hψ

= 〈w,ψj(x−1)Ajv〉Hψ

= 〈w,Ajφj(x−1)v〉Hψ

= 〈φj(x−1)∗A∗jw, v〉Hφ

= 〈φj(x)A∗jw, v〉Hφ.

Thereby A∗jAj ∈ Hom(φj , φj) is invertible. By Schur’s Lemma 6.3.25, A∗jAj = λjI,where λj = 0. Let v ∈ Hφj

such that ‖v‖Hφ= 1. Then

λ = λ‖v‖2Hφ= 〈λv, v〉Hφ

= 〈A∗jAjv, v〉Hφ= 〈Ajv,Ajv〉Hψ

= ‖Ajv‖2Hψ> 0,

so that we may define Bj := λ−1/2Aj ∈ Hom(φj , ψj). Then the mapping Bj :Hφj → Hψj is an isometry, B∗j Bj = I. Finally, define

B :=m⊕

j=1

Bj .

Clearly, B : Hφ → Hψ is an isometry, bijection, and B ∈ Hom(φ, ψ). �

We have now dealt with groups in general. In the sequel, by specialisingto certain classes of groups, we will obtain fruitful ground for further results inrepresentation theory.

Chapter 7

Topological Groups

A topological group is a natural amalgam of topological spaces and groups: it is aHausdorff space with continuous group operations. Topology adds a new flavourto representation theory. Especially interesting are compact groups, where group-invariant probability measures exist. Moreover, nice-enough functions on a com-pact group have Fourier series expansions, which generalise the classical Fourierseries of periodic functions.

7.1 Topological groups

Next we marry topology to groups.

Definition 7.1.1 (Topological groups). A group and a topological space G is calleda topological group if {e} ⊂ G is closed and if the group operations

((x, y) �→ xy) : G×G→ G,

(x �→ x−1) : G→ G

are continuous.

Remark 7.1.2. The reader may wonder why we assumed that {e} ⊂ G is closed –actually, this condition is left out in some other definitions for a topological group.Notice that the good property brought by this assumption is that the topologicalgroups become even Hausdorff spaces (see Exercise 7.1.3), which appeals to thosewho work in analysis.Example. In the following, when not specified, the topologies and the group oper-ations are the usual ones:

1. Any group G endowed with the so-called discrete topology P(G) = {U : U ⊂G} is a topological group.

2. Z, Q, R and C are topological groups when the group operation is the additionand the topology is as usual.

446 Chapter 7. Topological Groups

3. Q×, R×, C× are topological groups when the group operation is the multi-plication and the topology is as usual.

4. Topological vector spaces are topological groups with vector addition: sucha space is both a vector space and a topological Hausdorff space such thatthe vector space operations are continuous.

5. Let X be a Banach space. The set AUT(X) := Aut(X) ∩ L(X) of invertiblebounded linear operators X → X forms a topological group with respect tothe norm topology.

6. Subgroups of topological groups are topological groups.7. If G and H are topological groups then G×H is a topological group. Actually,

Cartesian products always preserve the topological group structure.

Exercise 7.1.3. Show that a topological group is a Hausdorff space.

Lemma 7.1.4. Let G be a topological group and y ∈ G. Then

x �→ xy, x �→ yx, x �→ x−1

are homeomorphisms G→ G.

Proof. The mapping

(x �→ xy) : Gx�→(x,y)→ G×G

(a,b)�→ab→ G

is continuous as a composition of continuous mappings. Its inverse mapping (x �→xy−1) : G → G is also continuous; hence this is a homeomorphism. Similarly,(x �→ yx) : G → G is a homeomorphism. By definition, the group operation(x �→ x−1) : G→ G is continuous, and it is its own inverse. �Corollary 7.1.5. If U ⊂ G is open and S ⊂ G then SU, US,U−1 ⊂ G are open.

Proposition 7.1.6. Let G be a topological group. If H < G then H < G. If H � Gthen H � G.

Proof. Let H < G. Trivially e ∈ H ⊂ H. Now

H H ⊂ HH = H,

where the inclusion is due to the continuity of the mapping ((x, y) �→ xy) : G×G→G. The continuity of the inversion (x �→ x−1) : G→ G gives

H−1 ⊂ H−1 = H.

Thus H < G.Let H � G, y ∈ G. Then

yH = yH = Hy = Hy;

notice how homeomorphisms (x �→ yx), (x �→ xy) : G→ G were used. �

7.1. Topological groups 447

Remark 7.1.7. Let H < G and S ⊂ G. For analysis on the quotient space G/H,let us recall Remark 6.2.15: the mapping (x �→ xH) : G→ G/H identifies the sets

SH = {sh : s ∈ S, h ∈ H} ⊂ G,

{sH : s ∈ S} = {{sh : h ∈ H} : s ∈ S} ⊂ G/H.

Definition 7.1.8 (Quotient topology on G/H). Let G be a topological group, H <G. The quotient topology of G/H is

τG/H := {{uH : u ∈ U} : U ⊂ G open} ;

in other words, τG/H is the strongest (i.e., largest) topology for which the quotientmap (x �→ xH) : G→ G/H is continuous. If U ⊂ G is open, we may identify setsUH ⊂ G and {uH : u ∈ H} ⊂ G/H.

Proposition 7.1.9. Let G be a topological group and H < G. Then a functionf : G/H → C is continuous if and only if (x �→ f(xH)) : G→ C is continuous.

Proof. If f ∈ C(G/H) then (x �→ f(xH)) ∈ C(G), since it is obtained by compos-ing f and the continuous quotient map (x �→ xH) : G→ G/H.

Now suppose (x �→ f(xH)) ∈ C(G). Take open V ⊂ C. Then U := (x �→f(xH))−1(V ) ⊂ G is open, so that U ′ := {uH : u ∈ U} ⊂ G/H is open. Trivially,f(U ′) = V . Hence f ∈ C(G/H). �Proposition 7.1.10 (When is G/H Hausdorff?). Let G be a topological group andH < G. Then G/H is a Hausdorff space if and only if H is closed.

Proof. If G/H is a Hausdorff space then H = (x �→ xH)−1({H}) ⊂ G is closed,because the quotient map is continuous and {H} ⊂ G/H is closed.

Next suppose H is closed. Take xH, yH ∈ G/H such that xH = yH. ThenS := ((a, b) �→ a−1b)−1(H) ⊂ G×G is closed, since H ⊂ G is closed and ((a, b) �→a−1b) : G×G→ G is continuous. Now (x, y) ∈ S. Take open sets U � x and V � ysuch that (U × V ) ∩ S = ∅. Then the sets

U ′ := {uH : u ∈ U} ⊂ G/H,

V ′ := {vH : v ∈ V } ⊂ G/H

are disjoint and open such that xH ∈ U ′ and yH ∈ V ′. Thus G/H is Hausdorff.�

Theorem 7.1.11 (When is G/H a topological group?). Let G be a topological groupand H � G. Then

((xH, yH) �→ xyH) : (G/H)× (G/H)→ G/H,

(xH �→ x−1H) : G/H → G/H

are continuous. Moreover, G/H is a topological group if and only if H is closed.


Proof. We know already that the operations in the theorem are well-defined groupoperations, because H is normal in G. Recall Remark 7.1.7, how we may identifycertain subsets of G with subsets of G/H. Then a neighbourhood of the pointxyH ∈ G/H is of the form UH for some open U ⊂ G, U � xy. Take open U1 � xand U2 � y such that U1U2 ⊂ U . Then

(xH)(yH) ⊂ (U1H)(U2H) = U1U2H ⊂ UH,

so that ((xH, yH) �→ xyH) : (G/H)× (G/H)→ G/H is continuous. A neighbour-hood of the point x−1H ∈ G/H is of the form V H for some open V ⊂ G, V � x−1.But V −1 � x is open, and (V −1)−1 = V , so that (xH �→ x−1H) : G/H → G/H iscontinuous.

Notice that eG/H = H. If G/H is a topological group, then

H = (x �→ xH)−1{eG/H

}⊂ G

is closed. On the other hand, if H � G is closed then

(G/H) \ {eG/H} ∼= (G \H)H ⊂ G

is open, i.e., {eG/H} ⊂ G/H is closed. �Definition 7.1.12 (Continuous homomorphisms). Let G1, G2 be topological groups.Let

HOM(G1, G2) := Hom(G1, G2) ∩ C(G1, G2),

i.e., the set of continuous homomorphisms G1 → G2.

Remark 7.1.13. By Theorem 7.1.11, closed normal subgroups of G correspondbijectively to continuous surjective homomorphisms from G to some other topo-logical group (up to isomorphism).Remark 7.1.14. Let us recall some topological concepts: A topological space isconnected if the only subsets which are both closed and open are the empty setand the whole space. A non-connected space is called disconnected. The componentof a point x in a topological space is the largest connected subset containing x.

Proposition 7.1.15. Let G be a topological group and Ce ⊂ G the component of e.Then Ce � G is closed.

Proof. Components are always closed, and e ∈ Ce by definition. Since Ce ⊂ Gis connected, also Ce × Ce ⊂ G × G and is connected. By the continuity of thegroup operations, CeCe ⊂ G and C−1

e ⊂ G are connected. Since e = ee ∈ CeCe,we have CeCe ⊂ Ce. And since e = e−1 ∈ C−1

e , also C−1e ⊂ Ce. Take y ∈ G.

Then y−1Cey ⊂ G is connected, by the continuity of (x �→ y−1xy) : G→ G. Nowe = y−1ey ∈ y−1Cey, so that y−1Cey ⊂ Ce; Ce is normal in G. �Proposition 7.1.16. Let ((x, p) �→ x · p) : G ×M → M be a continuous action ofG on M , and let q ∈M . If Gq and G/Gq are connected then G is connected.

7.2. Representations of topological groups 449

Proof. Suppose G is disconnected and Gq is connected. Then there are non-emptydisjoint open sets U, V ⊂ G such that G = U ∪ V . The sets

U ′ := {uGq : u ∈ U} ⊂ G/Gq,

V ′ := {vGq : v ∈ V } ⊂ G/Gq

are non-empty and open, and G/Gq = U ′ ∪ V ′. Take u ∈ U and v ∈ V . As acontinuous image of a connected set, uGq = (x �→ ux)(Gq) ⊂ G is connected;moreover u = ue ∈ uGq; thereby uGq ⊂ U . In the same way we see that vGq ⊂ V .Hence U ′ ∩ V ′ = ∅, so that G/Gq is disconnected. �Corollary 7.1.17 (When is a group connected?). If G is a topological group, H < Gis connected and G/H is connected then G is connected.

Proof. Using the notation of Proposition 7.1.16, let M = G/H, q = H and x · p =xp, so that Gq = H and G/Gq = G/H. �Exercise 7.1.18 (Groups SO(n), SU(n) and U(n) are connected). Show thatSO(n), SU(n) and U(n) are connected for every n ∈ Z+. How about O(n)?

Exercise 7.1.19 (Finiteness of connected components). Prove that a compact topo-logical group can have only finitely many connected components. Consequently,conclude that a discrete compact group is finite.

7.2 Representations of topological groups

Definition 7.2.1 (Strongly continuous representations). Let G be a topologicalgroup and H be a Hilbert space. A representation φ ∈ Hom(G,U(H)) is stronglycontinuous if

(x �→ φ(x)v) : G→ His continuous for all v ∈ H.

Remark 7.2.2. The strong continuity in Definition 7.2.1 means that the mapping(x �→ φ(x)) : G → L(H) is continuous, when L(H) ⊃ U(H) is endowed with thestrong operator topology:

Ajstrongly→ A

definition⇐⇒ ∀v ∈ H : ‖Ajv −Av‖H → 0.

Why should we not endow U(H) with the operator norm topology (which is evenstronger, i.e., a larger topology)? The reason is that there are interesting unitaryrepresentations, which are continuous in the strong operator topology, but not inthe operator norm topology. This phenomenon is exemplified by Exercise 7.2.3:

Exercise 7.2.3. Let us define πL : Rn → U(L2(Rn)) by

(πL(y)f)(x) := f(x− y)

for almost every x ∈ Rn. Show that πL is strongly continuous, but not normcontinuous.


Definition 7.2.4 (Topologically irreducible representations). A strongly continuousφ ∈ Hom(G,U(H)) is called topologically irreducible if the only closed φ-invariantsubspaces are the trivial ones {0} and H.

Exercise 7.2.5. Let V be a topological vector space and let W ⊂ V be an A-invariant subspace, where A ∈ Aut(V ) is continuous. Show that the closure W ⊂ Vis also A-invariant.

Definition 7.2.6 (Cyclic representations and cyclic vectors). A strongly continuousφ ∈ Hom(G,U(H)) is called a cyclic representation if

span φ(G)v ⊂ H

is dense for some v ∈ H; such v is called a cyclic vector.

Example. If φ ∈ Hom(G,U(H)) is topologically irreducible then any non-zerov ∈ H is cyclic. Indeed, if V := span φ(G)v then φ(G)V ⊂ V and consequentlyφ(G)V ⊂ V , so that V is φ-invariant. If v = 0 then V = H, because of thetopological irreducibility.

Definition 7.2.7 (Representation as a direct sum). A Hilbert space H is a directsum of closed subspaces (Hj)j∈J , denoted by

H =⊕j∈J

Hj

if the subspace family is pairwise orthogonal and the linear span of the set ∪j∈JHj

is dense in H. Then the vectors in H have a unique orthogonal series expansion,more precisely

∀x ∈ H ∀j ∈ J ∃!xj ∈ Hj : x =∑j∈J

xj , ‖x‖2H =∑j∈J

‖xj‖2H .

If φ ∈ Hom(G,U(H)) and each Hj is φ-invariant then φ is said to be the directsum

φ =⊕j∈J

φ|Hj

where φ|Hj= (x �→ φ(x)v) ∈ Hom(G,U(Hj)).

Proposition 7.2.8 (Decomposition of strongly continuous representations). Let φ ∈Hom(G,U(H)) be strongly continuous. Then

φ =⊕j∈J

φ|Hj,

where each φ|Hjis cyclic.

7.3. Compact groups 451

Proof. Let J be the family of all closed φ-invariant subspaces V ⊂ H for whichφ|V is cyclic. Let

S ={

s ⊂ J∣∣∣ ∀V,W ∈ s : V = W or V⊥W

}.

It is easy to see that {{0}} ∈ S, so that S = ∅. Let us introduce a partial orderon S by inclusion:

s1 ≤ s2definition⇐⇒ s1 ⊂ s2.

The chains in S have upper bounds: if R ⊂ S is a chain then r ≤ ∪s∈R s ∈ S forall r ∈ R. Therefore by Zorn’s Lemma, there exists a maximal element t ∈ S. Let

V :=⊕W∈t

W.

To get a contradiction, suppose V = H. Then there exists v ∈ V ⊥ \ {0}. Sincespan(φ(G)v) is φ-invariant, its closure W0 is also φ-invariant (see Exercise 7.2.5).Clearly W0 ⊂ V ⊥ = V ⊥, and φ|W0 has cyclic vector v, yielding

s := t ∪ {W0} ∈ S,

where t ≤ s ≤ t. This contradicts the maximality of t; thus V = H. �Exercise 7.2.9. Fill in the details in the proof of Proposition 7.2.8.

Exercise 7.2.10. Assuming thatH is separable, prove Proposition 7.2.8 by ordinaryinduction (without resorting to Zorn’s Lemma).

7.3 Compact groups

Definition 7.3.1 ((Locally) compact groups). A topological group is a (locally)compact group if it is (locally) compact as a topological space.

Remark 7.3.2. We have the following properties:

1. Any group G with the discrete topology is a locally compact group; then Gis a compact group if and only if it is finite.

2. Q, Q× are not locally compact groups;R, R×, C, C× are locally compact groups, but non-compact.

3. A normed vector space is a locally compact group if and only if it is finite-dimensional.

4. O(n), SO(n), U(n),SU(n) are compact groups.5. GL(n) is a locally compact group, but non-compact.6. If G, H are locally compact groups then G×H is a locally compact group.7. If {Gj}j∈J is a family of compact groups then

∏j∈J Gj is a compact group.

8. If G is a compact group and H < G is closed then H is a compact group.


Proposition 7.3.3. Let ((x, p) �→ x · p) : G×M → M be a continuous action of acompact group G on a Hausdorff space M . Let q ∈M . Then the mapping

f := (xGq �→ x · q) : G/Gq → G · q

is a homeomorphism.

Proof. We already know that f is a well-defined bijection. We need to show thatf is continuous. An open subset of G · q is of the form V ∩ (G · q), where V ⊂Mis open. Since the action is continuous, also (x �→ x · q) : G→M is continuous, sothat U := (x �→ x · q)−1(V ) ⊂ G is open. Thereby

f−1(V ∩ (G · q)) = {xGq : x ∈ U} ⊂ G/Gq

is open. Thus f is continuous. The space G is compact and the quotient map(x �→ xGq) : G → G/Gq is continuous and surjective, so that G/Gq is compact.From general topology we know that a continuous bijection from a compact spaceto a Hausdorff space is a homeomorphism (see Proposition A.12.7). �

Corollary 7.3.4. If G is compact, φ ∈ HOM(G, H) and K = Ker(φ) then

ψ := (xK �→ φ(x)) ∈ HOM(G/K, φ(G))

is a homeomorphism.

Proof. Using the notation of Proposition 7.3.3, we have M = H, q = eH , x · p =φ(x)p, so that Gq = K, G/Gq = G/K, G · q = φ(G), ψ = f . �

Remark 7.3.5. What could happen if we drop the compactness assumption inCorollary 7.3.4? If G and H are Banach spaces, φ ∈ L(G, H) is compact anddim(φ(G)) =∞ then ψ = (x + Ker(φ) �→ φ(x)) : G/Ker(φ)→ φ(G) is a boundedlinear bijection, but ψ−1 is not bounded! But if φ ∈ L(G, H) is a bijection thenφ−1 is bounded by the Open Mapping Theorem! (Theorem B.4.31)

Definition 7.3.6 (Uniform continuity on a topological group). Let G be a topolog-ical group. A function f : G → C is uniformly continuous if for every ε > 0 thereexists open U � e such that

∀x, y ∈ G : x−1y ∈ U ⇒ |f(x)− f(y)| < ε.

Exercise 7.3.7. Under what circumstances is a polynomial p : R → C uniformlycontinuous? Show that if a continuous function f : R → C is periodic or vanishesoutside a bounded set then it is uniformly continuous.

Theorem 7.3.8. If G is a compact group and f ∈ C(G) then f is uniformly con-tinuous.

7.4. Haar measure and integral 453

Proof. Take ε > 0. Define the open disk D(z, r) := {w ∈ C : |w − z| < r}, wherez ∈ C, r > 0. Since f is continuous, the set

Vx := f−1(D(f(x), ε)) � x

is open. Then x−1Vx � ee = e is open, so that there exist open sets U1,x, U2,x � esuch that U1,xU2,x ⊂ x−1Vx, by the continuity of the group multiplication. DefineUx := U1,x ∩ U2,x. Since {xUx : x ∈ G} is an open cover of the compact space G,there is a finite subcover {xjUxj}n

j=1. Now the set

U :=n⋂

j=1

Uxj � e

is open. Suppose x, y ∈ G such that x−1y ∈ U . There exists k ∈ {1, . . . , n} suchthat x ∈ xkUxk

, so that

x, y ∈ xU ⊂ xkUxkUxk

⊂ xkx−1k Vxk

= Vxk,

yielding|f(x)− f(y)| ≤ |f(x)− f(xk)|+ |f(xk)− f(y)| < 2ε. �

Exercise 7.3.9. Let G be a compact group, x ∈ G and A = {xn}∞n=1. Show thatA < G.

7.4 Haar measure and integral

On a group, it would be natural to integrate with respect to measures that areinvariant under the group operations: consider, e.g., the Lebesgue integral on Rn.However, it is not obvious whether there exist such invariant integrals in general.Next we will show that on a compact group there exists a unique probabilityfunctional, which corresponds to the so-called Haar measure.

Definition 7.4.1 (Positive functionals). Let X be a compact Hausdorff space andK ∈ {R, C}. Then C(X, K) is a Banach space over K with the norm

f �→ ‖f‖C(X,K) := maxx∈X

|f(x)|.

Its dual C(X, K)′ = L(C(X, K), K) consists of the bounded linear functionalsC(X, K)→ K, and is endowed with the Banach space norm

L �→ ‖L‖C(X,K)′ := supf∈C(X,K): ‖f‖C(X,K)≤1

|Lf |.

A functional L : C(X, K)→ C is called positive if Lf ≥ 0 whenever f ≥ 0.

Exercise 7.4.2. Let X be a compact Hausdorff space. Show that a positive linearfunctional L : C(X, R)→ R is bounded.


By the Riesz Representation Theorem (see Theorem C.4.65), if L ∈ C(X, K)′

is positive then there exists a unique positive Borel regular measure μ on X suchthat

Lf =∫

X

f dμ

for every f ∈ C(X, K); moreover, μ(X) = ‖L‖C(X,K)′ . For short, C(X) :=C(X,C).Note that this is different from Chapter A (see, e.g., Exercise A.6.6) where we wroteC(X) for C(X, R).

In the sequel, we shall construct a unique positive normalised translation-invariant measure on G. More precisely, we shall prove the following result:

Theorem 7.4.3 (Haar functional). Let G be a compact group. There exists a uniquepositive linear functional Haar ∈ C(G)′ such that

Haar(f) = Haar(x �→ f(yx)),Haar(1) = 1,

for all y ∈ G, where 1 = (x �→ 1) ∈ C(G). Moreover, this Haar functional satisfies

Haar(f) = Haar(x �→ f(xy))= Haar(x �→ f(x−1)).

Remark 7.4.4 (Haar measure and integral). By the Riesz Representation The-orem (see Theorem C.4.65), the Haar functional begets a unique Borel regularprobability measure μG such that

Haar(f) =∫

G

f dμG =∫

G

f(x) dμG(x).

This μG is called the Haar measure of G. Often the Haar measure is implicitlyassumed, and we may write, e.g.,∫

G

f(x) dx :=∫

G

f dμG.

Obviously, ∫G

1 dμG = μG(G) = 1,∫G

f(x) dx =∫

G

f(yx) dx

=∫

G

f(xy) dx =∫

G

f(x−1) dx.

Thus the Haar integral Haar(f) =∫

Gf(x) dx can be thought of as the most

natural average of f ∈ C(G). In practical applications we can know usually only


finitely many values of f , i.e., we are able to take only samples {f(x) : x ∈ S}for a finite set S ⊂ G. Then a natural idea for approximating Haar(f) would becomputing the finite sum ∑

x∈S

f(x) α(x),

where sampling weights α(x) ≥ 0 satisfy∑

x∈S α(x) = 1. Of course, such a sumis not usually invariant under the group operations. The problem is to find cleverchoices for sampling sets and weights, some sort of almost uniformly distributedunit mass on G is needed; for this end we shall introduce convolutions.

Example. If G is finite then∫G

f dμG =1|G|

∑x∈G

f(x).

Example (Haar measure on Tn). For Tn = Rn/Zn,∫Tn

f dμTn =∫

[0,1)n

f(x + Zn) dx,

i.e., integration with respect to the Lebesgue measure on [0, 1)n.

What follows is preparation for the proof of Theorem 7.4.3.

Definition 7.4.5 (Sampling measures). Let G be a compact group. A functionα : G→ [0, 1] is a sampling measure on G, α ∈ SMG, if

supp(α) := cl {a ∈ G : α(a) = 0} is finite and∑a∈G

α(a) = 1.

The set supp(α) ⊂ G is called the support of α. Since supp(α) is finite we also havesupp(α) = {a ∈ G : α(a) = 0} and, therefore, a sampling measure α ∈ SMG canbe regarded as a finitely supported probability measure on G, satisfying∫

G

f dα = α ∗ f(e) = f ∗ α(e),

where α(a) := α(a−1).

Remark 7.4.6. A sampling measure is nothing else but

α =∑

j

αjδaj,

where the sum is finite, aj ∈ G, δajis the Dirac measure at aj (i.e., a probability

measure supported at aj), and∑

j αj = 1.


Definition 7.4.7 (Convolutions). Let α, β ∈ SMG and f ∈ C(G, K). The convolu-tions

α ∗ β, α ∗ f, f ∗ β : G→ K

are defined by

α ∗ β(b) =∑a∈G

α(a)β(a−1b),

α ∗ f(x) =∑a∈G

α(a)f(a−1x),

f ∗ β(x) =∑b∈G

f(xb−1)β(b).

Notice that these summations are finite, as the sampling measures are supportedon finite sets.

Definition 7.4.8 (Semigroups and monoids). A semigroup is a non-empty set Swith an operation ((r, s) �→ rs) : S × S → S satisfying r(st) = (rs)t for allr, s, t ∈ S. A semigroup is commutative if rs = sr for all r, s ∈ S. Moreover, ifthere exists e ∈ S such that es = se = s for all s ∈ S then S is called a monoid.

Example. Z+ = {n ∈ Z : n > 0} is a commutative monoid with respect tomultiplication, and a commutative semigroup with respect to addition. If V is avector space then (End(V ), (A,B) �→ AB) is a monoid with e = I.

Lemma 7.4.9. The structure (SMG, (α, β) �→ α ∗ β) is a monoid.

Exercise 7.4.10. Prove Lemma 7.4.9. How is supp(α ∗ β) related to supp(α) andsupp(β)? In which case is SMG is a group? Show that SMG is commutative ifand only if G is commutative.

Lemma 7.4.11. If α ∈ SMG then (f �→ α ∗ f), (f �→ f ∗ α) ∈ L(C(G, K)) and

‖α ∗ f‖C(G,K) ≤ ‖f‖C(G,K),

‖f ∗ α‖C(G,K) ≤ ‖f‖C(G,K).

Moreover, α ∗ 1 = 1 = 1 ∗ α.

Proof. Trivially, α ∗ 1 = 1. Because (x �→ a−1x) : G → G is a homeomorphismand the summation is finite, α∗f ∈ C(G, K). Linearity of f �→ α∗f is clear. Next,

|α ∗ f(x)| ≤∑a∈G

α(a)|f(a−1x)| ≤∑a∈G

α(a)‖f‖C(G,K) = ‖f‖C(G,K).

Similar conclusions hold also for f ∗ α. �Definition 7.4.12. Let G be a compact group. Let us define a mapping pG :C(G, R)→ R by

pG(f) := max(f)−min(f).


Lemma 7.4.13. If f ∈ C(G, R) and α ∈ SMG then

min(f) ≤ min(α ∗ f) ≤ max(α ∗ f) ≤ max(f),

min(f) ≤ min(f ∗ α) ≤ max(f ∗ α) ≤ max(f),

so thatpG(α ∗ f) ≤ pG(f), pG(f ∗ α) ≤ pG(f).

Proof. Now

min(f) =∑a∈G

α(a) min(f) ≤ minx∈G

∑a∈G

α(a)f(a−1x) = min(α ∗ f),

max(α ∗ f) = maxx∈G

∑a∈G

α(a)f(a−1x) ≤∑a∈G

α(a) max(f) = max(f),

and clearly min(α ∗ f) ≤ max(α ∗ f). The proof for f ∗ α is symmetric. �

Exercise 7.4.14. Show that pG is a bounded seminorm on C(G, R).

Proposition 7.4.15. Let f ∈ C(G, R). For every ε > 0 there exist α, β ∈ SMG

such thatpG(α ∗ f) < ε, pG(f ∗ β) < ε.

Remark 7.4.16. This is the decisive stage in the construction of the Haar measure.The idea is that for a non-constant f ∈ C(G) we can find sampling measuresα, β that tame the oscillations of f so that α ∗ f and f ∗ β are almost constantfunctions. It will turn out that there exists a unique constant function Haar(f)1approximated by the convolutions of the type α∗f and f ∗β. In the sequel, noticehow compactness is exploited!

Proof. Let ε > 0. By Theorem 7.3.8, a continuous function on a compact group isuniformly continuous. Thus there exists an open set U ⊃ e such that |f(x)−f(y)| <ε, when x−1y ∈ U . We notice easily that if γ ∈ SMG then also |γ∗f(x)−γ∗f(y)| <ε, when x−1y ∈ U :

|γ ∗ f(x)− γ ∗ f(y)| =

∣∣∣∣∣∑a∈G

γ(a)(f(a−1x)− f(a−1y)

)∣∣∣∣∣≤

∑a∈G

γ(a)∣∣f(a−1x)− f(a−1y)

∣∣<

∑a∈G

γ(a) ε

= ε.


Now {xU : x ∈ G} is an open cover of the compact space G, hence having a finitesubcover {xjU}n

j=1. The set S := {xix−1j : 1 ≤ i, j ≤ n} has |S| ≤ n2 elements.

Define γ1 ∈ SMG by

γ1(a) =

{|S|−1, when a ∈ S,

0, otherwise.

Let γk+1 := γk ∗ γ1 ∈ SMG. Then

pG(γk+1 ∗ f) = max(γk+1 ∗ f)−min(γk+1 ∗ f)≤ max(γk+1 ∗ f)−min(γk ∗ f)

=1|S| max

x∈G

∑a∈S

γk ∗ f(a−1x)−min(γk ∗ f)

(�)<

1|S| [(|S| − 1) max(γk ∗ f) + [min(γk ∗ f) + ε]]−min(γk ∗ f)

=|S| − 1|S| pG(γk ∗ f) +

1|S|ε,

where the last inequality (�) was obtained by estimating |S| − 1 terms in the sumtrivially, and finally the remaining term was estimated by recalling the uniformcontinuity of γk ∗ f . Notice that (pG(γk ∗ f))∞k=1 ⊂ R is a non-increasing sequenceof non-negative numbers. Thus there exists the limit δ := lim

k→∞pG(γk ∗f) ≥ 0, and

δ ≤ |S| − 1|S| δ +

1|S|ε,

so that δ ≤ ε. Hence there exists k0 such that, say, pG(γk ∗ f) ≤ 2ε for everyk ≥ k0. This proves the claim. �Exercise 7.4.17. In the proof above, check the validity of inequality (�).

Definition 7.4.18. The following Corollary 7.4.19 defines the Haar functional Haar :C(G, R)→ R.

Corollary 7.4.19 (What is the Haar functional Haar(f)?). For f ∈ C(G, R) thereexists a unique constant function Haar(f)1 that belongs to the closure of the set

{α ∗ f : α ∈ SMG} ⊂ C(G, R).

Moreover, Haar(f)1 is the unique constant function that belongs to the closure ofthe set

{f ∗ β : β ∈ SMG} ⊂ C(G, R).

Proof. Pick any α1 ∈ SMG. Suppose we have chosen αk ∈ SMG. Let αk+1 :=γk ∗ αk, where γk ∈ SMG satisfies

pG(αk+1 ∗ f) = pG(γk ∗ (αk ∗ f)) < 2−k.


Nowmin(αk ∗ f) ≤ min(αk+1 ∗ f) ≤ max(αk+1 ∗ f) ≤ max(αk ∗ f),

so that there exists

limk→∞

min(αk ∗ f) = limk→∞

max(αk ∗ f) =: c1 ∈ R.

In the same way we may construct a sequence (βk)∞k=1 ⊂ SMG such that

limk→∞

min(f ∗ βk) = limk→∞

max(f ∗ βk) =: c2 ∈ R.

But

|c1 − c2| = ‖c11− c21‖C(G,R)

= ‖(c11− αk ∗ f) ∗ βk + αk ∗ (f ∗ βk − c21)‖C(G,R)

≤ ‖(c11− αk ∗ f) ∗ βk‖C(G,R) + ‖αk ∗ (f ∗ βk − c21)‖C(G,R)

≤ ‖c11− αk ∗ f‖C(G,R) + ‖f ∗ βk − c21‖C(G,R)

−−−−→k→∞

0.

Thus c1 = c2 ∈ R is unique, depending only on the function f . �Definition 7.4.20 (Haar functional of f ∈ C(G, C)). The Haar functional of f ∈C(G) is

Haar(f) := Haar(Re(f)) + i Haar(Im(f)),

where Re(f), Im(f) are the real and imaginary parts of f , respectively.

Let us now reformulate Theorem 7.4.3:

Theorem 7.4.21 (Haar). The Haar functional Haar : C(G) → C on a compactgroup G is the unique positive linear functional satisfying

Haar(1) = 1,

Haar(f) = Haar(x �→ f(yx)),

for all f ∈ C(G) and y ∈ G. Moreover,

Haar(f) = Haar(x �→ f(xy)) = Haar(x �→ f(x−1)).

Proof. By Definition 7.4.20 of Haar, it is enough to deal with real-valued functionshere. From the construction, it is clear that

f ≥ 0 ⇒ Haar(f) ≥ 0,

|Haar(f)| ≤ ‖f‖C(G),

Haar(λf) = λ Haar(f),Haar(1) = 1,

Haar(f) = Haar(x �→ f(yx)) = Haar(x �→ f(xy)).


Choose α, β ∈ SMG such that

‖α ∗ f −Haar(f)1‖C(G) < ε,

‖g ∗ β −Haar(g)1‖C(G) < ε.

Then

‖α ∗ (f + g) ∗ β − (Haar(f) + Haar(g))1‖C(G)

= ‖(α ∗ f −Haar(f)1) ∗ β + α ∗ (g ∗ β −Haar(g)1)‖C(G)

≤ ‖(α ∗ f −Haar(f)1) ∗ β‖C(G) + ‖α ∗ (g ∗ β −Haar(g)1)‖C(G)

≤ ‖α ∗ f −Haar(f)1‖C(G) + ‖g ∗ β −Haar(g)1‖C(G)

< 2ε,

so that Haar(f + g) = Haar(f) + Haar(g).Suppose L : C(G) → C is a positive linear functional such that L(1) = 1

and L(f) = L(x �→ f(yx)) for all f ∈ C(G) and y ∈ G. Let f ∈ C(G), ε > 0 andα ∈ SMG be as above. Then

|L(f)−Haar(f)| = |L(α ∗ f −Haar(f)1)|≤ ‖L‖C(G)′ ‖α ∗ f −Haar(f)1‖C(G)

< ‖L‖C(G)′ ε

yields the uniqueness L = Haar.Finally, (f �→ Haar(x �→ f(x−1))) : C(G)→ C is a positive linear translation-

invariant normalised functional, hence equal to Haar by the uniqueness. �Exercise 7.4.22. In the previous proof, many properties were declared clear, butthe reader is encouraged to verify the claims.

Definition 7.4.23 (Spaces Lp(μG)). For 1 ≤ p < ∞, the Lebesgue-p-space Lp(μG)on a topological group G is a special case of the Lebesgue-p-space from DefinitionC.4.6. Because the group is compact, by looking in local coordinates, we see fromExercise 1.3.33 that it is the completion of C(G) with respect to the norm

f �→ ‖f‖Lp(μG) :=(∫

G

|f |p dμG

)1/p

.

The space L∞(μG) is the usual Banach space of μG-essentially bounded functionswith the norm f �→ ‖f‖L∞(μG); on the closed subspace C(G) ⊂ L∞(μG) we have‖f‖C(G) = ‖f‖L∞(μG). Notice that Lp(μG) is a Banach space, but it is a Hilbertspace if and only if p = 2, having the inner product (f, g) �→ 〈f, g〉L2(μG) satisfying

〈f, g〉L2(μG) =∫

G

fg dμG

for f, g ∈ C(G).


Remark 7.4.24. We have now seen that for a compact group G there exists aunique translation-invariant probability functional on C(G), the Haar functional!We also know that it is enough to demand only either left- or right-invariance,since one follows from the other. Moreover, the Haar functional is also inversion-invariant. It must be noted that an inversion-invariant probability functional onC(G) is not necessarily translation-invariant: e.g., let us consider the Dirac pointmass δe at e ∈ G, for which the functional

f �→ f(e) =∫

G

f(x) dδe(x)

is inversion-invariant but clearly not translation-invariant (unless G = {e}). Nextwe observe that the Haar integral distinguishes continuous functions f, g ∈ C(G)in the sense that if

∫G|f − g| dμG = 0 then f = g:

Theorem 7.4.25. Let G be a compact group and f ∈ C(G). If∫

G|f | dμG = 0 then

f = 0.

Proof. The set U := f−1(C \ {0}) ⊂ G is open, since f is continuous and {0} ⊂ Cis closed. Suppose f = 0. Then U = ∅, and {xU : x ∈ G} is an open cover for G.By the compactness, there exists a subcover {xjU}n

j=1. Define g ∈ C(G) by

g(x) :=n∑

j=1

∣∣f(x−1j x)

∣∣ .

Now g(x) > 0 for all x ∈ G, so that there exists c := minx∈G g(x) > 0 by thecompactness. We use the normalisation, positivity and translation-invariance ofμG to obtain

0 < c =∫

G

c1 dμG ≤∫

G

g dμG = n

∫G

|f | dμG,

so that 0 <∫

G|f | dμG. �

Exercise 7.4.26. Let G, H be compact groups. Show that μG×H = μG×μH (i.e., theHaar measure of the product group is the product of the original Haar measures).

Exercise 7.4.27. LetMG denote the σ-algebra of the Haar-measurable sets on thecompact group G. Consider mappings m, p1, p2 : G×G→ G, where

m(x, y) = xy, p1(x, y) = x, p2(x, y) = y.

Show that they are Haar measurable (that is, (MG×G,MG)-measurable). More-over, show that

μG(E) = μG×G(m−1(E)) = μG×G(p−11 (E)) = μG×G(p−1

2 (E))

for all E ∈MG.


7.4.1 Integration on quotient spaces

We have already noticed that the good subgroups of a topological group are theclosed ones. Moreover, by now we know that a transitive action of a compacttopological group G on a Hausdorff space X begets a homeomorphism G/H ∼=X of compact Hausdorff spaces, where H is a closed subgroup of G; effectively,spaces G/H and X are the same. We are now about to show that for X thereexists a unique G-action-invariant probability functional on C(X), which mightbe called the Haar functional of the action; the corresponding measure on G/H willaccordingly be denoted by μG/H . We have seen that continuous functions on G/H(and hence on X) can be interpreted as continuous right-H-translation-invariantfunctions on G, i.e., f(xh) = f(x) for all x ∈ G and h ∈ H. Next we show howf ∈ C(G) casts a shadow fG/H ∈ C(G/H) in a natural way:

Lemma 7.4.28 (Projection PG/H). Let G be a compact group and H < G closed.If f ∈ C(G) then PG/Hf ∈ C(G) and fG/H ∈ C(G/H), where

fG/H(xH) = PG/Hf(x) :=∫

H

f(xh) dμH(h).

Furthermore, the projection PG/H : C(G) → C(G) is bounded, more precisely‖fG/H‖C(G/H) =

∥∥PG/Hf∥∥

C(G)≤ ‖f‖C(G).

Proof. First, H is a compact group having the Haar measure μH . The integrationin the definition is legitimate since fx := (h �→ f(xh)) ∈ C(H) for each x ∈ G. Ifx ∈ G and h0 ∈ H then

PG/Hf(xh0) =∫

H

fx(h0h) dμH(h) =∫

H

fx(h) dμH(h) = PG/Hf(x),

so that fG/H : G/H → C. Next we prove the continuity. Let ε > 0. A continuousfunction on a compact group is uniformly continuous, so that for f ∈ C(G) thereexists an open U � e such that

∀x, y ∈ G : xy−1 ∈ U ⇒ |f(x)− f(y)| < ε

(apparently, this slightly deviates from our definition of the uniform continuity;however, this is clearly equivalent). Suppose xy−1 ∈ U . Then

|PG/Hf(x)− PG/Hf(y)| =∣∣∣∣∫

H

f(xh)− f(yh) dμH(h)∣∣∣∣

≤∫

H

|f(xh)− f(yh)| dμH(h)

< ε,

so that PG/Hf ∈ C(G) and fG/H ∈ C(G/H). Finally,

|PG/Hf(x)| ≤∫

H

|f(xh)| dμH(h) ≤∫

H

‖f‖C(G) dμH(h) = ‖f‖C(G). �


Exercise 7.4.29. Show that the projection PG/H ∈ L(C(G)) extends uniquely toan orthogonal projection PG/H ∈ L(L2(μG)).

Theorem 7.4.30 (Existence of action-invariant measure on quotient spaces). Let((x, p) �→ x ·p) : G×M →M be a continuous transitive action of a compact groupG on a Hausdorff space M . Then there exists a unique Borel-regular probabilitymeasure μM on M which is action-invariant in the sense that∫

M

fM dμM =∫

M

fM (x · p) dμM (p)

for all fM ∈ C(M) and x ∈ G.

Proof. Given q ∈ M , we know that M ∼= G/Gq. Hence it is enough to dealwith M = G/H, where H < G is closed and the action is ((x, yH) �→ xyH) :G×G/H → G/H.

We first prove the existence of a G-action-invariant Borel regular proba-bility measure μG/H on the compact Hausdorff space G/H. Define HaarG/H :C(G/H)→ C by

HaarG/H(fG/H) :=∫

G

fG/H(xH) dμG(x).

Notice that

HaarG/H(fG/H) =∫

G

∫H

f(xh) dμH(h) dμG(x)

Fubini=∫

H

∫G

f(xh) dμG(x) dμH(h)

=∫

H

HaarG(f) dμH

= HaarG(f).

It is clear that HaarG/H is a bounded linear functional, and that

HaarG/H(1G/H) = HaarG(1G) = 1.

By the Riesz Representation Theorem (see Theorem C.4.65), there exists a uniqueBorel-regular probability measure μG/H on G/H such that

HaarG/H(fG/H) =∫

G/H

fG/H dμG/H

for all fG/H ∈ C(G/H). The action-invariance follows from the left-invariance ofthe functional HaarG: if g(y) := f(xy) for all y ∈ G then gG/H(yH) = fG/H(xyH)


and

HaarG/H(y �→ fG/H(xyH)) = HaarG/H(gG/H)= HaarG(g)= HaarG(f)= HaarG/H(fG/H).

Next we shall prove the uniqueness part. Suppose L : C(G/H) → C is anaction-invariant bounded linear functional for which L(1G/H) = 1. Recall themapping (f �→ fG/H) : C(G)→ C(G/H) in Lemma 7.4.28. Then

L(f) := L(fG/H)

defines a bounded linear functional L : C(G)→ C such that L(1G) = 1 and

L(y �→ f(xy)) = L(y �→ fG/H(xyH)) = L(fG/H) = L(f).

Hence L = HaarG by Theorem 7.4.21. Consequently,

L(fG/H) = L(f) = HaarG(f) = HaarG/H(fG/H),

yielding L = HaarG/H . �Remark 7.4.31. Let G be a compact group and H < G closed. From the proof ofTheorem 7.4.30 we see that∫

G

f dμG =∫

G/H

∫H

f(xh) dμH(h) dμG/H(xH)

for all f ∈ C(G).

Exercise 7.4.32. Let ωj(t) ∈ SO(3) denote the rotation of R3 by angle t ∈ R aroundthe jth coordinate axis, j ∈ {1, 2, 3}. Show that x ∈ SO(3) can be represented inthe form

x = x(φ, θ, ψ) = ω3(φ) ω2(θ) ω3(ψ)

where 0 ≤ φ, ψ < 2π and 0 ≤ θ ≤ π.

Exercise 7.4.33. Let the group G = SO(3) act on the sphere M = S2 by rotations.Let q = (0, 0, 1) ∈ M , i.e., q is the north pole. Show that Gq = {ω3(ψ) : 0 ≤ψ < 2π}. We know that the Lebesgue measure is rotation-invariant. Using thenormalised angular part of the Lebesgue measure of R3, deduce that here∫

G

f dμG =1

8π2

∫ 2π

0

∫ π

0

∫ 2π

0

f(x(φ, θ, ψ)) sin(θ) dψ dθ dφ,

i.e., dμSO(3) = 18π2 sin(θ) dψ dθ dφ.

We return to the example of SO(3) in Chapter 11.

7.5. Peter–Weyl decomposition of representations 465

7.5 Peter–Weyl decomposition of representations

In the sequel we apply the Haar integral in studying unitary representations ofcompact groups. The main result is the Peter–Weyl Theorem 7.5.14, leading to anatural Fourier series representation for functions on a compact group.

Exercise 7.5.1. Let φ ∈ Hom(G, Aut(H)) be a representation of a compact groupG on a finite-dimensional C-vector space H. Construct a G-invariant inner product((u, v) �→ 〈u, v〉H) : H×H → C, that is

〈φ(x)u, φ(x)v〉H = 〈u, v〉Hfor all x ∈ G and u, v ∈ H. Notice that now the representation φ is unitary withrespect to this inner product!

Lemma 7.5.2. Let G be a compact group and H be a Hilbert space with the innerproduct (u, v) �→ 〈u, v〉H. Let φ ∈ Hom(G,U(H)) be cyclic and w ∈ H a φ-cyclicvector with ‖w‖H = 1. Then

〈u, v〉φ :=∫

G

〈φ(x)u, w〉H 〈w, φ(x)v〉H dx

defines an inner product (u, v) �→ 〈u, v〉φ for H. Moreover, φ is unitary also withrespect to this new inner product, and ‖u‖φ ≤ ‖u‖H for all u ∈ H, where ‖u‖2φ :=〈u, u〉φ.

Proof. Defining fu(x) := 〈φ(x)u, w〉H, we notice that fu ∈ C(G), because

|fu(x)− fu(y)| = |〈(φ(x)− φ(y))u, w〉H|≤ ‖(φ(x)− φ(y))u‖H ‖w‖H−−−→x→y

0 ,

due to the strong continuity of φ. Thereby fufv is Haar integrable, justifying thedefinition of 〈u, v〉φ.

Let λ ∈ C and t, u, v ∈ H. Then it is easy to verify that

〈λu, v〉φ = λ〈u, v〉φ,

〈t + u, v〉φ = 〈t, v〉φ + 〈u, v〉φ,

〈u, v〉φ = 〈v, u〉φ,

‖u‖2φ =∫

G

|fu|2 dμG ≥ 0.

What if 0 = ‖u‖2φ =∫

G|fu|2dμG? Then fu = 0 by Theorem 7.4.25, i.e.,

0 = 〈φ(x)u, w〉H = 〈u, φ(x−1)w〉Hfor all x ∈ G. Since w is a cyclic vector, u = 0 follows. Thus (u, v) �→ 〈u, v〉φ is aninner product on H.


The original norm dominates the φ-norm, since

‖u‖2φ =∫

G

|〈φ(x)u, w〉H|2 dx

≤∫

G

‖φ(x)u‖2H ‖w‖2H dx

=∫

G

‖u‖2H dx

= ‖u‖2H.

The φ-unitarity of φ follows by

〈u, φ(y)∗v〉φ = 〈φ(y)u, v〉φ

=∫

G

〈φ(xy)u, w〉H 〈w, φ(x)v〉H dx

z=xy=

∫G

〈φ(z)u, w〉H 〈w, φ(zy−1)v〉H dz

= 〈u, φ(y)−1v〉φ,

where we applied the translation invariance of the Haar integral. �Exercise 7.5.3. Check the missing details in the proof of Lemma 7.5.2.

Lemma 7.5.4. Let 〈u, v〉φ be as above. Then

〈u, Av〉H := 〈u, v〉φdefines a compact self-adjoint operator A ∈ L(H). Furthermore, A is positivedefinite and A ∈ Hom(φ, φ).

Proof. By Lemma 7.5.2, if v ∈ H then Fv(u) := 〈u, v〉φ defines a linear functionalFv : H → C, which is bounded in both norms, since

|Fv(u)| = |〈u, v〉φ| ≤ ‖u‖φ ‖v‖φ ≤ ‖u‖H ‖v‖φ.

The Riesz Representation Theorem B.5.19 implies that Fv is represented by aunique vector A(v) ∈ H, i.e., Fv(u) = 〈u, A(v)〉H for all u ∈ H. Thus we have anoperator A : H → H, which is clearly linear. We obtain a bound ‖A‖L(H) ≤ 1from

‖Av‖2H = 〈Av,Av〉H = 〈Av, v〉φ ≤ ‖Av‖φ ‖v‖φ ≤ ‖Av‖H ‖v‖H.

Self-adjointness follows from

〈u, A∗v〉H = 〈Au, v〉H = 〈v,Au〉H = 〈v, u〉φ = 〈u, v〉φ = 〈u, Av〉H.

Moreover, A is positive definite, because 〈u, Au〉H = 〈u, u〉φ = ‖u‖2φ ≥ 0, where‖u‖φ = 0 if and only if u = 0.


The property that A ∈ Hom(φ, φ) is seen from

〈u, Aφ(y)v〉H = 〈u, φ(y)v〉φ= 〈φ(y)−1u, v〉φ= 〈φ(y)−1u, Av〉H= 〈u, φ(y)Av〉H.

Let B = {u ∈ H : ‖u‖H ≤ 1}, the closed unit ball of H. To show thatA ∈ L(H) is compact, we must show that A(B) ⊂ H is a compact set. So take asequence (vj)∞j=1 ⊂ A(B); we have to find a converging subsequence. Take a se-quence (uj)∞j=1 ⊂ B such that Auj = vj . By the Banach–Alaoglu Theorem B.5.30,the closed ball B is weakly compact: there exists a subsequence (ujk

)∞k=1 such thatujk

−−−−→k→∞

u ∈ B weakly, i.e.,

〈ujk, v〉H −−−−→

k→∞〈u, v〉H

for all v ∈ H. Then

‖vjk−Au‖2H = ‖A(ujk

− u)‖2H= 〈A(ujk

− u), ujk− u〉φ

=∫

G

gk dμG

wheregk(x) := 〈φ(x)A(ujk

− u), w〉H 〈w, φ(x)(ujk− u)〉H.

Let us show that∫

Ggk dμG → 0 as k → ∞. First, gk ∈ C(G) (hence gk is

integrable) and for each x ∈ G,

|gk(x)| =∣∣〈ujk

− u, A∗φ(x−1)w〉H∣∣ ∣∣〈φ(x−1)w, ujk

− u〉H∣∣

−−−−→k→∞

0

by the weak convergence. Second,

|gk(x)| ≤ ‖φ(x)‖2L(H) ‖A∗‖L(H) ‖w‖2H ‖ujk− u‖2H

≤ 4,

because ‖φ(x)‖L(H) = 1, ‖A‖L(H) = ‖A∗‖L(H) ≤ 1, ‖w‖H = 1 and ujk, u ∈ B.

Thus∫

Ggk dμG −−−−→

k→∞0 by the Lebesgue Dominated Convergence Theorem (see

Theorem C.3.22). Equivalently, vjk−−−−→k→∞

Au ∈ A(B). We have shown that the

set A(B) = A(B) ⊂ H is compact. �


Theorem 7.5.5 (Decomposition in finite-dimensional representations). Let G be acompact group and H a Hilbert space. Let φ ∈ Hom(G,U(H)) be strongly contin-uous. Then φ is a direct sum of finite-dimensional irreducible unitary representa-tions.

Proof. We know by Proposition 7.2.8 that φ is a direct sum of cyclic represen-tations. Therefore it is enough to assume that φ itself is cyclic. The operatorA ∈ Hom(φ, φ) in Lemma 7.5.4 is compact and self-adjoint. Hence by the Hilbert–Schmidt Spectral Theorem B.5.26 we have

H =⊕

λ∈σ(A)

Ker(λI −A),

where dim(Ker(λI − A)) < ∞ for each λ ∈ σ(A)\{0}. This can be extended toλ = 0 as well by Lemma 7.5.4 and the definition of A. Since A ∈ Hom(φ, φ), thesubspace Ker(λI −A) ⊂ H is φ-invariant for each λ. Thereby

φ =⊕

λ∈σ(A)

φ|Ker(λI−A),

where φ|Ker(λI−A) is finite-dimensional and unitary for all λ ∈ σ(A). The proofis concluded, since we know that a finite-dimensional unitary representation is adirect sum of irreducible unitary representations. �Corollary 7.5.6 (Finite dimensionality of representations!). Strongly continuousirreducible unitary representations of compact groups are finite-dimensional.

Definition 7.5.7 (Unitary dual G). The (unitary) dual G of a locally compact groupG is the set consisting of all equivalence classes of strongly continuous irreducibleunitary representations of G (for the definition of equivalent representations seeDefinition 6.3.18).

Remark 7.5.8 (Continuity is enough). For a compact group G, the set G consistsof the equivalence classes of continuous irreducible unitary representations (dueto the finite-dimensionality), i.e.,

G = {[φ] | φ is a continuous irreducible unitary representation of G} ,

where [φ] = {ψ | ψ ∼ φ} is the equivalence class of φ.

Remark 7.5.9 (Duals Rn and Tn). It can be proven that

Rn ={[eξ] | ξ ∈ Rn, eξ : Rn → U(1), eξ(x) := ei2πx·ξ} .

Noticing that eξeη = eξ+η and that [eξ] = [eη] for ξ = η, we may identify Rn ∼= Rn

as groups. Similarly, and in view of Theorem 3.1.17 and Remark 3.1.18, we have

Tn ={[eξ] | ξ ∈ Zn, eξ : Rn → U(1), eξ(x) := ei2πx·ξ} ,

so that Tn ∼= Zn as groups.


Remark 7.5.10 (Pontryagin duality). For a commutative locally compact groupG the unitary dual G has a natural structure of a commutative locally compact

group, and G ∼= G; this is the so-called Pontryagin duality. For a compact non-

commutative group G, the unitary dual G is never a group, but still has a sort ofweak algebraic structure; we do not consider this in the sequel.

Remark 7.5.11 (Matrix representations). Let G be a compact group. For theequivalence class ξ ∈ G there exists a unitary matrix representation φ ∈ ξ = [φ].That is, we have a homomorphism φ = (φij)

mi,j=1 : G → U(m), where functions

φij : G → C are continuous. We may find such a representation in the followingway: if ψ ∈ ξ, ψ ∈ Hom(G,U(H)) and {ej}m

j=1 ⊂ H is an orthonormal basis forH, then we can define

φij(x) := 〈ei, ψ(x)ej〉H.

Next we present an L2-orthogonality result for such functions φij : G→ C.

Lemma 7.5.12 (Orthogonality of representations). Let G be a compact group. Letξ, η ∈ G, where ξ � φ = (φij)m

i,j=1 ∈ Hom(G, U(m)) and η � ψ = (ψkl)nk,l=1 ∈

Hom(G, U(n)). Then

〈φij , ψkl〉L2(μG) =

{0, if ξ = η,1m δikδjl, if φ = ψ.

Proof. Fix 1 ≤ j ≤ m and 1 ≤ l ≤ n. Define the matrix E ∈ Cm×n by Epq := δpjδlq

(i.e., the matrix elements of E are zero except for the (j, l)-element, which is 1.)Define the matrix A ∈ Cm×n by

A :=∫

G

φ(y) E ψ(y−1) dy.

Now A ∈ Hom(ψ, φ), since

φ(x)A =∫

G

φ(xy) E ψ(y−1) dy

=∫

G

φ(z) E ψ(z−1x) dz

= Aψ(x).

Since φ, ψ are finite-dimensional irreducible unitary representations, Schur’s Lem-ma 6.3.25 implies that

A =

{0, if φ ∼ ψ,

λI, if φ = ψ


for some λ ∈ C. We notice that

Aik =∫

G

m∑p=1

n∑q=1

φip(y) Epq ψqk(y−1) dy

=∫

G

φij(y) ψkl(y) dy = 〈φij , ψkl〉L2(μG).

Now suppose φ = ψ. Then m = n and

〈φkj , ψkl〉L2(μG) = Akk = λ =1m

Tr(A)

=1m

∫G

Tr(φ(y) E φ(y−1)) dy

=1m

∫G

Tr(E) dy =1m

δjl,

where we used the property Tr(BC) = Tr(CB) of the trace functional. The resultscan be collected from above. �Definition 7.5.13 (Left and right regular representations). Let G be a compactgroup. Its left and right regular representations πL, πR : G → U(L2(μG)) aredefined, respectively, by

(πL(y) f)(x) := f(y−1x),

(πR(y) f)(x) := f(xy)

for μG-almost every x ∈ G.

The idea here is that G is represented as a natural group of operators onthe Hilbert space L2(μG), enabling the use of functional analytic techniques instudying G. And now for a major result in representation theory:

Theorem 7.5.14 (Peter–Weyl Theorem (1927)). Let G be a compact group. Then

B :={√

dim(φ)φij | φ = (φij)dim(φ)i,j=1 , [φ] ∈ G

}is an orthonormal basis for L2(μG). Let φ = (φij)m

i,j=1, φ ∈ [φ] ∈ G. Then

Hφi,· := span{φij | 1 ≤ j ≤ m} ⊂ L2(μG)

is πR-invariant andφ ∼ πR|Hφ

i,·,

L2(μG) =⊕

[φ]∈G

m⊕i=1

Hφi,·,

πR ∼⊕

[φ]∈G

m⊕i=1

φ.


Remark 7.5.15. Here⊕m

i=1 φ := φ⊕· · ·⊕φ, the m-fold direct sum of φ; in literature,this is sometimes denoted even by mφ.

Remark 7.5.16 (Left Peter–Weyl). We can formulate the Peter–Weyl Theorem7.5.14 analogously for the left regular representation, as follows: Let φ = (φij)m

i,j=1,where φ ∈ [φ] ∈ G. Then

Hφ·,j := span{φij | 1 ≤ i ≤ m} ⊂ L2(μG)

is πL-invariant andφ ∼ πL|Hφ

·,j,

L2(μG) =⊕

[φ]∈G

m⊕j=1

Hφ·,j ,

πL ∼⊕

[φ]∈G

m⊕j=1

φ.

Remark 7.5.17 (Peter–Weyl for Tn). Let G = Tn. Recall from Remark 7.5.9 that

Tn ={[eξ] | ξ ∈ Zn, eξ(x) = ei2πx·ξ} .

Now B = {eξ | ξ ∈ Zn} is an orthonormal basis for L2(μTn),

L2(μTn) =⊕ξ∈Zn

span{eξ},

πL ∼⊕ξ∈Zn

eξ ∼ πR.

Moreover, for f ∈ L2(μTn), we have

f =∑ξ∈Zn

f(ξ) eξ,

where the Fourier coefficients f(ξ) are calculated by

f(ξ) =∫

Tn

f eξ dμTn = 〈f, eξ〉L2(μTn ).

Analogously, the Peter–Weyl Theorem 7.5.14 provides Fourier series expansionsfor L2-functions on any compact group. We shall return to the Fourier seriestheme after the proof of the Peter–Weyl theorem.


Proof for the Peter–Weyl Theorem 7.5.14. The πR-invariance of Hφi,· follows due

to

πR(y)φij(x) = φij(xy) =dim(φ)∑

k=1

φik(x)φkj(y),

i.e., with λk(y) = φkj(y) we have

πR(y)φij =dim(φ)∑

k=1

λk(y) φik ∈ span{φik}dim(φ)k=1 = Hφ

i,·.

If {ej}dim(φ)j=1 ⊂ Cdim(φ) is the standard orthonormal basis then

φ(y)ej =dim(φ)∑

k=1

φkj(y)ek,

so that the equation

A

dim(φ)∑j=1

λjej :=dim(φ)∑

j=1

λjφij

defines an intertwining isomorphism A ∈ Hom(φ, πR|Hφi,·

), i.e., φ ∼ πR|Hφi,·

.

By Lemma 7.5.12, the set B ⊂ L2(μG) is orthonormal. Let

H :=⊕

[φ]∈G

dim(φ)⊕i=1

Hφi,·.

We assume that H = L2(μG), and show that this leads to a contradiction (sothat H = L2(μG) and B must be a basis). First, clearly H is πR-invariant. By ourassumption, H⊥ is a non-trivial πR-invariant closed subspace. Since πR|H⊥ is a di-rect sum of irreducible unitary representations, there exists a non-trivial subspaceE ⊂ H⊥ and a unitary matrix representation φ = (φij)m

i,j=1 ∈ HOM(G, U(m))such that φ ∼ πR|E . The subspace E has an orthonormal basis {fj}m

j=1 such that

πR(y)fj =m∑

i=1

φij(y)fi

for all y ∈ G and j ∈ {1, . . . ,m}. Notice that fj ∈ L2(μG) has pointwise valuesperhaps only μG-almost everywhere, so that

fj(xy) =m∑

i=1

φij(y)fi(x)


may hold for only μG-almost every x ∈ G. Let us define measurable sets

N(y) :=

{x ∈ G : fj(xy) =

m∑i=1

φij(y)fi(x)

},

M(x) :=

{y ∈ G : fj(xy) =

m∑i=1

φij(y)fi(x)

},

K :=

{(x, y) ∈ G×G : fj(xy) =

m∑i=1

φij(y)fi(x)

}.

By Exercise 7.4.27, we may utilise the Fubini Theorem to change the order ofintegration, to get∫

G

μG(M(x)) dμG(x) = μG×G(K) =∫

G

μG(N(y)) dμG(y)

=∫

G

0 dμG(y) = 0,

meaning that μG(M(x)) = 0 for almost every x ∈ G. But it is enough to pick justone x0 ∈ G such that μG(M(x0)) = 0. Then

fj(x0y) =m∑

i=1

φij(y)fi(x0)

for μG-almost every y ∈ G. If we denote z := x0y then

fj(z) =m∑

i=1

φij(x−10 z)fi(x0)

=m∑

i=1

m∑k=1

φik(x−10 )φkj(z)fi(x0)

=m∑

k=1

φkj(z)m∑

i=1

φik(x−10 )fi(x0)

for μG-almost every z ∈ G. Hence

fj ∈ span{φkj}mk=1 ⊂

m⊕k=1

Hφk,· ⊂ H

for all j ∈ {1, . . . ,m}. Thereby

E = span{fj}mj=1 ⊂ H;

at the same time E ⊂ H⊥, yielding E = {0}. This is a contradiction, since Eshould be non-trivial. Hence H = L2(μG) and B is a basis. �Exercise 7.5.18. Check the details of the proof of the Peter–Weyl theorem. Inparticular, pay attention to verify the conditions for applying the Fubini Theorem.


7.6 Fourier series and trigonometric polynomials

The classical Fourier series express a periodic function as an infinite sum of el-ementary waves that behave well under translations. This can be viewed as aspecial case of a more general phenomenon: a function on a compact group ad-mits an analogous series expansion, thanks to the Peter–Weyl Theorem 7.5.14. Westart by discussing the trigonometric polynomials because they play an importantrole as finite linear combinations of the basis elements of L2(μG) provided by thePeter–Weyl theorem.

Definition 7.6.1 (Trigonometric polynomials on groups). Let G be a compact groupand

B :={√

dim(φ)φij | φ = (φij)dim(φ)i,j=1 , [φ] ∈ G

}as in the Peter–Weyl Theorem 7.5.14. The space of trigonometric polynomials onG is

TrigPol(G) = span(B).

For instance, f ∈ TrigPol(Tn) is of the form

f(x) =∑ξ∈Zn

f(ξ)ei2πx·ξ,

where f(ξ) = 0 for only finitely many ξ ∈ Zn, see Remark 3.1.26. In the case of thetorus the following density statement was already verified in the proof of Theorem3.1.20.

Theorem 7.6.2 (Density I). TrigPol(G) is a dense subalgebra of C(G).

Proof. It is enough to verify that TrigPol(G) is an involutive subalgebra of C(G),because the Stone–Weierstrass Theorem A.14.4 provides then the density. Wealready know that TrigPol(G) is a subspace of C(G).

First, φ = (x �→ (1)) ∈ Hom(G, U(1)) is a continuous irreducible unitaryrepresentation, so that 1 = (x �→ 1) ∈ C(G) belongs to B ⊂ TrigPol(G).

Let [φ] ∈ G, φ = (φij)mi,j=1. Then [φ] ∈ G, where φ = (φij)m

i,j=1, as it is easy toverify. Thereby we get the involutivity: f ∈ TrigPol(G) whenever f ∈ TrigPol(G).

Let [ψ] ∈ G, ψ = (ψkl)nk,l=1. Then φ ⊗ ψ|G = (x �→ φ(x) ⊗ ψ(x)) ∈

Hom(G,U(Cm⊗Cn)). Let {ei}mi=1 ⊂ Cm and {fk}n

k=1 ⊂ Cn be orthonormal bases.Then {ei ⊗ fk | 1 ≤ i ≤ m, 1 ≤ k ≤ n} is an orthonormal basis for Cm ⊗Cn, andthe (ik)(jl)-matrix element of φ⊗ ψ|G is calculated by

(φ⊗ ψ|G)(ik)(jl)(x) = 〈(φ⊗ ψ|G)(x)(ej ⊗ fl), ei ⊗ fk〉Cm⊗Cn

= 〈φ(x)ej , ei〉Cm 〈ψ(x)fl, fk〉Cn = φij(x)ψkl(x).

Hence φijψkl is a matrix element of φ ⊗ ψ|G. Representation φ ⊗ ψ|G can bedecomposed as a finite direct sum of irreducible unitary representations. Hence thematrix elements of φ⊗ψ|G can be written as linear combinations of elements of B.Thus φijψkl ∈ TrigPol(G), so that fg ∈ TrigPol(G) for all f, g ∈ TrigPol(G). �

7.6. Fourier series and trigonometric polynomials 475

Corollary 7.6.3 (Density II). TrigPol(G) is dense in L2(μG).

Remark 7.6.4. Notice that we did not need the Peter–Weyl Theorem 7.5.14 toshow that TrigPol(G) ⊂ L2(μG) is dense. Therefore this density provides anotherproof for the Peter–Weyl Theorem 7.5.14.Remark 7.6.5. By now, we have encountered plenty of translation- and inversion-invariant function spaces on G. For instance, TrigPol(G), C(G) and Lp(G), andmore: namely, if [φ] ∈ G, φ = (φij)m

i,j=1, then

πL(y)φi0j0 , πR(y)φi0j0 ∈ span{φij}mi,j=1

for all y ∈ G (and inversion-invariance is clear!).

Exercise 7.6.6. Prove that f ∈ C(G) is a trigonometric polynomial if and only if

dim (span{πR(y)f : y ∈ G}) <∞.

As a direct consequence of knowing the basis of L2(μG) by the Peter–Weyltheorem, we obtain:

Corollary 7.6.7 (Fourier series and Plancherel (matrix form)). On a compact groupG, a Fourier series presentation of f ∈ L2(μG) is given by

f =∑

[φ]∈G

dim(φ)dim(φ)∑i,j=1

〈f, φij〉L2(μG) φij , (7.1)

where we pick just one unitary matrix representation φ = (φij)dim(φ)i,j=1 from each

equivalence class [φ] ∈ G. Moreover, there is the Plancherel identity (sometimescalled the Parseval identity)

‖f‖L2(μG) =∑

[φ]∈G


|〈f, φij〉L2(μG)|2. (7.2)

Remark 7.6.8. In L2(μG), also clearly

f =∑

[φ]∈G


〈f, φij〉L2(μG) φij .

A nice thing about the Fourier series is that the basis functions φij and φij arewell behaved under translations and inversions.

Definition 7.6.9 (Fourier coefficients and Fourier transform). Let G be a compactgroup, f ∈ L1(μG) and φ = (φij)m

i,j=1, [φ] ∈ G. The φ-Fourier coefficient of f is

f(φ) :=∫

G

f(x) φ(x)∗ dx ∈ Cm×m,


where the integration of the matrix-valued function is element-wise. The matrix-valued function f is called the Fourier transform of f ∈ L1(μG). We note thatthis definition immediately extends also to L2(μG) in view of L2(μG) ⊂ L1(μG),which follows, e.g., by the Holder’s inequality from the compactness of G.

Corollary 7.6.10 (Fourier series and Plancherel). On a compact group G, a Fourierseries presentation of f ∈ L2(μG) is given by

f(x) =∑

[φ]∈G

dim(φ) Tr(f(φ) φ(x)

)(7.3)

converging for μG-almost every x ∈ G, as well as in L2(μG). The Plancherelidentity takes the form

‖f‖2L2(μG) =∑

[φ]∈G

dim(φ) Tr(f(φ) f(φ)∗

).

If f, g ∈ L2(G), then we also have the Parseval identity

(f, g)L2(G) =∑

[ξ]∈G

dim(φ) Tr(f(φ) g(φ)∗

)= (f(φ), g(φ))L2(G),

with L2(G) as in Definition 7.6.11.

Proof. Now

f(φ)ij =∫

G

f(x) (φ(x)∗)ij dμG(x) =∫

G

f(x) φji(x) dx = 〈f, φji〉L2(μG),

so that

Tr(f(φ) φ(x)

)=

dim(φ)∑i=1

(f(φ) φ(x)

)ii

=dim(φ)∑i,j=1

f(φ)ij φji(x)

=dim(φ)∑i,j=1

〈f, φji〉L2(μG) φji(x).

Hence (7.3) follows from (7.1). Finally, if A = (Akl)mk,l=1 ∈ Cm×m then

‖A‖2HS := Tr(A∗A) =m∑

k,l=1

|Akl|2,

completing the proof, if we take A = f(φ) and use (7.2). The details of the proofof the Parceval identity will be given in Proposition 10.3.17. �

7.6. Fourier series and trigonometric polynomials 477

The convergence in L2(μG) is automatic, see Theorem B.5.32.

Definition 7.6.11 (Hilbert space L2(G)). Let G be a compact group. Let L2(G)be the space containing the mappings

F : G→∞⋃

m=1

Cm×m

satisfying F ([φ]) ∈ Cdim(φ)×dim(φ) such that∑[φ]∈G

dim(φ) ‖F ([φ])‖2Cdim(φ)×dim(φ) <∞.

Then L2(G) is a Hilbert space with the inner product

〈E,F 〉L2(G) :=∑

[φ]∈G

dim(φ) 〈E([φ]), F ([φ])〉Cdim(φ)×dim(φ) .

Exercise 7.6.12. Verify that L2(G) is indeed a Hilbert space.

Theorem 7.6.13 (Fourier transform is an isometry L2(μG) → L2(G)). Let G bea compact group. The Fourier transform f �→ f defines a surjective isometryL2(μG)→ L2(G).

Proof. Let us choose one unitary matrix representation φ from each [φ] ∈ G. If wedefine F ([φ]) := f(φ) then F ∈ L2(G), and f �→ F is isometric by the Plancherelequality.

Now take any F ∈ L2(G). We have to show that F ([φ]) = f(φ) for somef ∈ L2(μG), where φ ∈ [φ] ∈ G. Define

f(x) :=∑

[φ]∈G

dim(φ) Tr (F ([φ]) φ(x))

for μG-almost every x ∈ G. This can be done, since

f =∑

[φ]∈G


F ([φ])ij φji

belongs to L2(μG) by‖f‖2L2(μG) = ‖F‖2

L2(G)<∞.

Clearly f(φ) = F ([φ]), so that the Fourier transform is surjective. �

We will return to a more detailed analysis of the space L2(G) and of otherspaces of functions and distributions on the unitary dual G in Section 10.3.


7.7 Convolutions

For functions f and g on a group, their convolution f ∗ g can be thought as amodulation of one with the other. More precisely, the Fourier coefficients of f ∗ gare the pointwise products of the Fourier coefficients of f and g, as presented inProposition 7.7.5.

Definition 7.7.1 (Convolutions on compact groups). Let G be a compact group,and let f ∈ L1(μG) and g ∈ C(G) (or f ∈ C(G) and g ∈ L1(μG)). The convolutionf ∗ g : G→ C is defined by

f ∗ g(x) :=∫

G

f(y) g(y−1x) dy.

Remark 7.7.2. Now f ∗ g ∈ C(G). Indeed, due to the uniform continuity, for eachε > 0 there exists open U � e such that |g(x)−g(z)| < ε when z−1x ∈ U . Thereby

|f ∗ g(x)− f ∗ g(z)| ≤∫

G

|f(y)| |g(y−1x)− g(y−1z)| dy

≤ ‖f‖L1(μG) ε,

when z−1x ∈ U . Furthermore, the linear mapping g �→ f ∗ g satisfies

‖f ∗ g‖C(G) ≤ ‖f‖L1(μG) ‖g‖C(G),

‖f ∗ g‖C(G) ≤ ‖f‖C(G) ‖g‖L1(μG),

‖f ∗ g‖L1(μG) ≤ ‖f‖L1(μG) ‖g‖L1(μG).

Hence we can consider g �→ f ∗ g as a bounded operator on C(G) and L1(μG); ofcourse, we have symmetrical results for g �→ g ∗ f .

It is also easy to show other Lp-boundedness results, like

‖f ∗ g‖L∞(μG) ≤ ‖f‖L2(μG) ‖g‖L2(μG)

and so on. Notice that the convolution product is commutative if and only if G iscommutative.

Proposition 7.7.3. Let f, g, h ∈ L1(μG). Then f ∗ g ∈ L1(μG),

‖f ∗ g‖L1(μG) ≤ ‖f‖L1(μG) ‖g‖L1(μG),

and f ∗ g(x) =∫

Gf(y−1) g(yx) dy for almost every x ∈ G. Moreover, for μG-

almost every x ∈ G,

f ∗ g(x) =∫

G

f(xy−1) g(y) dy

=∫

G

f(y−1) g(yx) dy

=∫

G

f(xy) g(y−1) dy.

The convolution product is also associative: f ∗ (g ∗ h) = (f ∗ g) ∗ h.

7.8. Characters 479

Exercise 7.7.4. Prove Proposition 7.7.3.

Proposition 7.7.5. For f, g ∈ L1(μG) it holds that f ∗ g(φ) = g(φ) f(φ).

Proof. It is enough to assume that f, g ∈ C(G). Then

f ∗ g(φ) =∫

G

f ∗ g(x) φ(x)∗ dx

=∫

G

∫G

f(y) g(y−1x) dy φ(x)∗ dx

=∫

G

∫G

g(y−1x) φ(y−1x)∗ dx f(y) φ(y)∗ dy

=∫

G

g(z) φ(z)∗ dz

∫G

f(y) φ(y)∗ dy

= g(φ) f(φ),

completing the proof. �Remark 7.7.6. There are plenty of other interesting results concerning the Fouriertransform and convolutions on compact groups. For instance, one can study ap-proximate identities for L1(μG) and prove that the Fourier transform f �→ f isinjective on L1(μG).

7.8 Characters

Loosely speaking, a character is the trace of a representation, and it contains allthe essential information about the corresponding representation.

Definition 7.8.1 (Characters). Let φ : G→ Aut(H) be a representation of a groupG on a finite-dimensional Hilbert space H. The character of φ is the functionχφ : G→ C defined by

χφ(x) := Tr (φ(x)) .

Remark 7.8.2 (Purpose of characters). Notice that here G is just any group, andthat the character does not depend on the choice of the basis of H. It turns outthat on a compact group, characters provide a way of recognising equivalence ofrepresentations: namely, for finite-dimensional unitary representations, φ ∼ ψ ifand only if χφ = χψ, as we shall see.

Proposition 7.8.3 (Properties of characters). Let φ, ψ be finite-dimensional repre-sentations of a group G. Then the following hold:

(1) If φ ∼ ψ then χφ = χψ.(2) χφ(xyx−1) = χφ(y) for all x, y ∈ G.(3) χφ⊕ψ = χφ + χψ.(4) χφ⊗ψ|G = χφ χψ.(5) χφ(e) = dim(φ).


Proof. The results follow from the properties of the trace functionals, see, e.g.,Subsection B.5.1. �

Remark 7.8.4. Since the character depends only on the equivalence class of arepresentation, we may define χ[φ] := χφ, where [φ] is the equivalence class of φ.

Proposition 7.8.5 (Orthonormality of characters). Let G be a compact group andξ, η ∈ G. Then

〈χξ, χη〉L2(μG) =

{1 if ξ = η,

0 if ξ ∼ η.

Proof. Let φ = (φij)mi,j=1 ∈ ξ and ψ = (ψkl)n

k,l=1 ∈ η. Then

〈χξ, χη〉L2(μG) =m∑

j=1

n∑k=1

〈φjj , ψkk〉L2(μG)

=

{0 if φ ∼ ψ,

1 if φ = ψ

by Lemma 7.5.12. �

Theorem 7.8.6 (Irreducibility and equivalence characterisations). Let φ, ψ befinite-dimensional continuous unitary representations of a compact group G. Thenφ is irreducible if and only if ‖χφ‖L2(μG) = 1. Moreover, φ ∼ ψ if and only ifχφ = χψ.

Proof. We already know the “only if”-parts of the proof. So suppose φ is a finite-dimensional unitary representation. Then

φ ∼⊕[ξ]∈G

m[ξ]⊕j=1

ξ,

where m[ξ] ∈ N is non-zero for only finitely many [ξ] ∈ G, and with the conventionthat the empty sum gives zero. Then

χφ =∑

[ξ]∈G

m[ξ]χξ,

and if [η] ∈ G then

〈χφ, χη〉L2(μG) =∑

[ξ]∈G

m[ξ]〈χξ, χη〉L2(μG) = m[η].

7.8. Characters 481

This implies that the multiplicities m[ξ] ∈ N can be uniquely obtained by knowingonly χφ; hence if χφ = χψ then φ ∼ ψ. Moreover,

‖χφ‖2L2(μG) = 〈χφ, χφ〉L2(μG)

=∑

[ξ],[η]∈G

m[ξ] m[η] 〈χξ, χη〉L2(μG)

=∑

[ξ]∈G

m2[ξ],

so that φ is irreducible if and only if ‖χφ‖L2(μG) = 1. �Exercise 7.8.7. If f ∈ L2(μG) then

f =∑

[ξ]∈G

dim(ξ) f ∗ χξ =∑

[ξ]∈G

dim(ξ) Tr(ξ(x) f(ξ)

).

Thus, the projection of f ∈ L2(G) to Hξ is given by f �→ f ∗ χξ. The solution ofthis exercise can be found in Corollary 10.11.6.

We note that the restriction of the representation and the characters to themaximal torus of the group determine them completely:

Theorem 7.8.8 (Cartan’s maximal torus theorem). Let Tn ↪→ G be an injectivegroup homomorphism with the largest possible n. Then two representations φ andψ of G are equivalent if and only if their restrictions to Tn are equivalent. Inparticular, the restriction χφ|Tn of χφ to Tn determines the class [φ].

Remark 7.8.9 (Tensor products of representations). According to Proposition7.8.3, (4), we have χφ⊗ψ|G = χφ χψ for any two finite-dimensional representationsφ and ψ of G. By Theorem 7.5.5 the representation φ⊗ψ|G = (x �→ φ(x)⊗ψ(x)) ∈Hom(G,U(Hφ ⊗ Hψ)) can be decomposed as a direct sum of irreducible unitaryrepresentations:

φ⊗ ψ|G =⊕[ξ]∈G

mφ,ψ([ξ])⊕1

ξ,

where mφ,ψ([ξ]) is the multiplicity of [ξ] in φ ⊗ ψ|G, and only finitely many ofmφ,ψ([ξ]) are non-zero in view of the finite dimensionality. We also have

χφ χψ = χφ⊗ψ|G =∑

[ξ]∈G

mφ,ψ([ξ]) χξ

in view of Proposition 7.8.3, (3). The multiplicities mφ,ψ([ξ]) can be analysedusing Theorem 7.8.8 because we have, in particular, χφ|Tn χψ|Tn = χφ⊗ψ|Tn =∑

[ξ]∈Gmφ,ψ([ξ]) χξ|Tn .


7.9 Induced representations

A group representation trivially gives a representation of its subgroup: if H < Gand ψ ∈ Hom(G, Aut(V )) then the restriction

ResGHψ := (h �→ ψ(h)) ∈ Hom(H, Aut(V )). (7.4)

In the sequel, we show how a representation of a subgroup sometimes induces arepresentation for the whole group. This induction process has also plenty of niceproperties. Induced representations were defined and studied by Ferdinand GeorgFrobenius in 1898 for finite groups, and by George Mackey in 1949 for (most ofthe) locally compact groups.

The technical assumptions here are that G is a compact group, H < G isclosed and φ ∈ Hom(H,U(H)) is strongly continuous; then φ induces a stronglycontinuous unitary representation

IndGHφ ∈ Hom

(G,U(IndG

φH))

,

where the notation will be explained in the sequel. We start by a lengthy definitionof the induced representation space IndG

φH.Remark 7.9.1 (Uniformly continuous Hilbert space-valued mappings). Since G isa compact group, continuous mappings G → H are uniformly continuous in thefollowing sense: Let f ∈ C(G,H) and ε > 0. Then there exists open U � e suchthat ‖f(x)−f(y)‖H < ε when xy−1 ∈ U (or x−1y ∈ U); the proof of this fact is asin the scalar-valued case. We shall also need to integrate H-valued functions in theweak sense: that is, we need the concept of the Pettis integral, the details of whichcan be found from exercises related to Definition B.3.28 (see also Remark 7.9.3).

Proposition 7.9.2. If f ∈ C(G,H) then fφ ∈ C(G,H), where

fφ(x) :=∫

H

φ(h)f(xh) dμH(h), (7.5)

defined in the weak sense as the Pettis integral. Moreover, we have fφ(xh) =φ(h)∗fφ(x) for all x ∈ G and h ∈ H.

Remark 7.9.3 (Pettis integral). The weak (Pettis) integration in (7.5) means thatfor every f ∈ C(G,H) there exists a unique fφ ∈ C(G,H) such that for allu ∈ H′ = H we have

〈u, fφ〉H =∫

H

〈u, φ(h)f(xh)〉H dμH(h).

We denote this fφ as weak integral (7.5). The Riesz Representation TheoremB.5.19 gives the correctness of this integral definition since fφ is clearly a boundedlinear functional acting on u ∈ H. For a more general version of the Pettis integralwe refer to Definition B.3.28.

7.9. Induced representations 483

Proof of Proposition 7.9.2. Let {ej}j∈J ⊂ H be an orthonormal basis. Then

fφ(x) =∑j∈J

〈fφ(x), ej〉Hej ∈ H

is the unique vector defined by inner products

〈fφ(x), ej〉H =∫

H

〈φ(h)f(xh), ej〉H dμH(h).

It is easy to prove that the integrals here are sound, since

(h �→ 〈φ(h)f(xh), ej〉H) ∈ C(H)

because f ∈ C(G,H) and φ is strongly continuous. If h0 ∈ H then

fφ(xh0) =∫

H

φ(h) f(xh0h) dμH(h)

=∫

H

φ(h−10 h)f(xh) dμH(h)

= φ(h0)∗fφ(x).

Take ε > 0. By the uniform continuity of f ∈ C(G,H) mentioned in Remark 7.9.1,there exists an open set U � e such that ‖f(a)− f(b)‖H < ε whenever ab−1 ∈ U .If x ∈ yU then

‖fφ(x)− fφ(y)‖2H =∥∥∥∥∫

H

φ(h)(f(xh)− f(yh)) dμH(h)∥∥∥∥2

H

≤(∫

H

‖f(xh)− f(yh)‖H dμH(h))2

≤ ε2,

proving the continuity of fφ. �Lemma 7.9.4. If f, g ∈ C(G,H) then (xH �→ 〈fφ(x), gφ(x)〉H) ∈ C(G/H).

Proof. Let x ∈ G and h ∈ H. Then

〈fφ(xh), gφ(xh)〉H = 〈φ(h)∗fφ(x), φ(h)∗gφ(x)〉H= 〈fφ(x), gφ(x)〉H,

so that (xH �→ 〈fφ(x), gφ(x)〉H) : G/H → C is well defined. There exists a constantC < ∞ such that ‖fφ(y)‖H, ‖gφ(x)‖H ≤ C because G is compact and fφ, gφ ∈C(G,H). Thereby

|〈fφ(x), gφ(x)〉H − 〈fφ(y), gφ(y)〉H|≤ |〈fφ(x)− fφ(y), gφ(x)〉H|+ |〈fφ(y), gφ(x)− gφ(y)〉H|≤ C (‖fφ(x)− fφ(y)‖H + ‖gφ(x)− gφ(y)‖H)−−−→x→y

0

by the continuities of fφ and gφ. �


Definition 7.9.5 (Induced representation space IndGφH). Let us endow the vector

space

Cφ(G,H) := {fφ | f ∈ C(G,H)}= {e ∈ C(G,H) | ∀x ∈ G ∀h ∈ H : e(xh) = φ(h)∗e(x)}

with the inner product defined by

〈fφ, gφ〉IndGφH :=

∫G/H

〈fφ(x), gφ(x)〉H dμG/H(xH).

Let IndGφH be the completion of Cφ(G,H) with respect to the corresponding norm

fφ �→ ‖fφ‖IndGφH :=

√〈fφ, fφ〉IndG

φH;

this Hilbert space is called the induced representation space.

Remark 7.9.6. If H = {0} then {0} = Cφ(G,H) ⊂ IndGφH. Why? Let 0 = u ∈ H.

Due to the strong continuity of φ, we can choose open U ⊂ G such that e ∈ U and‖(φ(h)− φ(e))u‖H < ‖u‖H for all h ∈ H ∩ U . Choose w ∈ C(G) such that w ≥ 0,w|G\U = 0 and

∫H

w(h) dμH(h) = 1. Let f(x) := w(x)u for all x ∈ G. Then

‖fφ(e)− u‖H =∥∥∥∥∫

H

w(h) (φ(h)− φ(e))u dμH(h)∥∥∥∥H

=∫

H

w(h) ‖(φ(h)− φ(e))u‖H dμH(h)

< ‖u‖H,

so that fφ(e) = 0, yielding fφ = 0.

Theorem 7.9.7 (Induced representations). If x, y ∈ G and fφ ∈ Cφ(G,H), let(IndG

Hφ(y)fφ

)(x) := fφ(y−1x).

This begets a unique strongly continuous IndGHφ ∈ Hom

(G,U(IndG

φH)), called the

representation of G induced by φ.

Proof. If y ∈ G and fφ ∈ Cφ(G,H) then IndGHφ(y)fφ = gφ ∈ Cφ(G,H), where

g ∈ C(G,H) is defined by g(x) := f(y−1x). Thus we have a linear mappingIndG

Hφ(y) : Cφ(G,H)→ Cφ(G,H). Clearly

IndGHφ(yz)fφ = IndG

Hφ(y) IndGHφ(z)fφ.

Hence IndGHφ ∈ Hom (G, Aut(Cφ(G,H))).


If f, g ∈ C(G,H) then⟨IndG

Hφ(y)fφ, gφ

⟩IndG

φH=

∫G/H

〈fφ(y−1x), gφ(x)〉H dμG/H(xH)

=∫

G/H

〈fφ(z), gφ(yz)〉H dμG/H(zH)

=⟨fφ, IndG

Hφ(y)−1gφ

⟩IndG

φH;

hence we have an extension IndGHφ ∈ Hom

(G,U(IndG

φH)). Next we exploit the

uniform continuity of f ∈ C(G,H): Let ε > 0. Take an open set U � e such that‖f(a)− f(b)‖H < ε when ab−1 ∈ U . Thereby, if y−1z ∈ U then∥∥∥(

IndGHφ(y)− IndG

Hφ(z))

fφ

∥∥∥2

IndGφH

=∫

G/H

∥∥fφ(y−1x)− fφ(z−1x)∥∥2

H dμG/H(xH)

≤ ε2.

This shows the strong continuity of the induced representation. �

Remark 7.9.8. In the sequel, some elementary properties of induced representa-tions are deduced. Briefly: induced representations of equivalent representationsare equivalent, and the induction process can be taken in stages leading to thesame result modulo equivalence.

Proposition 7.9.9. Let G be a compact group and H < G a closed subgroup. Letφ ∈ Hom(H,U(Hφ)) and ψ ∈ Hom(H,U(Hψ)) be strongly continuous. If φ ∼ ψ

then IndGHφ ∼ IndG

Hψ.

Proof. Since φ ∼ ψ, there is an isometric isomorphism A ∈ Hom(φ, ψ). Then

(Bfφ)(x) := A(fφ(x))

defines a linear mapping B : Cφ(G,Hφ) → Cψ(G,Hψ), because if x ∈ G andh ∈ H then

(Bfφ)(xh) = A(fφ(xh))= A(φ(h)∗fφ(x))= A(φ(h)∗A∗A(fφ(x)))= A(A∗ψ(h)∗A(fφ(x)))= ψ(h)∗A(fφ(x))= ψ(h)∗(Bfφ)(x).


Furthermore, B can be extended to a unique linear isometry C : IndGφHφ →

IndGψHψ, since

‖Bfφ‖2IndGψHψ

=∫

G/H

‖(Bfφ)(x)‖2HψdμG/H(xH)

=∫

G/H

‖A(fφ(x))‖2HψdμG/H(xH)

=∫

G/H

‖fφ(x)‖2HφdμG/H(xH)

= ‖fφ‖2IndGφHφ

.

Next, C is a surjection: if F ∈ Cψ(G,Hψ) then (y �→ A−1(F (y)) ∈ Cφ(G,Hφ) and(C

(y �→ A−1(F (y))

))(x) = AA−1(F (x)) = F (x), and this is enough due to the

density of Cψ(G,Hψ) in IndGφH. Finally,

(C IndGHφ(y)fφ)(x) = A(IndG

Hφ(y)fφ(x))= A(fφ(y−1x))= (Cfφ)(y−1x)

= (IndGHφ(y)Cfφ)(x),

so that C ∈ Hom(IndG

Hφ, IndGHψ

)is an isometric isomorphism. �

Corollary 7.9.10. Let G be a compact group and H < G closed. Let φ1 andφ2 be strongly continuous unitary representations of H. Then IndG

H(φ1 ⊕ φ2) ∼(IndG

Hφ1

)⊕

(IndG

Hφ2

).

Exercise 7.9.11. Prove Corollary 7.9.10.

Corollary 7.9.12. IndGHφ is irreducible only if φ is irreducible.

Exercise 7.9.13. Let G1, G2 be compact groups and H1 < G1,H2 < G2 be closed.Let φ1, φ2 be strongly continuous unitary representations of H1,H2, respectively.Show that

IndG1×G2H1×H2

(φ1 ⊗ φ2) ∼(IndG1

H1φ1

)⊗

(IndG2

H2φ2

).

Theorem 7.9.14 (Inducing representations in steps). Let G be a compact group andH < K < G, where H,K are closed. If φ ∈ Hom(H,U(H)) is strongly continuousthen IndG

Hφ ∼ IndGKIndK

Hφ.

Proof. In this proof, x ∈ G, k, k0 ∈ K and h ∈ H. Let ψ := IndKHφ and Hψ :=

IndKφ H. Let fφ ∈ Cφ(G,H). Since (k �→ fφ(xk)) : K → H is continuous and

fφ(xkh) = φ(h)∗fφ(xk), we obtain (k �→ fφ(xk)) ∈ Cφ(K,H) ⊂ Hψ. Let us definefK

φ : G→ Hψ byfK

φ (x) := (k �→ fφ(xk)).


If x ∈ G and k0 ∈ K then

fKφ (xk0)(k) = fφ(xk0k)

= fKφ (x)(k0k)

=(ψ(k0)∗fK

φ (x))(k),

i.e., fKφ (xk0) = ψ(k0)∗fK

φ (x). Let ε > 0. By the uniform continuity of fφ, takeopen U � e such that ‖fφ(a) − fφ(b)‖H < ε if ab−1 ∈ U . Thereby if xy−1 ∈ Uthen ∥∥fK

φ (x)− fKφ (y)

∥∥2

Hψ=

∫K/H

∥∥fKφ (x)(k)− fK

φ (y)(k)∥∥2

H dμK/H(kH)

=∫

K/H

‖fφ(xk)− fφ(yk)‖2H dμK/H(kH)

≤ ε2.

Hence fKφ ∈ Cψ(G,Hψ) ⊂ IndG

ψHψ, so that we indeed have a mapping (fφ �→fK

φ ) : Cφ(G,H)→ Cψ(G,Hψ).

Next, we claim that fφ �→ fKφ defines a surjective linear isometry IndG

φH →IndG

ψHψ. Isometricity follows by∥∥fKφ

∥∥2

IndGψHψ

=∫

G/K

∥∥fKφ (x)

∥∥2

HψdμG/K(xK)

=∫

G/K

∫K/H

∥∥fKφ (x)(k)

∥∥2

H dμK/H(kH) dμG/K(xK)

=∫

G/K

∫K/H

‖fφ(xk)‖2H dμK/H(kH) dμG/K(xK)

=∫

G/H

‖fφ(x)‖2H dμG/H(xH)

= ‖fφ‖2IndGφH .

How about the surjectivity? The representation space IndGψHψ is the closure

of Cψ(G,Hψ), and Hψ is the closure of Cφ(K,H). Consequently, IndGψHψ is the

closure of the vector space

Cψ(G, Cφ(K,H)) := {g ∈ C(G, C(K,H)) | ∀x ∈ G ∀k ∈ K ∀h ∈ H :g(xk) = ψ(k)∗g(x), g(x)(kh) = φ(h)∗g(x)(k)}.

Given g ∈ Cψ(G, Cφ(K,H)), define fφ ∈ Cφ(G,H) by fφ(x) := g(x)(e). ThenfK

φ = g, because

fKφ (x)(k) = fφ(xk) = g(xk)(e) = ψ(k)∗g(x)(e) = g(x)(k).


Thus (fφ �→ fKφ ) : Cφ(G,H) → Cψ(G, Cφ(K,H)) is a linear isometric bijection.

Hence this mapping can be extended uniquely to a linear isometric bijection A :IndG

φH → IndGψHψ.

Finally, A ∈ Hom(IndG

Hφ, IndGKIndK

Hφ), since

A(IndG

Hφ(y)fφ(x))

= Afφ(y−1x)

= fKφ (y−1x)

= IndGKψ(y)fK

φ (x)

= IndGKψ(y)Afφ(x).


Exercise 7.9.15. Let H be a closed subgroup of a compact group G. Let φ = (h �→I) ∈ Hom(H,U(H)), where I = (u �→ u) : H → H.a) Show that IndG

φH ∼= L2(G/H,H), where the L2(G/H,H) inner product isgiven by

〈fG/H , gG/H〉L2(G/H,H) :=∫

G/H

〈fG/H(xH), gG/H(xH)〉H dμG/H(xH),

when fG/H , gG/H ∈ C(G/H,H).b) Let K < G be closed. Let πK and πG be the left regular representations of Kand G, respectively. Prove that πG ∼ IndG

KπK .

Remark 7.9.16 (Multiplicity of a representation). A fundamental result for in-duced representations is the Frobenius Reciprocity Theorem 7.9.17, stated belowwithout a proof. Let G be a compact group and φ ∈ Hom(G,U(H)) be stronglycontinuous. Let n ([ξ], φ) ∈ N denote the multiplicity of [ξ] ∈ G in φ, defined asfollows: if φ =

⊕kj=1 φj , where each φj is a continuous irreducible unitary repre-

sentation, thenn([ξ], φ) := |{j ∈ {1, . . . , k} : [φj ] = [ξ]}| .

That is, n([ξ], φ) is the number of times ξ may occur as an irreducible componentin a direct sum decomposition of φ.

Theorem 7.9.17 (Frobenius Reciprocity Theorem). Let G be a compact group andH < G be closed. Let ξ, η be continuous such that [ξ] ∈ G and [η] ∈ H. Then

n([ξ], IndG

Hη)

= n([η],ResG

Hξ),

where ResGHξ is the restriction1 of ξ to H.

1see (7.4) for the definition of ResGHξ.


Example. Let [ξ] ∈ G, H = {e} and η = (e �→ I) ∈ Hom(H,U(C)). Then πL ∼IndG

Hη by Exercise 7.9.15, and H = {[η]}, so that

n([ξ], IndG

Hη)

= n ([ξ], πL)

Peter−Weyl= dim(ξ)= dim(ξ) n ([η], η)

= n

⎛⎝[η],dim(ξ)⊕

j=1

η

⎞⎠= n

([η],ResG

Hξ).

As it should be, this is in accordance with the Frobenius Reciprocity Theo-rem 7.9.17.Example. Let [ξ], [η] ∈ G. Then by the Frobenius Reciprocity Theorem 7.9.17,

n([ξ], IndG

Gη)

= n([η],ResG

Gξ)

= n ([η], ξ)

=

{1, when [ξ] = [η],0, when [ξ] = [η].

Let φ be a finite-dimensional continuous unitary representation of G. Then φ =⊕kj=1 ξk, where each ξk is irreducible. Thereby

IndGGφ ∼

k⊕j=1

IndGGξj ∼

k⊕j=1

ξj ∼ φ;

in other words, induction practically does nothing in this case.

Chapter 8

Linear Lie Groups

In this chapter we study linear Lie groups, i.e., Lie groups which are closed sub-groups of GL(n, C). But first some words about the general Lie groups:

Definition 8.0.1 (Lie groups). A Lie group is a C∞-manifold which is also a groupsuch that the group operations are C∞-smooth.

We will be mostly interested in the non-commutative Lie groups in view ofthe following:

Remark 8.0.2 (Commutative Lie groups). In the introduction to Part II we men-tioned that in the case of commutative groups it is sufficient to study cases of Tn

and Rn. Indeed, we have the following two facts:

• Any compact commutative Lie group is isomorphic to the product of a toruswith a finite commutative group.

• Any connected commutative Lie group is isomorphic to the product of a torusand the Euclidean space. In other words, if G is a connected commutativeLie group then G ∼= Tn × Rm for some n, m.

We will not prove these facts here but refer to, e.g., [20, p. 25] for further details.

Definition 8.0.3 (Linear Lie groups). A linear Lie group is a Lie group which is aclosed subgroup of GL(n, C).

There is a result stating that any compact Lie group is diffeomorphic to alinear Lie group, and thereby the matrix groups are especially interesting. In fact,we have:

Corollary 8.0.4 (Universality of unitary groups). Let G be a compact Lie group.Then there is some n ∈ N such that G is isomorphic to a subgroup of U(n).

492 Chapter 8. Linear Lie Groups

8.1 Exponential map

The fundamental tool for studying linear Lie groups is the matrix exponentialmap, treated below.

Let us endow Cn with the Euclidean inner product

(x, y) �→ 〈x, y〉Cn :=n∑

j=1

xjyj .

The corresponding norm is x �→ ‖x‖Cn := 〈x, x〉1/2Cn . We identify the matrix algebra

Cn×n with L(Cn), the algebra of linear operators Cn → Cn. Let us endow Cn×n ∼=L(Cn) with the operator norm

Y �→ ‖Y ‖L(Cn) := supx∈Cn: ‖x‖Cn≤1

‖Y x‖Cn .

Notice that ‖XY ‖L(Cn) ≤ ‖X‖L(Cn)‖Y ‖L(Cn). For a matrix X ∈ Cn×n, the expo-nential exp(X) ∈ Cn×n is defined by the power series

exp(X) :=∞∑

k=0

1k!

Xk,

where X0 := I; this series converges in the Banach space Cn×n ∼= L(Cn), because

∞∑k=0

1k!

∥∥Xk∥∥L(Cn)

≤∞∑

k=0

1k!‖X‖k

L(Cn) = e‖X‖L(Cn) <∞.

Proposition 8.1.1. Let X, Y ∈ Cn×n. If XY = Y X then

exp(X + Y ) = exp(X) exp(Y ).

Therefore exp : Cn×n → GL(n, C) satisfies exp(−X) = exp(X)−1.

Proof. Now

exp(X + Y ) = liml→∞

2l∑k=0

1k!

(X + Y )k

XY =Y X= liml→∞

2l∑k=0

1k!

k∑i=0

k!i! (k − i)!

XiY k−i

= liml→∞

⎛⎜⎜⎝ l∑i=0

1i!

Xil∑

j=0

1j!

Y j +∑

i,j: i+j≤2l,max(i,j)>l

1i! j!

XiY j

⎞⎟⎟⎠

8.1. Exponential map 493

= liml→∞

⎛⎝ l∑i=0

1i!

Xil∑

j=0

1j!

Y j

⎞⎠= exp(X) exp(Y ),

since the remainder term satisfies∥∥∥∥∥∥∥∥∑


1i! j!

XiY j

∥∥∥∥∥∥∥∥L(Cn)

≤∑


1i! j!

‖X‖iL(Cn)‖Y ‖j

L(Cn)

≤ l(l + 1)1

(l + 1)!c2l

−−−→l→∞

0,

where c := max(1, ‖X‖L(Cn), ‖Y ‖L(Cn)

).

Consequently, I = exp(0) = exp(X) exp(−X) = exp(−X) exp(X), so thatwe get exp(−X) = exp(X)−1. �Exercise 8.1.2. Verify the estimates and the ranges of the summation indices inthe proof of Proposition 8.1.1.

Lemma 8.1.3. Let X ∈ Cn×n and P ∈ GL(n, C). Then

exp(XT

)= exp(X)T,

exp(X∗) = exp(X)∗,exp(PXP−1) = P exp(X)P−1.

Proof. For the adjoint X∗,

exp(X∗) =∞∑

k=0

1k!

(X∗)k =∞∑

k=0

1k!

(Xk)∗ =

( ∞∑k=0

1k!

Xk

)∗= exp(X)∗,

and similarly for the transpose XT. Finally,

exp(PXP−1) =∞∑

k=0

1k!

(PXP−1)k =∞∑

k=0

1k!

PXkP−1 = P exp(X)P−1. �

Proposition 8.1.4. If λ ∈ C is an eigenvalue of X ∈ Cn×n then eλ is an eigenvalueof exp(X). Consequently

det(exp(X)) = eTr(X).

Proof. Choose P ∈ GL(n, C) such that Y := PXP−1 ∈ Cn×n is upper triangular;the eigenvalues of X and Y are the same, and for triangular matrices the eigen-values are the diagonal elements. Since Y k is upper triangular for every k ∈ N,


exp(Y ) is upper triangular. Moreover, (Y k)jj = (Yjj)k, so that (exp(Y ))jj = eYjj .The eigenvalues of exp(X) and exp(Y ) = P exp(X)P−1 are the same. The deter-minant of a matrix is the product of its eigenvalues; the trace of a matrix is thesum of its eigenvalues; this implies the last claim. �Remark 8.1.5. Recall that HOM(G, H) is the set of continuous homomorphismsfrom G to H, see Definition 7.1.12.

Theorem 8.1.6 (The form of HOM(R, GL(n, C))). We have

HOM(R, GL(n, C)) ={t �→ exp(tX) | X ∈ Cn×n

}.

Proof. It is clear that (t �→ exp(tX)) ∈ HOM(R, GL(n, C)), since it is continuousand exp(sX) exp(tX) = exp((s + t)X).

Let φ ∈ HOM(R, GL(n, C)). Then φ(s + t) = φ(s)φ(t) implies that(∫ h

0

φ(s) ds

)φ(t) =

∫ h

0

φ(s + t) ds =∫ t+h

t

φ(u) du.

Recall that if ‖I −A‖L(Cn) < 1 then A ∈ Cn×n is invertible; now∥∥∥∥∥I − 1h

∫ h

0

φ(s) ds

∥∥∥∥∥L(Cn)

=

∥∥∥∥∥ 1h

∫ h

0

(I − φ(s)) ds

∥∥∥∥∥L(Cn)

≤ sups: |s|≤|h|

‖I − φ(s)‖L(Cn)

< 1

when |h| is small enough, because φ(0) = I and φ is continuous. Therefore∫ h

0φ(s) ds is invertible for small |h|, and we get

φ(t) =

(∫ h

0

φ(s) ds

)−1 ∫ t+h

t

φ(u) du.

Since φ is continuous, this formula states that φ is differentiable. Now

φ′(t) = lims→0

φ(s + t)− φ(t)s

= lims→0

φ(s)− φ(0)s

φ(t) = X φ(t),

where X := φ′(0). Hence the initial value problem{ψ′(t) = X ψ(t), ψ : R→ GL(n, C),ψ(0) = I

has the solutions ψ = φ and ψ = φX := (t �→ exp(tX)). Define α : R → GL(n, C)by α(t) := φ(t) φX(−t). Then α(0) = φ(0) φX(0) = I and

α′(t) = φ′(t) φX(−t)− φ(t) φ′X(−t)= X φ(t) φX(−t)− φ(t) X φX(−t) = 0,

since X φ(t) = φ(t) X. Thus α(t) = I for all t ∈ R, so that φ = φX . �

8.1. Exponential map 495

Proposition 8.1.7 (Logarithms). Let A ∈ Cn×n be such that ‖I−A‖L(Cn) < 1. Thelogarithm

log(A) := −∞∑

k=1

1k

(I −A)k

is well defined, and exp(log(A)) = A. Moreover, there exists r > 0 such thatlog(exp(X)) = X if ‖X‖L(Cn) < r.

Proof. Let c := ‖I −A‖ < 1 for a matrix A ∈ Cn×n. Then

∞∑k=1

1k

∥∥(I −A)k∥∥L(Cn)

≤∞∑

k=1

1k‖I −A‖k

L(Cn) ≤∞∑

k=1

ck =c

1− c<∞,

so that log(A) is well defined. Noticing that I and A commute, we have

exp(log(A)) =∞∑

k=0

1k!

(−

∞∑l=1

1l(I −A)l

)k

= A,

because if |1− a| < 1 for a number a ∈ C, then

eln a =∞∑

k=0

1k!

(−

∞∑l=1

1l(1− a)l

)k

= a. (8.1)

Due to the continuity of the exponential function, there exists r > 0 such that|1− ex| < 1 if x ∈ C satisfies |x| < r, and then

ln(ex) = −∞∑

l=1

1l(1− ex)l = −

∞∑l=1

1l

(1−

∞∑k=0

1k!

xk

)l

= x, (8.2)

so that if X ∈ Cn×n satisfies ‖X‖L(C) < r then

log(exp(X)) = −∞∑

l=1

1l(I − exp(X))l = −

∞∑l=1

1l

(I −

∞∑k=0

1k!

Xk

)l

= X. �

Exercise 8.1.8. Find an estimate for r in Proposition 8.1.7.

Exercise 8.1.9. Justify formulae (8.1) and (8.2) and their matrix forms.

Corollary 8.1.10. Let r be as above and B :={X ∈ Cn×n : ‖X‖L(Cn) < r

}. Then

(X �→ exp(X)) : B → exp(B) is a diffeomorphism (i.e., a bijective C∞-smoothmapping).

Proof. As exp and log are defined by power series, they are not just C∞-smoothbut also analytic. �


Lemma 8.1.11. Let X, Y ∈ Cn×n. Then

exp(X + Y ) = limm→∞ (exp(X/m) exp(Y/m))m

andexp([X, Y ]) = lim

m→∞ {exp(X/m), exp(Y/m)}m2

,

where [X, Y ] := XY − Y X and {a, b} := aba−1b−1.

Proof. As t→ 0,

exp(tX) exp(tY ) =(

I + tX +t2

2X2 +O(t3)

) (I + tY +

t2

2Y 2 +O(t3)

)= I + t(X + Y ) +

t2

2(X2 + 2XY + Y 2) +O(t3),

so that

{exp(tX), exp(tY )} =(

I + t(X + Y ) +t2

2(X2 + 2XY + Y 2) +O(t3)

)×

(I − t(X + Y ) +

t2

2(X2 + 2XY + Y 2) +O(t3)

)= I + t2(XY − Y X) +O(t3)

= I + t2[X, Y ] +O(t3).

Since exp is an injection in a neighbourhood of the origin 0 ∈ Cn×n, we have

exp(tX) exp(tY ) = exp(t(X + Y ) +O(t2)

),

{exp(tX), exp(tY )} = exp(t2[X, Y ] +O(t3)

)as t→ 0. Notice that exp(X)m = exp(mX) for all m ∈ N. Therefore we get

limm→∞ (exp(X/m) exp(Y/m))m = lim

m→∞ exp(X + Y +O(m−1)

)= exp(X + Y ),

limm→∞ {exp(X/m), exp(Y/m)}m2

= limm→∞ exp

([X, Y ] +O(m−1)

)= exp([X, Y ]). �

8.2 No small subgroups for Lie, please

Definition 8.2.1 (“No small subgroups” property). A topological group is said tohave the “no small subgroups” property if there exists a neighbourhood of theneutral element containing no non-trivial subgroups.

8.2. No small subgroups for Lie, please 497

We shall show that this property characterises Lie groups among compactgroups.Example. Let {Gj}j∈J be an infinite family of compact groups each having morethan one element. Let us consider the compact product group G :=

∏j∈J Gj . Let

Hj := {x ∈ G | ∀i ∈ J \ {j} : xi = eGi} .

Then Gj∼= Hj < G, and Hj is a non-trivial subgroup of G. If V ⊂ G is a

neighbourhood of e ∈ G then it contains all but perhaps finitely many Hj , due tothe definition of the product topology. Hence in this case G “has small subgroups”(i.e., has not the “no small subgroups” property).

Theorem 8.2.2 (Kernels of representations). Let G be a compact group and V ⊂ Gopen such that e ∈ V . Then there exists φ ∈ HOM(G, U(n)) for some n ∈ Z+ suchthat Ker(φ) ⊂ V .

Proof. First, {e} ⊂ G and G \ V ⊂ G are disjoint closed subsets of a compactHausdorff space G. By Urysohn’s Lemma (Theorem A.12.11), there exists f ∈C(G) such that f(e) = 1 and f(G \ V ) = {0}. Since trigonometric polynomialsare dense in C(G) by Theorem 7.6.2, we may take p ∈ TrigPol(G) such that‖p− f‖C(G) < 1/2. Then

H := span {πR(x)p | x ∈ G} ⊂ L2(μG)

is a finite-dimensional vector space, andH inherits the inner product from L2(μG).Let A : H → Cn be a linear isometry, where n = dim(H). Let us identify U(Cn)with U(n). Define φ ∈ Hom(G, U(n)) by

φ(x) := A πR(x)|H A−1.

Then φ is clearly a continuous unitary representation. For every x ∈ G \ V ,

|p(x)− 0| = |p(x)− f(x)| ≤ ‖p− f‖C(G) < 1/2,

so that p(x) = p(e), because

|p(e)− 1| = |p(e)− f(e)| ≤ ‖p− f‖C(G) < 1/2;

consequently πR(x)p = p. Thus Ker(φ) ⊂ V . �Corollary 8.2.3 (Characterisation of linear Lie groups). Let G be a compact group.Then G has no small subgroups if and only if it is isomorphic to a linear Lie group.

Proof. Let G be a compact group without small subgroups. By Theorem 8.2.2, forsome n ∈ Z+ there exists an injective φ ∈ HOM(G, U(n)). Then (x �→ φ(x)) : G→φ(G) is an isomorphism, and a homeomorphism by Proposition A.12.7, because φis continuous, G is compact and U(n) is Hausdorff. Thus φ(G) < U(n) < GL(n, C)is a compact linear Lie group.


Conversely, suppose G < GL(n, C) is closed. Recall that the mapping (X �→exp(X)) : B→ exp(B) is a homeomorphism, where

B ={X ∈ Cn×n : ‖X‖L(Cn) < r

}for some small r > 0. Hence V := exp(B/2)∩G is a neighbourhood of I ∈ G. In thesearch for a contradiction, suppose there exists a nontrivial subgroup H < G suchthat A ∈ H ⊂ V and A = I. Then 0 = log(A) ∈ B/2, so that m log(A) ∈ B\ (B/2)for some m ∈ Z+. Thereby

exp(m log(A)) = exp(log(A))m = Am ∈ H ⊂ V ⊂ exp(B/2),

but alsoexp(m log(A)) ∈ exp(B \ (B/2)) = exp(B) \ exp(B/2);

this is a contradiction. �Remark 8.2.4. Actually, it is shown above that Lie groups have no small subgroups;compactness played no role in this part of the proof.

Exercise 8.2.5. Use the Peter–Weyl Theorem 7.5.14 to provide an alternative prooffor Theorem 8.2.2. Hint: For each x ∈ G \ V there exists φx ∈ HOM(G, U(nx))such that x ∈ Ker(φx), because. . .

8.3 Lie groups and Lie algebras

Next we deal with representation theory of Lie groups. We introduce Lie alge-bras, which sometimes still bear the archaic label “infinitesimal groups”, quiteadequately describing their essence: a Lie algebra is a sort of locally linearisedversion of a Lie group.

Definition 8.3.1 (Lie algebras). A K-Lie algebra is a K-vector space V endowedwith a bilinear mapping ((a, b) �→ [a, b]V = [a, b]) : V × V → V satisfying

[a, a] = 0 and [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0

for all a, b, c ∈ V ; the second identity is called the Jacobi identity. Notice that here[a, b] = −[b, a] for all a, b ∈ V . A vector subspace W ⊂ V of a Lie algebra V iscalled a Lie subalgebra if [a, b] ∈ W for all a, b ∈ W (and thus W is a Lie algebrain its own right). A linear mapping A : V1 → V2 between Lie algebras V1, V2 iscalled a Lie algebra homomorphism if [Aa,Ab]V2 = A[a, b]V1 for all a, b ∈ V1.

Example.

1. For a K-vector space V , the trivial Lie product [a, b] := 0 gives a trivial Liealgebra.

2. A K-algebra A can be endowed with the canonical Lie product

(a, b) �→ [a, b] := ab− ba;

8.3. Lie groups and Lie algebras 499

this Lie algebra is denoted by LieK(A). Important special cases of such Liealgebras are

LieK(Cn×n) ∼= LieK(End(Cn)), LieK(End(V )), LieK(L(X)),

where X is a normed space and End(V ) is the algebra of linear operatorsV → V on a vector space V . For short, let

gl(V ) := LieR(End(V )).

3. (Derivations of algebras). Let D(A) be the K-vector space of derivations ofa K-algebra A; that is, D ∈ D(A) if it is a linear mapping A → A satisfyingthe Leibniz property

D(ab) = D(a) b + a D(b)

for all a, b ∈ A. Then D(A) has a Lie algebra structure given by [D,E] :=DE − ED. An important special case is A = C∞(M), where M is a C∞-manifold; if C∞(M) is endowed with the topology of local uniform conver-gence for all derivatives, then D ∈ D(C∞(M)) is continuous if and only ifit is a linear first-order partial differential operator with smooth coefficients(alternatively, a smooth vector field on M).

Definition 8.3.2. The Lie algebra Lie(G) = g of a linear Lie group G is introducedin the following Theorem 8.3.3:

Theorem 8.3.3 (Lie algebras of linear Lie groups). Let G < GL(n, C) be closed.The R-vector space

Lie(G) = g :={X ∈ Cn×n | ∀t ∈ R : exp(tX) ∈ G

}is a Lie subalgebra of the R-Lie algebra LieR(Cn×n) ∼= gl(Cn).

Proof. Let X, Y ∈ g and λ ∈ R. Trivially, exp(tλX) ∈ G for all t ∈ R, yieldingλX ∈ g. Since G is closed and exp is continuous,

G � (exp(tX/m) exp(tY/m))m −−−−→m→∞ exp (t(X + Y )) ∈ G

G � {exp(tX/m), exp(Y/m)}m2

−−−−→m→∞ exp (t[X, Y ]) ∈ G

by Lemma 8.1.11. Thereby X + Y, [X, Y ] ∈ g. �Exercise 8.3.4. Let X ∈ Cn×n be such that exp(tX) = I for all t ∈ R. Show thatX = 0.

Exercise 8.3.5. Let g ⊂ Cn×n be the Lie algebra of a linear Lie group G <GL(n, R). Show that g ⊂ Rn×n.

Definition 8.3.6 (Dimension of a linear Lie group). Let G be a linear Lie groupand g = Lie(G). The dimension of G is dim(G) := dim(g) = k, hence g ∼= Rk as avector space.


Remark 8.3.7 (Exponential coordinates). From Theorem 8.1.6 it follows that

HOM(R, G) = {t �→ exp(tX) | X ∈ g} .

The mapping (X �→ exp(X)) : g → G is a diffeomorphism in a small neighbour-hood of 0 ∈ g. Hence, given a vector space basis for g ∼= Rk, a small neighbourhoodof exp(0) = I ∈ G is endowed with the so-called exponential coordinates. If G iscompact and connected then exp(g) = G, so that the exponential map may “wrapg around G”; we shall not prove this.Remark 8.3.8. Informally speaking, if X, Y ∈ g are near 0 ∈ g, x := exp(X) andy := exp(Y ) then x, y ∈ G are near I ∈ G and

exp(X + Y ) ≈ xy, exp([X, Y ]) ≈ {x, y} = xyx−1y−1.

In a sense, the Lie algebra g is the infinitesimally linearised G near I ∈ G.Remark 8.3.9 (Lie algebra as invariant vector fields). The Lie algebra g can beidentified with the tangent space of G at the identity I ∈ G. Using left-translations(resp. right-translations), g can be identified with the set of left-invariant (resp.right-invariant) vector fields on G, and vector fields have a natural interpretationas first-order partial differential operators on G: For x ∈ G, X ∈ g and f ∈ C∞(G),define

LXf(x) :=ddt

f (x exp(tX))∣∣∣∣t=0

,

RXf(x) :=ddt

f (exp(tX) x)∣∣∣∣t=0

.

Then πL(y)LXf = LXπL(y)f and πR(y)RXf = RXπR(y)f for all y ∈ G, whereπL, πR are the left and right regular representations of G, respectively.

Definition 8.3.10 (Abbreviations for Lie algebras). Some usual abbreviations are

gl(n, K) = Lie(GL(n, K)),sl(n, K) = Lie(SL(n, K)),

o(n) = Lie(O(n)),so(n) = Lie(SO(n)),u(n) = Lie(U(n)),

su(n) = Lie(SU(n)),

and so on.

Exercise 8.3.11. Calculate the dimensions of the linear Lie groups mentioned inDefinition 8.3.10.

Proposition 8.3.12. Let G, H be linear Lie groups having the respective Lie algebrasg, h. Let ψ ∈ HOM(G, H). Then for every X ∈ g there exists a unique Y ∈ h suchthat ψ(exp(tX)) = exp(tY ) for all t ∈ R.


Proof. Let X ∈ g. Then φ := (t �→ ψ(exp(tX))) : R → H is a continuous homo-morphism, so that φ = (t �→ exp(tY )), where Y = φ′(0) ∈ h. �Proposition 8.3.13. Let F,G, H be closed subgroups of GL(n, C), with their respec-tive Lie algebras f, g, h. Then

(a) H < G⇒ h ⊂ g,(b) the Lie algebra of F ∩G is f ∩ g,(c) the Lie algebra cI of the component CI < G of the neutral element I is g.

Proof. (a) If H < G and X ∈ h then exp(tX) ∈ H ⊂ G for all t ∈ R, so thatX ∈ g.

(b) Let e be the Lie algebra of F ∩ G. By (a), e ⊂ f ∩ g. If X ∈ f ∩ g thenexp(tX) ∈ F ∩G for all t ∈ R, so that X ∈ e. Hence e = f ∩ g.

(c) By (a), cI ⊂ g. Let X ∈ g. Now the connectedness of R (Theorem A.16.9)and the continuity of t �→ exp(tX) by Proposition A.16.3 imply the connectednessof

{exp(tX) : t ∈ R} � exp(0) = I.

Thereby {exp(tX) : t ∈ R} ⊂ CI , so that X ∈ cI . �Example (Lie algebra of SL(n, K)). Let us compute the Lie algebra sl(n, K) of thelinear Lie group

SL(n, K) = {A ∈ GL(n, K) | det(A) = 1} .

Notice that sl(n, K) ⊂ Kn×n by Exercise 8.3.5. Hence

sl(n, K) :={X ∈ Kn×n | ∀t ∈ R : exp(tX) ∈ SL(n, K)

}=

{X ∈ Kn×n | ∀t ∈ R : exp(tX) ∈ Kn×n, det(exp(tX)) = 1

}.

Let {λj}nj=1 ⊂ C be the set of eigenvalues of X ∈ Kn×n. The characteristic

polynomial (z �→ det(zI −X)) : C→ C of X satisfies

det(zI −X) =n∏

j=1

(z − λj)

= zn − zn−1n∑

j=1

λj + · · ·+ (−1)nn∏

j=1

λj

= zn − zn−1Tr(X) + · · ·+ (−1)ndet(X),

We know that X is similar to an upper triangular matrix Y = PXP−1 for someP ∈ GL(n, K). Since

det(zI − PXP−1) = det(P (zI −X)P−1)= det(P ) det(zI −X) det(P−1)= det(zI −X),


the eigenvalues of X and Y are the same, and they are on the diagonal of Y .Evidently, {eλj}n

j=1 ⊂ C is the set of the eigenvalues of both exp(Y ) and exp(X) =P−1 exp(Y )P . Since the determinant is the product of the eigenvalues and thetrace is the sum of the eigenvalues, we have

det(exp(X)) =n∏

j=1

eλj = e∑ n

j=1 λj = eTr(X)

(see also Proposition 8.1.4). Therefore X ∈ sl(n, K) if and only if Tr(X) = 0 andexp(tX) ∈ Kn×n for all t ∈ R. Thus

sl(n, K) ={X ∈ Kn×n | Tr(X) = 0

}as the reader may check.

Next we ponder the relationship between Lie group and Lie algebra homo-morphisms.

Definition 8.3.14 (Differential homomorphisms). Let G, H be linear Lie groupswith respective Lie algebras g, h. The differential homomorphism of ψ ∈HOM(G, H) is the mapping ψ′ = Lie(ψ) : g→ h defined by

ψ′(X) :=ddt

ψ(exp(tX))∣∣∣∣t=0

.

Remark 8.3.15. Above, ψ′ is well defined since

f := (t �→ ψ(exp(tX))) ∈ HOM(R,H)

is of the form t �→ exp(tY ) for some Y ∈ h, as a consequence of Theorem 8.1.6.Moreover, Y = f ′(0) = ψ′(X) holds, so that

ψ(exp(tX)) = exp(tψ′(X)).

Theorem 8.3.16. Let F,G, H be linear Lie groups with respective Lie algebras f, g, h.Let φ ∈ HOM(F,G) and ψ ∈ HOM(G, H). The mapping ψ′ : g → h defined inDefinition 8.3.14 is a Lie algebra homomorphism. Moreover,

(ψ ◦ φ)′ = ψ′φ′ and Id′G = Idg,

where IdG = (x �→ x) : G→ G and Idg = (X �→ X) : g→ g.

Proof. Let X, Y ∈ g and λ ∈ R. Then

ψ′(λX) =ddt

ψ(exp(tλX))|t=0

= λddt

ψ(exp(tX))|t=0

= λψ′(X).


If t ∈ R then

exp (tψ′(X + Y )) = ψ (exp(tX + tY ))

= ψ(

limm→∞ (exp(tX/m) exp(tY/m))m

)= lim

m→∞ (ψ(exp(tX/m)) ψ(exp(tY/m)))m

= limm→∞ (exp(tψ′(X)/m) exp(tψ′(Y )/m))m

= exp(t(ψ′(X) + ψ′(Y ))),

so that tψ′(X +Y ) = t (ψ′(X) + ψ′(Y )) for small enough |t|, as we recall that expis injective in a small neighbourhood of 0 ∈ g. Consequently, ψ′ : g → h is linear.Next,

exp (tψ′([X, Y ])) = ψ (exp(t[X, Y ]))

= ψ(

limm→∞ {exp(tX/m), exp(tY/m)}m2

)= lim

m→∞ {exp(tψ′(X)/m), exp(tψ′(Y )/m)}m2

= exp (t[ψ′(X), ψ′(Y )]) ,

so that we get ψ′([X, Y ]) = [ψ′(X), ψ′(Y )]. Thus ψ′ : g → h is a Lie algebrahomomorphism.

If Z ∈ f then

(ψ ◦ φ)′(Z) =ddt

ψ (φ(exp(tZ))) |t=0

=ddt

ψ (exp(tφ′(Z))) |t=0

= ψ′(φ′(Z)).

Finally, ddt exp(tX)|t=0 = X, yielding Id′G = Idg. �

Remark 8.3.17. Notice that isomorphic linear Lie groups must have isomorphicLie algebras. Now we know that a continuous Lie group homomorphism ψ cannaturally be linearised to get a Lie algebra homomorphism ψ′, so that we havethe commutative diagram

Gψ−−−−→ H,

exp

F⏐⏐ F⏐⏐exp

gψ′

−−−−→ h.

If we are given a Lie algebra homomorphism f : g → h, does there exist φ ∈HOM(G, H) such that φ′ = f? This problem is studied in the following exercises.


Definition 8.3.18 (Simply connected spaces). A topological space X is said to besimply connected if X is path-connected and if every closed curve in X can beshrunken to a point continuously in the set X.

Exercise 8.3.19. Show that the groups SU(n) and SL(n, C) are both connectedand simply connected.

Exercise 8.3.20. Show that the groups U(n) and GL(n, C) are connected but notsimply connected.

Exercise 8.3.21. Let G, H be linear Lie groups such that G is simply connected.Let f : g → h be a Lie algebra homomorphism. Show that there exists φ ∈HOM(G, H) such that φ′ = f . (This is a rather demanding task unless one knowsthat exp : g→ G is surjective and uses Lemma 8.1.11. A proof can be found, e.g.,in [37].)

Exercise 8.3.22. Related to Exercise 8.3.21, give an example of a non-simply-connected G and a homomorphism f : g→ h which is not of the form f = φ′.

Lemma 8.3.23. Let g be the Lie algebra of a linear Lie group G, and

S :={exp(X1) · · · exp(Xm) | m ∈ Z+, {Xj}m

j=1 ⊂ g}

.

Then S = CI , the component of I ∈ G.

Proof. Now S < G is path-connected, since

(t �→ exp(tX1) · · · exp(tXm)) : [0, 1]→ S

is continuous, connecting I ∈ S to the point exp(X1) · · · exp(Xm) ∈ S. For asmall enough neighbourhood U ⊂ g of 0 ∈ g, we have a homeomorphism (X �→exp(X)) : U → exp(U). Because of

exp(X1) · · · exp(Xm) ∈ exp(X1) · · · exp(Xm) exp(U) ⊂ S,

it follows that S < G is open. But open subgroups are always closed, as the readercan easily verify. Thus S � I is connected, closed and open, so that S = CI . �Corollary 8.3.24. Let G, H be linear Lie groups and φ, ψ ∈ HOM(G, H). Then:

(a) Lie(Ker(ψ)) = Ker(ψ′).(b) If G is connected and φ′ = ψ′ then φ = ψ.(c) Let H be connected; then ψ′ is surjective if and only if ψ is surjective.

Proof. (a) Ker(ψ) < G < GL(n, C) is a closed subgroup, since ψ is a continuoushomomorphism. Thereby

Lie(Ker(ψ)) ={X ∈ Cn×n | ∀t ∈ R : exp(tX) ∈ Ker(ψ)

}=

{X ∈ Cn×n | ∀t ∈ R : exp(tψ′(X)) = ψ(exp(tX)) = I

}=

{X ∈ Cn×n | ψ′(X) = 0

}= Ker(ψ′).


(b) Take A ∈ G. Then A = exp(X1) · · · exp(Xm) for some {Xj}mj=1 ⊂ g by

Lemma 8.3.23, so that

φ(A) = exp (φ′(X1)) · · · exp (φ′(Xm))= exp (ψ′(X1)) · · · exp (ψ′(Xm))= ψ(A).

(c) Suppose ψ′ : g→ h is surjective. Let B ∈ H. Now H is connected, so thatLemma 8.3.23 says that B = exp(Y1) · · · exp(Ym) for some {Yj}m

j=1 ⊂ h. Exploitthe surjectivity of ψ′ to obtain Xj ∈ g such that ψ′(Xj) = Yj . Then

ψ (exp(X1) · · · exp(Xm)) = ψ (exp(X1)) · · ·ψ (exp(Xm))= exp(Y1) · · · exp(Ym)= B.

Conversely, suppose ψ : G→ H is surjective. Trivially, ψ′(0) = 0 ∈ h; let 0 = Y ∈h. Let r0 := r/‖Y ‖, where r is as in Proposition 8.1.7; notice that if |t| < r0 thenlog(exp(tY )) = tY . The surjectivity of ψ guarantees that for every t ∈ R thereexists At ∈ G such that ψ(At) = exp(tY ). The set R := {At : 0 < t < r0} isuncountable, so that it has an accumulation point x ∈ Cn×n; and x ∈ G, becauseR ⊂ G and G ⊂ Cn×n is closed. Let ε > 0. Then there exist s, t ∈]0, r0[ such thats = t and

‖As − x‖ < ε, ‖At − x‖ < ε,∥∥A−1

s − x−1∥∥ < ε.

Thereby ∥∥A−1s At − I

∥∥ =∥∥A−1

s (At −As)∥∥

≤∥∥A−1

s

∥∥ (‖At − x‖+ ‖x−As‖)≤

(‖x−1‖+ ε

)2ε.

Hence we demand ‖A−1s At − I‖ < 1 and ‖ψ(A−1

s At)− I‖ < 1, yielding

ψ(A−1s At) = ψ(As)−1ψ(At) = exp((t− s)Y ).

Consequentlyψ′

(log(A−1

s At))

= (t− s)Y.

Therefore ψ′(

1t−s log(A−1

s At))

= Y . �

Definition 8.3.25 (Adjoint representation of Lie groups). The adjoint representa-tion of a linear Lie group G is the mapping Ad ∈ HOM(G, Aut(g)) defined by

Ad(A)X := AXA−1,

where A ∈ G and G ∈ g.


Remark 8.3.26. Indeed, Ad : G→ Aut(g), because

exp (tAd(A)X) = exp(tAXA−1

)= A exp (tX) A−1

belongs to G if A ∈ G, X ∈ g and t ∈ R. It is a homomorphism, since

Ad(AB)X = ABXB−1A−1 = Ad(A)(BXB−1) = Ad(A) Ad(B) X,

and Ad is trivially continuous.

Exercise 8.3.27. Let g be a Lie algebra. Consider Aut(g) as a linear Lie group.Show that Lie(Aut(g)) and gl(g) are isomorphic as Lie algebras.

Definition 8.3.28 (Adjoint representation of Lie algebras). The adjoint represen-tation of the Lie algebra g of a linear Lie group G is the differential representation

ad = Ad′ : g→ Lie(Aut(g)) ∼= gl(g),

that is ad(X) := Ad′(X), so that

ad(X)Y =ddt

(exp(tX)Y exp(−tX)) |t=0

=((

ddt

exp(tX))

Y exp(−tX) + exp(tX)Yddt

exp(−tX))|t=0

= XY − Y X

= [X, Y ].

Remark 8.3.29. Notice that the diagram commutes:

GAd−−−−→ Aut(G)

exp

F⏐⏐ F⏐⏐exp

gAd′=ad−−−−−→ Lie(Aut(g)).

8.3.1 Universal enveloping algebra

Here we discuss the universal enveloping algebra.Remark 8.3.30 (Universal enveloping algebra informally). We are going to studyhigher-order partial differential operators on G. Let g be the Lie algebra of alinear Lie group G. Next we construct a natural associative algebra U(g) generatedby g modulo an ideal, enabling embedding g into U(g). Recall that g can beinterpreted as the vector space of first-order left- (or right-) translation invariantpartial differential operators on G. Consequently, U(g) can be interpreted as thevector space of finite-order left- (or right-) translation invariant partial differentialoperators on G.


Definition 8.3.31 (Universal enveloping algebra). Let g be a K-Lie algebra. Let

T :=∞⊕

m=0

⊗mg

be the tensor product algebra of g, where ⊗mg denotes the m-fold tensor productg⊗ · · · ⊗ g; that is, T is the linear span of the elements of the form

λ001 +M∑

m=1

Km∑k=1

λmk Xmk1 ⊗ · · · ⊗Xmkm,

where 1 is the formal unit element of T , λmk ∈ K, Xmkj ∈ g and M,Km ∈ Z+;the product of T is begotten by the tensor product, i.e.,

(X1 ⊗ · · · ⊗Xp)(Y1 ⊗ · · · ⊗ Yq) := X1 ⊗ · · · ⊗Xp ⊗ Y1 ⊗ · · · ⊗ Yq

is extended to a unique bilinear mapping T × T → T . Let J be the (two-sided)ideal in T spanned by the set

O := {X ⊗ Y − Y ⊗X − [X, Y ] : X, Y ∈ g} ;

i.e., J ⊂ T is the smallest vector subspace such that O ⊂ J and DE, ED ∈ Jfor every D ∈ J and E ∈ T (in a sense, J is a “huge zero” in T ). The quotientalgebra

U(g) := T /Jis called the universal enveloping algebra of g.

Definition 8.3.32 (Canonical mapping of a Lie algebra). Let ι : T → U(g) = T /Jbe the quotient mapping t �→ t + J . A natural interpretation is that g ⊂ T . Therestricted mapping ι|g : g→ U(g) is called the canonical mapping of g.

Remark 8.3.33. Notice that ι|g : g→ LieK(U(g)) is a Lie algebra homomorphism:it is linear and

ι|g([X, Y ]) = ι([X, Y ])= ι(X ⊗ Y − Y ⊗X)= ι(X)ι(Y )− ι(Y )ι(X)= ι|g(X)ι|g(Y )− ι|g(Y )ι|g(X)= [ι|g(X), ι|g(Y )].

Theorem 8.3.34 (Universality of the enveloping algebra). Let g be a K-Lie algebra,ι|g : g→ U(g) its canonical mapping, A an associative K-algebra, and

σ : g→ LieK(A)


a Lie algebra homomorphism. Then there exists a unique algebra homomorphism

σ : U(g)→ A

satisfying σ (ι|g(X)) = σ(X) for all X ∈ g, i.e.,

U(g) σ−−−−→ A

ι|gF⏐⏐ ∥∥∥g

σ−−−−→ LieK(A).

Proof. Let us define a linear mapping σ0 : T → A by

σ0(X1 ⊗ · · · ⊗Xm) := σ(X1) · · ·σ(Xm). (8.3)

Then σ0(J ) = {0}, since

σ0(X ⊗ Y − Y ⊗X − [X, Y ]) = σ(X)σ(Y )− σ(Y )σ(X)− σ([X, Y ])= σ(X)σ(Y )− σ(Y )σ(X)− [σ(X), σ(Y )]= 0.

Hence if t, u ∈ T and t − u ∈ J then σ0(t) = σ0(u). Thereby we may defineσ := (t + J �→ σ0(t)) : U(g) → A. Finally, it is clear that σ is an algebrahomomorphism making the diagram above commute. The uniqueness is clear byconstruction since (8.3) must hold. �

Corollary 8.3.35 (Ado–Iwasawa Theorem). Let g be the Lie algebra of a linear Liegroup G. Then the canonical mapping ι|g : g→ U(g) is injective.

Proof. Let σ = (X �→ X) : g → gl(n, C). Due to the universality of U(g) thereexists an R-algebra homomorphism σ : U(g)→ Cn×n such that σ(X) = σ (ι|g(X))for all X ∈ G, i.e.,

U(g) σ−−−−→ Cn×n

ι|gF⏐⏐ ∥∥∥g

σ−−−−→ gl(n, C).

Then ι|g is injective because σ is injective. �

Remark 8.3.36. By the Ado–Iwasawa Theorem (Corollary 8.3.35), the Lie algebrag of a linear Lie group can be considered as a Lie subalgebra of LieR (U(g)).

Definition 8.3.37 (ad). Let g be a K-Lie algebra. Let us define the linear mappingad : g→ End(g) by ad(X)Z := [X, Z].


Remark 8.3.38. Let g be a K-Lie algebra and X, Z ∈ G. Since

0 = [[X, Y ], Z] + [[Y,Z], X] + [[Z,X], Y ]= [[X, Y ], Z]− ([X, [Y, Z]]− [Y, [X, Z]])= ad([X, Y ])Z − [ad(X), ad(Y )]Z,

we notice thatad([X, Y ]) = [ad(X), ad(Y )],

i.e., ad is a Lie algebra homomorphism g→ gl(g).

Definition 8.3.39 (Killing form and semisimple Lie groups). The Killing form ofthe Lie algebra g is the bilinear mapping B : g× g→ K, defined by

B(X, Y ) := Tr (ad(X) ad(Y ))

(recall that by Exercise B.5.41, on a finite-dimensional vector space the trace canbe defined independent of any inner product). A (R- or C-)Lie algebra g is calledsemisimple if its Killing form is non-degenerate, i.e., if

∀X ∈ g \ {0} ∃Y ∈ g : B(X, Y ) = 0;

equivalently, B is non-degenerate if the matrix (B(Xi, Xj))ni,j=1 is invertible, where

{Xj}nj=1 ⊂ g is a vector space basis. A connected linear Lie group is called semisim-

ple if its Lie algebra is semisimple.

Example. Linear Lie groups SL(n, K) and SO(n) are semisimple, but GL(n) is notsemisimple.Remark 8.3.40. Since Tr(ab) = Tr(ba), we have

B(X, Y ) = B(Y, X).

We also haveB(X, [Y,Z]) = B([X, Y ], Z),

because

Tr(a(bc− cb)) = Tr(abc)− Tr(acb) = Tr(abc)− Tr(bac) = Tr((ab− ba)c)

yields

B(X, [Y,Z]) = Tr (ad(X) ad([Y, Z]))= Tr (ad(X) [ad(Y ), ad(Z)])= Tr ([ad(X), ad(Y )] ad(Z))= Tr (ad([X, Y ]) ad(Z))= B([X, Y ], Z).

It can be proven that the Killing form of the Lie algebra of a compact linear Liegroup is negative semi-definite, i.e., B(X, X) ≤ 0. On the other hand, if the Killingform of a Lie group is negative definite, i.e., B(X, X) < 0 whenever X = 0, thenthe group is compact.


8.3.2 Casimir element and Laplace operator

Here we discuss some properties of the Casimir element and the correspondingLaplace operator.

Definition 8.3.41 (Casimir element). Let g be a semisimple K-Lie algebra with avector space basis {Xj}n

j=1 ⊂ g. Let B : g× g → K be the Killing form of g, anddefine the matrix R ∈ Kn×n by Rij := B(Xi, Xj). Let

Xi :=n∑

j=1

(R−1

)ij

Xj ,

so that {Xi}ni=1 is another vector space basis for g. Then the Casimir element

Ω ∈ U(g) of g is defined by

Ω :=n∑

i=1

XiXi.

Remark 8.3.42. The Casimir element Ω ∈ U(g) for the Lie algebra g of a compactsemisimple linear Lie group G can be considered as an elliptic linear second-order(left and right) translation invariant partial differential operator. In a sense, theCasimir operator is an analogy of the Euclidean Laplace operator

L =n∑

j=1

∂2

∂x2j

: C∞(Rn)→ C∞(Rn).

Such a Laplace operator can be constructed for any compact Lie group G, andwith it we may define Sobolev spaces on G nicely, etc.

Theorem 8.3.43 (Properties of Casimir element). The Casimir element of a finite-dimensional semisimple K-Lie algebra g is independent of the choice of the vectorspace basis {Xj}n

j=1 ⊂ g. Moreover,

DΩ = ΩD

for all D ∈ U(g).

Proof. Let {Xj}nj=1 ⊂ g, Rij = B(Xi, Xj) and Ω be as in Definition 8.3.41. To

simplify notation, we consider only the case K = R. Let {Yi}ni=1 ⊂ g be a vector

space basis of g. Then there exists A = (Aij)ni,j=1 ∈ GL(n, R) such that⎧⎨⎩Yi :=

n∑j=1

AijXj

⎫⎬⎭n

i=1

.


Then

S := (B(Yi, Yj))ni,j=1

=

(B

( n∑k=1

AikXk,

n∑l=1

AjlXl

))n

i,j=1

=

⎛⎝ n∑k,l=1

Aik B(Xk, Xl) Ajl

⎞⎠n

i,j=1

= ARAT;

henceS−1 = ((S−1)ij)n

i,j=1 = (AT)−1R−1A−1.

Let us now compute the Casimir element of g with respect to the basis {Yj}nj=1:

n∑i,j=1

(S−1)ijYiYj =n∑

i,j=1

(S−1)ij

n∑k=1

AikXk

n∑l=1

AjlXl

=n∑

k,l=1

XkXl

n∑i,j=1

Aik(S−1)ijAjl

=n∑

k,l=1

XkXl

n∑i,j=1

(AT)ki((AT)−1R−1A−1)ijAjl

=n∑

k,l=1

XkXl(R−1)kl.

Thus the definition of the Casimir element does not depend on the choice of avector space basis.

We still have to prove that Ω commutes with every D ∈ U(g). Since

B(Xi, Xj) =n∑

k=1

(R−1)ikB(Xk, Xj) =n∑

k=1

(R−1)ikRkj = δij ,

we can extend (Xi, Xj) �→ 〈Xi, Xj〉g := B(Xi, Xj) uniquely to an inner product

((X, Y ) �→ 〈X, Y 〉g) : g× g→ R,

with respect to which the collection {Xi}ni=1 is an orthonormal basis. For the Lie

product (x, y) �→ [x, y] := xy − yx of LieR(U(g)) we have

[x, yz] = [x, y]z + y[x, z],


so that for D ∈ g we get

[D,Ω] = [D,

n∑i=1

XiXi] =

n∑i=1

([D,Xi]Xi + Xi[D,Xi]

).

Let cij , dij ∈ R be defined by

[D,Xi] =n∑

j=1

cijXj , [D,Xi] =n∑

j=1

dijXj .

Then

cij = 〈Xj , [D,Xi]〉g = B(Xj , [D,Xi]) = B([Xj , D], Xi) = B(−[D,Xj ], Xi)

= B(−n∑

k=1

djkXk, Xi) = −n∑

k=1

djkB(Xk, Xi)

= −n∑

k=1

djk〈Xk, Xi〉g = −dji,

so that

[D,Ω] =n∑

i,j=1

(cijXjXi + dijXiX

j)

=n∑

i,j=1

(cij + dji)XjXi

= 0,

i.e., DΩ = ΩD for all D ∈ g. By induction, we may prove that

[D1D2 · · ·Dm,Ω] = D1[D2 · · ·Dm,Ω] + [D1,Ω]D2 · · ·Dm = 0

for every {Dj}mj=1 ⊂ g, so that DΩ = ΩD for all D ∈ U(g). �

Exercise 8.3.44. How should the proof of Theorem 8.3.43 be modified if K = Cinstead of K = R?

Definition 8.3.45 (Laplace operator on G). The Casimir element from Definition8.3.41, also denoted by

LG := Ω ∈ U(g),

and viewed as a second-order partial differential operator on G is also called theLaplace operator on G. Here a vector field Y ∈ g is viewed as a differential operatorY ≡ DY : C∞(G)→ C∞(G), defined by

Y f(x) ≡ DY f(x) =ddt

f(x exp(tY ))∣∣∣∣t=0

.


Remark 8.3.46. The Laplace operator LG is a negative definite bi-invariant oper-ator on G, by Theorem 8.3.43. If G is equipped with the unique (up to a constant)bi-invariant Riemannian metric, LG is its Laplace–Beltrami operator.

In the notation of the right and left Peter–Weyl theorem in Theorem 7.5.14and Remark 7.5.16, we write

Hφ :=dim φ⊕i=1

Hφi,· =

dim φ⊕j=1

Hφ·,j .

Theorem 8.3.47 (Eigenvalues of the Laplacian on G). For every [φ] ∈ G the spaceHφ is an eigenspace of LG and −LG|Hφ = λφI, for some λφ ≥ 0.

Proof. We will use the notation of Theorem 7.5.14. Note that by Theorem 8.3.43the Laplace operator LG is bi-invariant, so that it commutes with both πR(x) andπL(x), for all x ∈ G. Therefore, by the Peter–Weyl theorem it commutes withall φ ∈ G. Thus LG(Hφ

·,j) ⊂ Hφ·,j and LG(Hφ

i,·) ⊂ Hφi,·, for all 1 ≤ i, j ≤ dim(φ).

It follows that LGφij ∈ Hφi,·

⋂Hφ·,j = span(φij), so that LGφij = cijφij for some

constants cij . Let us now determine these constants. We have

(LGπR(y)φij)(x) = LG(φij(xy))

= LG

⎛⎝dim(φ)∑k=1

φik(x)φkj(y)

⎞⎠=

dim(φ)∑k=1

cikφik(x)φkj(y).

On the other hand we have

(πR(y)LGφij)(x) = cijφij(xy)

=dim(φ)∑

k=1

cijφik(x)φkj(y).

It follows now from the orthogonality Lemma 7.5.12 that cikφkj(y) = cijφkj(y), orthat cik = cij for all 1 ≤ i, j, k ≤ dim(φ). A similar calculation with the left regularaction πL(y) shows that ckj = cij for all 1 ≤ i, j, k ≤ dim(φ). Hence LGφij = cφij

for all 1 ≤ i, j ≤ dim(φ), and since LG is negative definite, we obtain the statementwith λφ := −c ≥ 0. �

Chapter 9

Hopf Algebras

Instead of studying a compact group G, we may consider the algebra C(G) ofcontinuous functions G→ C. The structure of the group is encoded in the functionalgebra, but we shall see that this approach paves the way for a more generalfunctional analytic theory of Hopf algebras, which possess nice duality properties.

9.1 Commutative C∗-algebras

Let A := C(X), where X is a compact Hausdorff space. We present1 some funda-mental results:

• All the algebra homomorphisms A → C are of the form

f �→ f(x),

where x ∈ X.• All the closed ideals of A are of the form

I(K) := {f ∈ A | f(K) = {0}} ,

where K ⊂ X (with convention I(∅) := C(X)). Moreover, K = V (I(K)),where

V (J) =⋂f∈J

f−1({0});

these results follow by Urysohn’s Lemma (Theorem A.12.11).• Linear functionals A → C are of the form

f �→∫

X

f dμ, (9.1)

where μ is a Borel-regular measure on X; this is the Riesz RepresentationTheorem C.4.60.

1These statements follow essentially from the results in Part I

516 Chapter 9. Hopf Algebras

• Probability functionals A → C are then of the form (9.1), where μ is aBorel-regular probability measure on X.

All in all, we might say that the topology and measure theory of a compact Haus-dorff space X is encoded in the algebra A = C(X), with a dictionary:

Space X Algebra A = C(X)

homeomorphism φ : X → X isomorphism(f �→ f ◦ φ) : A → A

point x ∈ X algebra functional(f �→ f(x)) : A → C

closed set in X closed ideal in AX metrisable A separable

Borel-regular measure on X linear functionalBorel-regular probability measure on X probability functional

......

Remark 9.1.1. In the light of the dictionary above, we are bound to ask:

1. If X is a group, how is this reflected in C(X)?2. Could we study non-commutative algebras just like the commutative ones?

We might call the traditional topology and measure theory by the name “commuta-tive geometry”, referring to the commutative function algebras; “non-commutativegeometry” would refer to the study of non-commutative algebras. Let us now tryto deal with the two questions posed above.

Answering to question 1. Let G be a compact group. By Urysohn’s Lemma (The-orem A.12.11), C(G) separates the points of X, so that the associativity of thegroup operation ((x, y) �→ xy) : G×G→ G is encoded by

∀x, y, z ∈ G ∀f ∈ C(G) : f((xy)z) = f(x(yz)).

Similarly,

∃e ∈ G ∀x ∈ G ∀f ∈ C(G) : f(xe) = f(x) = f(ex)

encodes the neutral element e ∈ G. Finally,

∀x ∈ G ∃x−1 ∈ G ∀f ∈ C(G) : f(x−1x) = f(e) = f(xx−1)

encodes the inversion (x �→ x−1) : G→ G. Thereby let us define linear operators

Δ : C(G)→ C(G×G), Δf(x, y) := f(xy),ε : C(G)→ C, εf := f(e),

S : C(G)→ C(G), Sf(x) := f(x−1);

9.2. Hopf algebras 517

the interactions of these algebra homomorphisms contain all the information aboutthe structure of the underlying group! This is a key ingredient in the Hopf algebratheory.

Answering to question 2. Our algebras always have a unit element 1. An involutiveC-algebra A is a C∗-algebra if it has a Banach space norm satisfying

‖ab‖ ≤ ‖a‖ ‖b‖ and ‖a∗a‖ = ‖a‖2

for all a, b ∈ A. By Gelfand and Naimark (1943), see Theorem D.5.3, up to anisometric ∗-isomorphism a C∗-algebra is a closed involutive subalgebra of L(H),where H is a Hilbert space; moreover, if A is a commutative unital C∗-algebrathen A ∼= C(X) for a compact Hausdorff space X, as explained below:

The spectrum of A is the set Spec(A) of the algebra homomorphisms A → C(automatically bounded functionals!), endowed with the Gelfand topology, whichis the relative weak∗-topology of L(A, C). It turns out that Spec(A) is a compactHausdorff space. For a ∈ A we define the Gelfand transform

a : Spec(A)→ C, a(x) := x(a).

It turns out that a is continuous, and that

(a �→ a) : A → C(Spec(A))

is an isometric ∗-algebra isomorphism!If B is a non-commutative C∗-algebra, it still has plenty of interesting com-

mutative C∗-subalgebras so that the Gelfand transform provides the nice tools ofclassical analysis on compact Hausdorff spaces in the study of the algebra. Namely,if a ∈ B is normal, i.e., a∗a = aa∗, then the closure of the algebraic span (polyno-mials) of {a, a∗} is a commutative C∗-subalgebra. E.g., b∗b ∈ B is normal for allb ∈ B.

Synthesis of questions 1 and 2. By the Gelfand–Naimark Theorem D.5.11, thearchetypal commutative C∗-algebra is C(X) for a compact Hausdorff space X.In the sequel, we introduce Hopf algebras. In a sense, they are a not-necessarily-commutative analogy of C(G), where G is a compact group. We begin by formallydualising the category of algebras, to obtain the category of co-algebras. By mar-rying these concepts in a subtle way, we obtain the category of Hopf algebras.

9.2 Hopf algebras

The definition of a Hopf algebra is a lengthy one, yet quite natural. In the sequel,notice the evident dualities in the commutative diagrams.

For C-vector spaces V,W , we define τV W : V ⊗W → W ⊗ V by the linearextension of

τV W (v ⊗ w) := w ⊗ v.


Moreover, in the sequel the identity operation (v �→ v) : V → V for any vectorspace V is denoted by I. We constantly identify C-vector spaces V and C⊗V (andrespectively V ⊗C), since (λ⊗ v) �→ λv defines a linear isomorphism C⊗ V → V .

In the usual definition of an algebra, the multiplication is regarded as a bilin-ear map. In order to use dualisation techniques for algebras, we want to linearisethe multiplication. Let us therefore give a new, equivalent definition for an algebra:

Definition 9.2.1 (Reformulation of algebras). The triple

(A, m, η)

is an algebra (more precisely, an associative unital C-algebra) if A is a C-vectorspace, and

m : A⊗A → A,

η : C→ A

are linear mappings such that the following diagrams commute: the associativitydiagram

A⊗A⊗A I⊗m−−−−→ A⊗A

m⊗I

⏐⏐( ⏐⏐(m

A⊗A m−−−−→ Aand the unit diagrams

A⊗ CI⊗η−−−−→ A⊗A

a⊗λ�→λa

⏐⏐( ⏐⏐(m

A A,

A⊗A η⊗I←−−−− C⊗A

m

⏐⏐( ⏐⏐(λ⊗a�→λa

A A.

The mapping m is called the multiplication and η the unit mapping; the algebraA is said to be commutative if mτAA = m. The unit of an algebra (A, m, η) is

1A := η(1),

and the usual abbreviation for the multiplication is ab := m(a ⊗ b). For algebras(A1,m1, η1) and (A2,m2, η2) the tensor product algebra (A1⊗A2, m, η) is definedby

m := (m1 ⊗m2)(I ⊗ τA1A2 ⊗ I),

i.e., (a1 ⊗ a2)(b1 ⊗ b2) = (a1b1)⊗ (a2b2), and

η(1) := 1A1 ⊗ 1A2 .

Remark 9.2.2. If an algebra A = (A, m, η) is finite-dimensional, we can formallydualise its structural mappings m and η; this inspires the concept of the co-algebra:


Definition 9.2.3 (Co-algebras). The triple

(C,Δ, ε)

is a co-algebra (more precisely, a co-associative co-unital C-co-algebra) if C is aC-vector space and

Δ : C → C ⊗ C,ε : C → C

are linear mappings such that the following diagrams commute: the co-associativitydiagram (notice the duality to the associativity diagram)

C ⊗ C ⊗ C I⊗Δ←−−−− C ⊗ C

Δ⊗I

F⏐⏐ F⏐⏐Δ

C ⊗ C Δ←−−−− C

and the co-unit diagrams (notice the duality to the unit diagrams)

C ⊗ CI⊗ε←−−−− C ⊗ C

λc�→c⊗λ

F⏐⏐ Δ

F⏐⏐C C,

C ⊗ C ε⊗I−−−−→ C⊗ C

Δ

F⏐⏐ F⏐⏐λc�→λ⊗c

C C.

The mapping Δ is called the co-multiplication and ε the co-unit mapping; theco-algebra C is co-commutative if τCCΔ = Δ. For co-algebras (C1,Δ1, ε1) and(C2,Δ2, ε2) the tensor product co-algebra (C1 ⊗ C2,Δ, ε) is defined by

Δ := (I ⊗ τC1C2 ⊗ I)(Δ1 ⊗Δ2)

andε(c1 ⊗ c2) := ε1(c1)ε2(c2).

Example. A trivial co-algebra example: if (A, m, η) is a finite-dimensional algebrathen the vector space dual A′ = L(A, C) has a natural co-algebra structure. In-deed, let us identify (A⊗A)′ and A′ ⊗A′ naturally, so that m′ : A′ → A′ ⊗A′ isthe dual mapping to m : A⊗A → A. Let us identify C′ and C naturally, so thatη′ : A′ → C is the dual mapping to η : C→ A. Then

(A′,m′, η′)

is a co-algebra (draw the commutative diagrams!). We shall give more interestingexamples of co-algebras after the definition of Hopf algebras.


Definition 9.2.4 (Convolution of linear operators). Let (B, m, η) be an algebraand (B,Δ, ε) be a co-algebra. Let L(B) denote the vector space of linear operatorsB → B. Let us define the convolution A∗B ∈ L(B) of linear operators A,B ∈ L(B)by

A ∗B := m(A⊗B)Δ.

Exercise 9.2.5. Show that L(B) in Definition 9.2.4 is an algebra, when endowedwith the convolution product of operators.

Definition 9.2.6 (Hopf algebras). A structure

(H, m, η,Δ, ε, S)

is a Hopf algebra if

• (H, m, η) is an algebra,

• (H,Δ, ε) is a co-algebra,

• Δ : H → H⊗H and ε : H → C are algebra homomorphisms, i.e.,

Δ(fg) = Δ(f)Δ(g), Δ(1H) = 1H⊗H,

ε(fg) = ε(f)ε(g), ε(1H) = 1,

• and S : H → H is a linear mapping, called the antipode, satisfying

S ∗ I = ηε = I ∗ S;

i.e., I ∈ L(H) and S ∈ L(H) are inverses to each other in the convolutionalgebra L(H).

For Hopf algebras (H1,m1, η1,Δ1, ε1, S1) and (H2,m2, η2,Δ2, ε2, S2) we define thetensor product Hopf algebra (H1 ⊗H2, m, η,Δ, ε, S) such that

(H1 ⊗H2, m, η)

is the usual tensor product algebra,

(H1 ⊗H2,Δ, ε)

is the usual tensor product co-algebra, and

S := SH1 ⊗ SH2 .

Exercise 9.2.7 (Uniqueness of the antipode). Let (H, m, η,Δ, ε, Sj) be Hopf alge-bras, where j ∈ {1, 2}. Show that S1 = S2.


Remark 9.2.8 (Commutative diagrams for Hopf algebras). Notice that we nowhave the multiplication and co-multiplication diagram

H⊗H Δm−−−−→ H⊗H

Δ⊗Δ

⏐⏐( F⏐⏐m⊗m

H⊗H⊗H⊗H I⊗τHH⊗I−−−−−−−→ H⊗H⊗H⊗H,

the co-multiplication and unit diagram

H η←−−−− C

Δ

⏐⏐( ∥∥∥H⊗H η⊗η←−−−− C⊗ C,

the multiplication and co-unit diagram

H ε−−−−→ C

m

F⏐⏐ ∥∥∥H⊗H ε⊗ε−−−−→ C⊗ C

and the “everyone with the antipode” diagrams

H ηε−−−−→ H

Δ

⏐⏐( F⏐⏐m

H⊗H I⊗S−−−−→S⊗I

H⊗H.

Example (A monoid co-algebra example). Let G be a finite group and F(G) bethe C-vector space of functions G→ C. Notice that F(G)⊗F(G) and F(G×G)are naturally isomorphic by

m∑j=1

(fj ⊗ gj)(x, y) :=m∑

j=1

fj(x)gj(y).

Then we can define mappings Δ : F(G)→ F(G)⊗F(G) and ε : F(G)→ C by

Δf(x, y) := f(xy), εf := f(e).

In the next example we show that (F(G),Δ, ε) is a co-algebra. But there is stillmore structure in the group to exploit: let us define an operator S : F(G)→ F(G)by (Sf)(x) := f(x−1). . .


Example (Hopf algebra for finite group). Let G be a finite group. Now F(G)from the previous example has a structure of a commutative Hopf algebra; it isco-commutative if and only if G is a commutative group. The algebra mappingsare given by

η(λ)(x) := λ, m(f ⊗ g)(x) := f(x)g(x)

for all λ ∈ C, x ∈ G and f, g ∈ F(G). Notice that the identification F(G×G) ∼=F(G)⊗F(G) gives the interpretation (ma)(x) = a(x, x) for a ∈ F(G×G). Clearly(F(G), m, η) is a commutative algebra. Let x, y, z ∈ G and f, g ∈ F(G). Then

((Δ⊗ I)Δf)(x, y, z) = (Δf)(xy, z)= f((xy)z)= f(x(yz))= (Δf)(x, yz)= ((I ⊗Δ)Δf)(x, y, z),

so that (Δ⊗ I)Δ = (I ⊗Δ)Δ. Next, (ε⊗ I)Δ ∼= I ∼= (I ⊗ ε)Δ, because

(m(ηε⊗ I)Δf)(x) = ((ηε⊗ I)Δf)(x, x)= Δf(e, x)= f(ex) = f(x) = f(xe)= · · · = (m(I ⊗ ηε)Δf)(x).

Thereby (F(G),Δ, ε) is a co-algebra. Moreover,

ε(fg) = (fg)(e) = f(e)g(e) = ε(f)ε(g),

ε(1F(G)) = 1F(G)(e) = 1,

so that ε : F(G) → C is an algebra homomorphism. The co-multiplication Δ :F(G)→ F(G)⊗F(G) ∼= F(G×G) is an algebra homomorphism, because

Δ(fg)(x, y) = (fg)(xy) = f(xy) g(xy) = (Δf)(x, y) (Δg)(x, y),

Δ(1F(G))(x, y) = 1F(G)(xy) = 1 = 1F(G×G)(x, y) ∼= (1F(G) ⊗ 1F(G))(x, y).

Finally,

((I ∗ S)f)(x) = (m(I ⊗ S)Δf)(x)= ((I ⊗ S)Δf)(x, x)= (Δf)(x, x−1)= f(xx−1) = f(e) = εf

= · · · = ((S ∗ I)f)(x),

so that I ∗ S = ηε = S ∗ I. Thereby F(G) can be endowed with a Hopf algebrastructure.


Example (Hopf algebra for a compact group). Let G be a compact group. Weshall endow the dense subalgebra H := TrigPol(G) ⊂ C(G) of trigonometric poly-nomials with a natural structure of a commutative Hopf algebra; H will be co-commutative if and only if G is commutative. Actually, if G is a finite group thenF(G) = TrigPol(G) = C(G). For a compact group G, it can be shown that hereH⊗H ∼= TrigPol(G×G), where the isomorphism is given by

m∑j=1

(fj ⊗ gj)(x, y) :=m∑

j=1

fj(x)gj(y).

The algebra structure(H, m, η)

is the usual one for the trigonometric polynomials, i.e., m(f⊗g) := fg and η(λ) =λ1, where 1(x) = 1 for all x ∈ G. By the Peter–Weyl Theorem 7.5.14, the C-vectorspace H is spanned by {

φij : φ = (φij)dim(φ)i,j , [φ] ∈ G

}.

Let us define the co-multiplication Δ : H → H⊗H by

Δφij :=dim(φ)∑

k=1

φik ⊗ φkj ;

we see that then

(Δφij)(x, y) =dim(φ)∑

k=1

(φik ⊗ φkj)(x, y)

=dim(φ)∑

k=1

φik(x)φkj(y)

= φij(xy).

The co-unit ε : H → C is defined by

εf := f(e),

and the antipode S : H → H by

(Sf)(x) := f(x−1).

Exercise 9.2.9. In the Example about H = TrigPol(G) above, check the validityof the Hopf algebra axioms.


Theorem 9.2.10 (Commutative C∗-algebras and Hopf algebras). Let H be a com-mutative C∗-algebra. If (H, m, η,Δ, ε, S) is a finite-dimensional Hopf algebra thenthere exists a Hopf algebra isomorphism H ∼= C(G), where G is a finite group andC(G) is endowed with the Hopf algebra structure given above.

Proof. Let G := Spec(H) = HOM(H, C). As H is a commutative C∗-algebra, it isisometrically ∗-isomorphic to the C∗-algebra C(G) via the Gelfand transform

(f �→ f) : H → C(G), f(x) := x(f).

The space G must be finite, because dim(C(G)) = dim(H) <∞.Now

e := ε ∈ G,

because ε : H → C is an algebra homomorphism. This e ∈ G will turn out to bethe neutral element of our group.

Let x, y ∈ G. We identify the spaces C ⊗ C and C, and get an algebrahomomorphism x⊗ y : H⊗H → C⊗ C ∼= C. Now Δ : H → H⊗H is an algebrahomomorphism, so that (x ⊗ y)Δ : H → C is an algebra homomorphism! Let usdenote

xy := (x⊗ y)Δ,

so that xy ∈ G. This defines the group operation ((x, y) �→ xy) : G×G→ G!Inversion x �→ x−1 will be defined via the antipode S : H → H. We shall

show that for a commutative Hopf algebra, the antipode is an algebra isomorphism.First we prove that S(1H) = 1H:

S1H = m(1H ⊗ S1H)= m(I ⊗ S)(1H ⊗ 1H)= m(I ⊗ S)Δ1H= (I ∗ S)1H = ηε1H= 1H.

Then we show that S(gh) = S(h)S(g), where g, h ∈ H, gh := m(g ⊗ h). Let ususe the so-called Sweedler notation

Δf =:∑

f(1) ⊗ f(2) =: f(1) ⊗ f(2);

consequently

(Δ⊗ I)Δf = (Δ⊗ I)(f(1) ⊗ f(2)) = f(1)(1) ⊗ f(1)(2) ⊗ f(2),

(I ⊗Δ)Δf = (I ⊗Δ)(f(1) ⊗ f(2)) = f(1) ⊗ f(2)(1) ⊗ f(2)(2),

and due to the co-associativity we may re-index as follows:

(Δ⊗ I)Δf =: f(1) ⊗ f(2) ⊗ f(3) := (I ⊗Δ)Δf


(notice that, e.g., f(2) appears in different meanings above – this is just a matterof notation!). Then

S(gh) = S(ε((gh)(1))(gh)(2))= ε((gh)(1)) S((gh)(2))= ε(g(1)h(1)) S(g(2)h(2))= ε(g(1)) ε(h(1)) S(g(2)h(2))= ε(g(1)) S(h(1)(1)) h(1)(2) S(g(2)h(2))= ε(g(1)) S(h(1)) h(2) S(g(2)h(3))= S(h(1)) ε(g(1)) h(2) S(g(2)h(3))= S(h(1)) S(g(1)(1)) g(1)(2) h(2) S(g(2)h(3))= S(h(1)) S(g(1)) g(2) h(2) S(g(3)h(3))= S(h(1)) S(g(1)) (gh)(2) S((gh)(3))= S(h(1)) S(g(1)) ε((gh)(2))= S(h(1)) S(g(1)) ε(g(2)h(2))= S(h(1)) S(g(1)) ε(g(2)) ε(h(2))= S(h(1)ε(h(2))) S(g(1)ε(g(2)))= S(h) S(g);

this computation can be compared to

(xy)−1 = e(xy)−1

= y−1y(xy)−1

= y−1ey(xy)−1

= y−1x−1xy(xy)−1

= y−1x−1e

= y−1x−1

for x, y ∈ G! Since H is commutative, we have proven that S : H → H is analgebra homomorphism. Thereby xS : H → C is an algebra homomorphism. Letus denote

x−1 := xS ∈ G,

which is the inverse of x ∈ G!

We leave it for the reader to show that (G, (x, y) �→ xy, x �→ x−1) is indeeda group. �

Exercise 9.2.11. Finish the proof of Theorem 9.2.10.


Exercise 9.2.12 (Universal enveloping algebra as a Hopf algebra). Let g be a Liealgebra and U(g) its universal enveloping algebra. Let X ∈ g; extend definitions

ΔX := X ⊗ 1U(g) + 1U(g) ⊗X, εX := 0, SX := −X

so that you obtain a Hopf algebra structure (U(g), m, η,Δ, ε, S).

Exercise 9.2.13. Let (H, m, η,Δ, ε, S) be a finite-dimensional Hopf algebra.

(a) Endow the dual H′ = L(H, C) with a natural Hopf algebra structure via theduality

(f, φ) �→ 〈f, φ〉H := φ(f)

where f ∈ H, φ ∈ H′.(b) If G is a finite group and H = F(G), what are the Hopf algebra operations

for H′?(c) With a suitable choice for H, give an example of a non-commutative non-co-

commutative Hopf algebra H⊗H′.Exercise 9.2.14 (M.E. Sweedler’s example). Let (H, m, η) be the algebra spannedby the set {1, g, x, gx}, where 1 is the unit element and g2 = 1, x2 = 0 andxg = −gx. Let us define algebra homomorphisms ε : H → C and Δ : H → H⊗Hby

Δ(g) := g ⊗ g, Δ(x) := x⊗ 1 + g ⊗ x,

ε(g) := 1, ε(x) := 0.

Let us define a linear mapping S : H → H by

S(1) := 1, S(g) := g, S(x) := −gx, S(gx) := −x.

Show that (H, m, η,Δ, ε, S) is a non-commutative non-co-commutative Hopf alge-bra.

Remark 9.2.15. In Exercise 9.2.14, a nice concrete matrix example can be given.Let us define A ∈ C2×2 by

A :=(

0 11 0

).

Let g, x ∈ C4×4 be given by

g :=(

A 00 −A

), x :=

(0 IC2

0 0

).

Then it is easy to see that H = span{IC4 , g, x, gx} is a four-dimensional subalgebraof C4×4 such that g2 = IC4 , x2 = 0 and xg = −gx.

Part IV

Non-commutative Symmetries

In this part, we develop a non-commutative quantization of pseudo-differentialoperators on compact Lie groups. The idea is that it can be constructed in a wayto run more or less parallel to the Kohn–Nirenberg quantization of operators onRn that was presented in Chapter 2, and to the toroidal quantization of operatorson Tn that was developed in Chapter 4. The main advantage of such an approachis that once the basic notions and definitions are understood, one can see andenjoy a lot of features which are already familiar from the commutative analysis.

The introduced matrix-valued full symbols turn out to have a number ofinteresting properties. The main difference with the toroidal quantization here isthat, due to the non-commutativity of the group, symbols become matrix-valuedwith sizes depending on the dimensions of the unitary irreducible representationsof the group, which are all finite-dimensional because the group is compact.

Among other things, the introduced approach provides a characterisation ofthe Hormander class of pseudo-differential operators on a compact Lie group Gusing a global quantization of operators, thus relying on the representation theoryrather than on the usual expressions in local coordinate charts. This yields a notionof the full symbol of an operator as a mapping defined globally on G× G, where Gis the unitary dual of G. As such, this presents an advantage over the local theorywhere only the notion of the principal symbol can be defined globally. In the caseof the torus G = Tn, we naturally have G × G ∼= Tn × Zn, and we recapture thenotion of a toroidal symbol introduced in Chapter 4, where symbols are scalar-valued (or 1×1 matrix-valued) because all the unitary irreducible representationsof the torus are one-dimensional.

As an important example, the approach developed here will give us quite de-tailed information on the global quantization of operators on the three-dimensionalsphere S3. More generally, we note that if we have a closed simply-connected three-dimensional manifold M , then by the recently resolved Poincare conjecture thereis a global diffeomorphism M � S3 � SU(2) that turns M into a Lie group with

528

a group structure induced by S3 (or by SU(2)). Thus, we can use the approachdeveloped for SU(2) to immediately obtain the corresponding global quantizationof operators on M with respect to this induced group product. In fact, all theformulae remain completely the same since the unitary dual of SU(2) (or S3 inthe quaternionic R4) is mapped by this diffeomorphism as well; for an exampleof this construction in the case of S3 � SU(2) see Section 12.5. The choice of thegroup structure on M may be not unique and is not canonical, but after usingthe machinery that we develop for SU(2), the corresponding quantization can bedescribed entirely in terms of M ; for an example compare Theorem 12.5.3 for S3

and Theorem 12.4.3 for SU(2). In this sense, as different quantizations of oper-ators exist already on Rn depending on the choice of the underlying structure(e.g., Kohn–Nirenberg quantization, Weyl quantizations, etc.), the possibility tochoose different group products on M resembles this. Due to space limitations,we postpone the detailed analysis of operators on the higher-dimensional spheresSn � SO(n+1)/SO(n) viewed as homogeneous spaces. However, we will introducea general machinery on which to obtain the global quantization on homogeneousspaces using the one on the Lie group that acts on the space. Although we do nothave general analogues of the diffeomorphic Poincare conjecture in higher dimen-sions, this will cover cases when M is a convex surface or a surface with positivecurvature tensor, as well as more general manifolds in terms of their Pontryaginclass, etc.

Thus, the cases of the three-dimensional sphere S3 and Lie group SU(2) areanalysed in detail in Chapter 12. There we show that pseudo-differential operatorsfrom Hormander’s classes Ψm(SU(2)) and Ψm(S3) have matrix-valued symbolswith a remarkable rapid off-diagonal decay property.

In Chapter 11 we develop the necessary foundations of this analysis on SU(2)which together with Chapter 12 provides a more detailed example of the quan-tization from Chapter 10. Finally, in Chapter 13 we give an application of theseconstructions to analyse pseudo-differential operators on homogeneous spaces.

Chapter 10

Pseudo-differential Operatorson Compact Lie Groups

10.1 Introduction

In this chapter we develop a global theory of pseudo-differential operators ongeneral compact Lie groups. As usual, Sm

1,0(Rn × Rn) ⊂ C∞(Rn × Rn) refers to

the Euclidean space symbol class, defined by the symbol inequalities∣∣∂αξ ∂β

xp(x, ξ)∣∣ ≤ C (1 + |ξ|)m−|α|, (10.1)

for all multi-indices α, β ∈ Nn0 , N0 = {0}∪N, where the constant C is independent

of x, ξ ∈ Rn but may depend on α, β, p, m. On a compact Lie group G we definethe class Ψm(G) to be the usual Hormander class of pseudo-differential operatorsof order m. Thus, the operator A belongs to Ψm(G) if in (all) local coordinatesoperator A is a pseudo-differential operator on Rn with some symbol p(x, ξ) sat-isfying estimates (10.1), see Definition 5.2.11. Of course, symbol p depends on thelocal coordinate systems.

It is a natural idea to build pseudo-differential operators out of smooth fam-ilies of convolution operators on Lie groups. In this work, we strive to develop theconvolution approach into a symbolic quantization, which always provides a muchmore convenient framework for the analysis of operators. For this, our analysis ofoperators and their symbols is based on the representation theory of Lie groups.This leads to the description of the full symbols of pseudo-differential operatorson Lie groups as sequences of matrices of growing size equal to the dimension ofthe corresponding representation of the group. Moreover, the analysis is globaland is not confined to neighbourhoods of the neutral element since it does notrely on the exponential map and its properties. We also characterise, in terms ofthe introduced quantizations, standard Hormander classes Ψm(G) on Lie groups.One of the advantages of the presented approach is that we obtain a notion of full(global) symbols compared with only principal symbols available in the standardtheory via localisations.

530 Chapter 10. Pseudo-differential Operators on Compact Lie Groups

In our analysis on a Lie group G, at some point we have to make a choicewhether to work with left- or right-convolution kernels. Since left-invariant oper-ators on C∞(G) correspond to right-convolutions f �→ f ∗ k, once we decide toidentify the Lie algebra g of G with the left-invariant vector fields on G, it becomesmost natural to work with right-convolution kernels in the sequel, and to definesymbols as we do in Definition 10.4.3.

It is also known that globally defined symbols of pseudo-differential operatorscan be introduced on manifolds in the presence of a connection which allows one touse a suitable globally defined phase function, see, e.g., [151, 100, 109]. However,on compact Lie groups the use of the group structure allows one to develop atheory parallel to those of Rn and Tn owing to the fact that the Fourier analysisis well adopted to the underlying representation theory. Some elements of such atheory were discussed in [128, 129] as well as in the PhD thesis of V. Turunen [139].However, here we present the finite-dimensional symbols and we do not rely on theexponential mapping thus providing a genuine global analysis in terms of the Liegroup itself. We also note that the case of the compact commutative group Tn isalso recovered from this general point of view, however a more advanced analysisis possible in this case because of the close relation between Tn and Rn.

Unless specified otherwise, in this chapter G will stand for a general compactLie group, and dμG will stand for the (normalised) Haar measure on G, i.e., theunique regular Borel probability measure which is left-translation-invariant:∫

G

f(x) dμG(x) =∫

G

f(yx) dμG(x)

for all f ∈ C(G) and y ∈ G. Then also∫G

f(x) dμG(x) =∫

G

f(xy) dμG(x) =∫

G

f(x−1) dμG(x),

see Remark 7.4.4. Usually we abbreviate dμG(x) to dx since this should causeno confusion.

10.2 Fourier series on compact Lie groups

We begin with the Fourier series on a compact group G.

Definition 10.2.1 (Rep(G) and G). Let Rep(G) denote the set of all stronglycontinuous irreducible unitary representations of G. In the sequel, whenever wemention unitary representations (of a compact Lie group G), we always meanstrongly continuous irreducible unitary representations, which are then automat-ically smooth. Let G denote the unitary dual of G, i.e., the set of equivalenceclasses of irreducible unitary representations from Rep(G), see Definitions 6.3.18and 7.5.7. Let [ξ] ∈ G denote the equivalence class of an irreducible unitary repre-sentation ξ : G → U(Hξ); the representation space Hξ is finite-dimensional sinceG is compact (see Corollary 7.5.6), and we set dim(ξ) = dimHξ.

10.2. Fourier series on compact Lie groups 531

We will always equip a compact Lie group G with the Haar measure μG,i.e., the uniquely determined bi-invariant Borel regular probability measure, seeRemark 7.4.4. For simplicity, we will write Lp(G) for Lp(μG),

∫G

f dx for∫

Gf dμG,

etc. First we collect several definitions scattered over previous chapters in differentforms.

Definition 10.2.2 (Fourier coefficients). Let us define the Fourier coefficient f(ξ) ∈End(Hξ) of f ∈ L1(G) by

f(ξ) :=∫

G

f(x) ξ(x)∗ dx; (10.2)

more precisely,

(f(ξ)u, v)Hξ=

∫G

f(x) (ξ(x)∗u, v)Hξdx =

∫G

f(x) (u, ξ(x)v)Hξdx

for all u, v ∈ Hξ, where (·, ·)Hξis the inner product of Hξ.

Remark 10.2.3. Notice that ξ(x)∗ = ξ(x)−1 = ξ(x−1).Remark 10.2.4 (Fourier coefficients on Tn as a group). Let G = Tn. Let us natu-rally identify End(C) with C, and U(C) with {z ∈ C : |z| = 1}. For each k ∈ Zn,we define ek : G→ U(C) by ek(x) := ei2πx·k. Then

f(k) := f(ek) =∫

Tn

f(x) e−i2πx·k dx

is the usual Fourier coefficient of f ∈ L1(Tn).Remark 10.2.5 (Intertwining isomorphisms). Let U ∈ Hom(η, ξ) be an intertwin-ing isomorphism, i.e., let U : Hη → Hξ be a bijective unitary linear mapping suchthat Uη(x) = ξ(x)U for all x ∈ G. Then we have

f(η) = U−1f(ξ) U ∈ End(Hη). (10.3)

Proposition 10.2.6 (Inner automorphisms). For u ∈ G, consider the inner auto-morphisms

φu = (x �→ u−1xu) : G→ G.

Then for all ξ ∈ Rep(G) we have

f ◦ φu(ξ) = ξ(u) f(ξ) ξ(u)∗. (10.4)

Proof. We can calculate

f ◦ φu(ξ) =∫

G

f(u−1xu) ξ(x)∗ dx =∫

G

f(x) ξ(uxu−1)∗ dx

= ξ(u)∫

G

f(x) ξ(x)∗ dx ξ(u)∗ = ξ(u) f(ξ) ξ(u)∗,

which gives (10.4). �


Proposition 10.2.7 (Convolutions). If f, g ∈ L1(G) then

f ∗ g = g f .

Proof. If ξ ∈ Rep(G) then

f ∗ g(ξ) =∫

G

(f ∗ g)(x) ξ(x)∗ dx

=∫

G

∫G

f(xy−1)g(y) dy ξ(x)∗ dx

=∫

G

g(y) ξ∗(y)∫

G

f(xy−1) ξ(xy−1)∗ dx dy

= g(ξ) f(ξ),

completing the proof. �Remark 10.2.8. The product g f in Proposition 10.2.7 usually differs from f gbecause f(ξ), g(ξ) ∈ End(Hξ) are operators unless G is commutative when they arescalars (see Corollary 6.3.26) and hence commute. This order exchange is due to thedefinition of the Fourier coefficients (10.2), where we chose the integration of thefunction with respect to ξ(x)∗ instead of ξ(x). This choice actually serves us well,as we chose to identify the Lie algebra g with left-invariant vector fields on the Liegroup G: namely, a left-invariant continuous linear operator A : C∞(G)→ C∞(G)can be presented as a right-convolution operator Ca = (f �→ f ∗ a), resulting inconvenient expressions like

CaCbf = a b f .

However, in Remark 10.4.13 we will still explain what would happen had we chosenanother definition for the Fourier transform.

Proposition 10.2.9 (Differentiating the convolution). Let Y ∈ g and let DY :C∞(G)→ C∞(G) be defined by

DY f(x) =ddt


.

Let f, g ∈ C∞(G). Then DY (f ∗ g) = f ∗DY g.

Proof. We have

DY (f ∗ g)(x) =∫

G

f(y)ddt

g(y−1x exp(tY ))∣∣∣∣t=0

dy = f ∗DY g(x).

�

We now summarise properties of the Fourier series as a corollary to thePeter–Weyl Theorem 7.5.14:

10.2. Fourier series on compact Lie groups 533

Corollary 10.2.10 (Fourier series). If ξ : G → U(d) is a unitary matrix represen-tation then

f(ξ) =∫

G

f(x) ξ(x)∗ dx ∈ Cd×d

has matrix elements

f(ξ)mn =∫

G

f(x) ξ(x)nm dx ∈ C, 1 ≤ m,n ≤ d.

If here f ∈ L2(G) then

f(ξ)mn = (f, ξ(x)nm)L2(G), (10.5)

and by the Peter–Weyl Theorem 7.5.14 we have

f(x) =∑

[ξ]∈G

dim(ξ) Tr(ξ(x) f(ξ)

)

=∑

[ξ]∈G

dim(ξ)d∑

m,n=1

ξ(x)nm f(ξ)mn (10.6)

for almost every x ∈ G, where the summation is understood so that from each class[ξ] ∈ G we pick just (any) one representative ξ ∈ [ξ]. The particular choice of arepresentation from the representation class is irrelevant due to formula (10.3) andthe presence of the trace in (10.6). The convergence in (10.6) is not only pointwisealmost everywhere on G but also in the space L2(G).

Example. For f ∈ L2(Tn), we get

f(x) =∑

k∈Zn

ei2πx·k f(k),

where f(k) =∫

Tn f(x) e−i2πx·k dx is as in Remark 10.2.4. Here C was identifiedwith C1×1.

Finally, we record a useful formula for representations:Remark 10.2.11. Let e ∈ G be the neutral element of G and let ξ be a unitary ma-trix representation of G. The unitarity of the representation ξ implies the identity

δmn = ξ(e)mn = ξ(x−1x)mn =∑

k

ξ(x−1)mk ξ(x)kn =∑

k

ξ(x)km ξ(x)kn.

Similarly,δmn =

∑k

ξ(x)mk ξ(x)nk.

Here, as usual, δmn is the Kronecker delta: δmn = 1 for m = n, and δmn = 0otherwise.


10.3 Function spaces on the unitary dual

In this section we lay down a functional analytic foundation concerning the func-tion spaces that will be useful in the sequel. In particular, distribution space S ′(G)is of importance since it provides a distributional interpretation for series on G.

10.3.1 Spaces on the group G

We recall Definition 8.3.45 of the Laplace operator LG on a Lie group G:Remark 10.3.1 (Laplace operator L = LG). The Laplace operator LG is a second-order negative definite bi-invariant partial differential operator on G correspondingto the Casimir element of the universal enveloping algebra U(g) of the Lie algebra gof G. If G is equipped with the unique (up to a constant) bi-invariant Riemannianmetric, LG is its Laplace–Beltrami operator. We will often denote the Laplaceoperator simply by L if there is no need to emphasize the group G in the notation.We refer to Section 8.3.2 for a discussion of its main properties.

In Definition 5.2.4 we defined Ck mappings on a manifold. These can be alsocharacterised globally:

Exercise 10.3.2. Let n = dim G and let {Yj}nj=1 be a basis of the Lie algebra g of

G. Show that f ∈ Ck(G) if and only if ∂αf ∈ C(G) for all ∂α = Y α11 · · ·Y αn

n forall |α| ≤ k, or if and only if Lf ∈ C(G) for all L ∈ U(g) of degree ≤ k.

Exercise 10.3.3. Show that f ∈ C∞(G) if and only if (−LG)kf ∈ C(G) for allk ∈ N. Show that f ∈ C∞(G) if and only if Lf ∈ C(G) for all L ∈ U(g). (Hint:use a priori estimates from Theorem 2.6.9.)

We can recall from Remark 5.2.15 the definition of the space D′(M) of distri-butions on a compact manifold M . Let us be more precise in the case of a compactLie group G:

Definition 10.3.4 (Distributions D′(G)). We define the space of distributions D′(G)as the space of all continuous linear functionals on C∞(G). This means that u ∈D′(G) if it is a functional u : C∞(G)→ C such that

1. u is linear, i.e., u(αϕ + βψ) = αu(ϕ) + βu(ψ) for all α, β ∈ C and all ϕ, ψ ∈C∞(G);

2. u is continuous, i.e., u(ϕj) → u(ϕ) in C whenever ϕj → ϕ in C∞(G), asj →∞.

Here1 ϕj → ϕ in C∞(G) if, e.g.,2 ∂αϕj → ∂αϕ for all ∂α ∈ U(g), as j →∞.We define the convergence in the space D′(G) as follows. Let uj , u ∈ D′(G).

We will say that uj → u in D′(G) as j →∞ if uj(ϕ) → u(ϕ) in C as j →∞, forall ϕ ∈ C∞(G).1For a general setting on compact manifolds see Remark 5.2.15.2Exercise 10.3.2 provides more options.

10.3. Function spaces on the unitary dual 535

Definition 10.3.5 (Duality 〈·, ·〉G). Let u ∈ D′(G) and ϕ ∈ C∞(G). We write

〈u, ϕ〉G := u(ϕ).

If u ∈ Lp(G), 1 ≤ p ≤ ∞, we can identify u with a distribution in D′(G) (whichwe will continue to denote by u) in a canonical way by

〈u, ϕ〉G = u(ϕ) :=∫

G

u(x) ϕ(x) dx,

where dx is the Haar measure on G.

Exercise 10.3.6. Let 1 ≤ p ≤ ∞. Show that if uj → u in Lp(G) as j → ∞ thenuj → u in D′(G) as j →∞.

Remark 10.3.7 (Derivations in D′(G)). Similar to operations on distributions inRn described in Section 1.3.2, we can define different operations on distributionson G. For example, for Y ∈ g, we can differentiate u ∈ D′(G) with respect to thevector field Y by defining

(Y u)(ϕ) := −u(Y ϕ),

for all ϕ ∈ C∞(G). Here the derivative Y ϕ = DY ϕ is as in Proposition 10.2.9.Similarly, if ∂α ∈ U(g) is a differential operator of order |α|, we define

(∂αu)(ϕ) := (−1)|α|u(∂αϕ),

for all ϕ ∈ C∞(G).

Exercise 10.3.8. Show that ∂αu ∈ D′(G) and that ∂α : D′(G)→ D′(G) is contin-uous.

Definition 10.3.9 (Sobolev space Hs(G)). First let us note that the Laplacian L =LG is symmetric and I −L is positive. Set Ξ := (I −L)1/2. Then Ξs ∈ L(C∞(G))and Ξs ∈ L(D′(G)) for every s ∈ R. Let us define

(f, g)Hs(G) := (Ξsf,Ξsg)L2(G) (f, g ∈ C∞(G)).

The completion of C∞(G) with respect to the norm f �→ ‖f‖Hs(G) = (f, f)1/2Hs(G)

gives us the Sobolev space Hs(G) of order s ∈ R. This is the same space as that inDefinition 5.2.16 on general manifolds, or as the Sobolev space obtained using anysmooth partition of unity on the compact manifold G, by Corollary 5.2.18. Theoperator Ξr is a Sobolev space isomorphism Hs(G)→ Hs−r(G) for every r, s ∈ R.

Exercise 10.3.10. Show that if Y ∈ g, then the differentiation with respect to Y isa bounded linear operator from Hk(G) to Hk−1(G) for all k ∈ N. Extend this tok ∈ R.


Remark 10.3.11 (Sobolev spaces and C∞(G)). We have⋂k∈N

H2k(G) = C∞(G). (10.7)

This can be seen locally, since H2k(G) = domain((−LG)k) and LG is elliptic, sothat (10.7) follows from the local a priori estimates in Theorem 2.6.9.

Since the analysis of Sobolev spaces is closely intertwined with spaces on G,we study them also in the next section, in particular we refer to Remark 10.3.24for their characterisation on the Fourier transform side.

10.3.2 Spaces on the dual G

Since we will be mostly using the “right” Peter–Weyl Theorem (Theorem 7.5.14),we may simplify the notation slightly, also adopting it to the analysis of pseudo-differential operators from the next sections. Thus, the space{√

dim(ξ) ξij : ξ = (ξij)dim(ξ)i,j=1 , [ξ] ∈ G

}is an orthonormal basis for L2(G), and the space

Hξ := span{ξij : 1 ≤ i, j ≤ dim(ξ)} ⊂ L2(G)

is πR-invariant, ξ ∼ πR|Hξ , and

L2(G) =⊕[ξ]∈G

Hξ.

By choosing a unitary matrix representation from each equivalence class [ξ] ∈ G,we can identify Hξ with Cdim(ξ)×dim(ξ) by choosing a basis in the linear space Hξ.Remark 10.3.12 (Spaces Hξ and Hξ). We would like to point out a differencebetween spaces Hξ and Hξ to eliminate any confusion. Recall that if ξ ∈ Rep(G),then ξ is a mapping ξ : G → U(Hξ), where Hξ is the representation space of ξ,with dimHξ = dim(ξ). On the other hand, the space Hξ ⊂ L2(G) is the span ofthe matrix elements of ξ and dimHξ = (dim(ξ))2. In the notation of the right andleft Peter–Weyl Theorems in Theorem 7.5.14 and Remark 7.5.16, we have

Hξ =dim(ξ)⊕

i=1

Hξi,· =

dim(ξ)⊕j=1

Hξ·,j .

Informally, spacesHξ can be viewed as “columns/rows” ofHξ, because for exampleξ(x)v ∈ Hξ for every v ∈ Hξ.

We recall the important property of the Laplace operator on spaces Hξ fromTheorem 8.3.47:


Theorem 10.3.13 (Eigenvalues of the Laplacian on G). For every [ξ] ∈ G the spaceHξ is an eigenspace of LG and −LG|Hξ = λξI, for some λξ ≥ 0.

Exercise 10.3.14. Show that Hξ ⊂ C∞(G) for all ξ ∈ Rep(G). (Hint: Use Theorem10.3.13 and the ellipticity of LG.)

From Definition 7.6.11 we recall the Hilbert space L2(G) which we can nowdescribe as follows. But first, we can look at the space of all mappings on G:

Definition 10.3.15 (Space M(G)). The space M(G) consists of all mappings

F : G→⋃

[ξ]∈G

L(Hξ) ⊂∞⋃

m=1

Cm×m,

satisfying F ([ξ]) ∈ L(Hξ) for every [ξ] ∈ G. In matrix representations, we can viewF ([ξ]) ∈ Cdim(ξ)×dim(ξ) as a dim(ξ)× dim(ξ) matrix.

The space L2(G) consists of all mappings F ∈M(G) such that

||F ||2L2(G)

:=∑

[ξ]∈G

dim(ξ) ‖F ([ξ])‖2HS <∞,

where||F ([ξ])||HS =

√Tr (F ([ξ]) F ([ξ])∗)

stands for the Hilbert–Schmidt norm of the linear operator F ([ξ]), see DefinitionB.5.43. Thus, the space

L2(G) =

⎧⎨⎩F : G→⋃

[ξ]∈G

L(Hξ), F : [ξ] �→ L(Hξ) :

||F ||2L2(G)

:=∑

[ξ]∈G

dim(ξ) ‖F ([ξ])‖2HS <∞

⎫⎬⎭ (10.8)

is a Hilbert space with the inner product

(E,F )L2(G) :=∑

[ξ]∈G

dim(ξ) Tr (E([ξ]) F ([ξ])∗).

Remark 10.3.16 (Mappings on G or on Rep(G)?). We note that because therepresentations in G are all unitary, and because of the Hilbert–Schmidt normthat we use, we can write F (ξ) instead of F ([ξ]), so that F is defined on Rep(G)instead of G. Thus, to simplify the notation we can write F (ξ) with a conventionthat F ∈M(G) if F (ξ) = F (η) whenever ξ ∼ η (i.e., whenever [ξ] = [η]).


Proposition 10.3.17 (Parseval’s identity). Let f, g ∈ L2(G). Then we have

(f, g)L2(G) =∑

[ξ]∈G

dim(ξ) Tr(f(ξ) g(ξ)∗

)= (f(ξ), g(ξ))L2(G).

Consequently,||f ||L2(G) = ||f ||L2(G).

Proof. Writing f and g as Fourier series (7.3) in Corollary 7.6.10, we obtain

(f, g)L2(G) =∑

[ξ],[η]∈G

dim(ξ) dim(η)∫

G

Tr(ξ(x) f(ξ)

)Tr (η(x) g(η)) dx

=∑

[ξ],[η]∈G

dim(ξ) dim(η)∫

G

Tr(ξ(x) f(ξ)

)Tr (η(x)∗ g(η)∗) dx

=∑

[ξ],[η]∈G

dim(ξ) dim(η)∫

G

dim(ξ)∑i,j=1

ξ(x)ij f(ξ)ji

dim(η)∑k,l=1

η(x)lk g(η)∗lk dx

=∑

[ξ],[η]∈G

dim(ξ) dim(η)dim(ξ)∑i,j=1

dim(η)∑k,l=1

(ξ(x)ij , η(x)lk)L2(G)f(ξ)ji g(η)∗lk.

From this, by Lemma 7.5.12, we obtain

(f, g)L2(G) =∑

[ξ]∈G

dim(ξ)dim(ξ)∑i,j=1

f(ξ)ji g(ξ)∗ij

=∑

[ξ]∈G

dim(ξ) Tr(f(ξ) g(ξ)∗

),

finishing the proof. �

We now define a scale 〈ξ〉 on Lie groups that is useful in measuring the growthon G and that is associated to the eigenvalues of the Laplace operator L = LG

from Remark 10.3.1 and Theorem 10.3.13.

Definition 10.3.18 (Definition of 〈ξ〉 on Lie groups). Let ξ ∈ Rep(G), so thatξ : G→ U(Hξ). Given v, w ∈ Hξ, the function ξvw : G→ C defined by

ξvw(x) := 〈ξ(x)v, w〉Hξ

is not only continuous but even C∞-smooth3. Let span(ξ) denote the linear spanof {ξvw : v, w ∈ Hξ} . If ξ ∼ η then span(ξ) = span(η); consequently4, we maywrite

span[ξ] := span(ξ) ⊂ C∞(G).3See Exercise 10.3.14.4In matrix representations this is the space Hξ.


It follows from Theorem 10.3.13 that

−Lξvw(x) = λ[ξ]ξvw(x),

where λ[ξ] ≥ 0, and we write

〈ξ〉 := (1 + λ[ξ])1/2. (10.9)

A relation between the dimension of the representation ξ and the weight 〈ξ〉 isgiven by

Proposition 10.3.19 (Dimension and eigenvalues). There exists a constant C > 0such that the inequality

dim(ξ) ≤ C〈ξ〉dim G

2

holds for all ξ ∈ Rep(G).

Proof. We note by Theorem 10.3.13 that 〈ξ〉 is an eigenvalue of the first-orderelliptic operator (1−LG)1/2. The corresponding eigenspace Hξ has the dimensiondim(ξ)2. Denoting by n = dim G, the Weyl formula for the counting function ofthe eigenvalues of (1− LG)1/2 yields∑

〈ξ〉≤λ

dim(ξ)2 = C0λn + O(λn−1)

as λ→∞. This implies the estimate dim(ξ)2 ≤ C〈ξ〉n for large 〈ξ〉, implying thestatement. �

Definition 10.3.20 (Space S(G)). The space S(G) consists of all mappings H ∈M(G) such that for all k ∈ N we have

pk(H) :=∑

[ξ]∈G

dim(ξ) 〈ξ〉k ||H(ξ)||HS <∞. (10.10)

We will say that Hj ∈ S(G) converges to H ∈ S(G) in S(G), and write Hj → H

in S(G) as j →∞, if pk(Hj −H)→ 0 as j →∞ for all k ∈ N, i.e., if∑[ξ]∈G

dim(ξ) 〈ξ〉k ||Hj(ξ)−H(ξ)||HS → 0 as j →∞,

for all k ∈ N.

We can take different families of seminorms on S(G). In particular, the fol-lowing equivalence will be of importance since it provides a more direct relationwith Sobolev spaces on G:


Proposition 10.3.21 (Seminorms on S(G)). For H ∈ M(G), let pk(H) be as in(10.10), and let us define a family qk(H) by

qk(H) :=

⎛⎝ ∑[ξ]∈G

dim(ξ) 〈ξ〉k ||H(ξ)||2HS

⎞⎠1/2

.

Then pk(H) <∞ for all k ∈ N if and only if qk(H) <∞ for all k ∈ N .

Proof. Let H ∈M(G). We claim that pk(H) ≤ Cq4k(H) if k is large enough, andthat q2k(H) ≤ pk(H). Indeed, by the Cauchy–Schwartz inequality (i.e., Holder’sinequality similar to that in Lemma 3.3.28 where we use the discreteness of G) wecan estimate

pk(H) =∑

[ξ]∈G

{(dim(ξ))1/2 〈ξ〉−k

}{(dim(ξ))1/2 〈ξ〉2k ||H(ξ)||HS

}

≤

⎛⎝ ∑[ξ]∈G

dim(ξ) 〈ξ〉−2k

⎞⎠1/2⎛⎝ ∑[ξ]∈G

dim(ξ) 〈ξ〉4k ||H(ξ)||2HS

⎞⎠1/2

≤ Cq4k(H),

if we choose k large enough and use Proposition 10.3.19. Conversely, we have

pk(H)2 =∑

[ξ]=[η]∈G

(dim(ξ))2 〈ξ〉2k ||H(ξ)||2HS

+∑

[ξ]�=[η]

dim(ξ) dim(η) 〈ξ〉k 〈η〉k||H(ξ)||HS ||H(η)||HS

≥∑

[ξ]∈G

dim(ξ) 〈ξ〉2k ||H(ξ)||2HS

= q2k(H)2. �

As a corollary of Proposition 10.3.21 we have

Corollary 10.3.22. For H ∈M(G), we have∑[ξ]∈G

dim(ξ) 〈ξ〉−k ||H(ξ)||HS <∞

for some k ∈ N if and only if∑[ξ]∈G

dim(ξ) 〈ξ〉−l ||H(ξ)||2HS <∞

for some l ∈ N.


Let us now summarise properties of the Fourier transform on L2(G).

Theorem 10.3.23 (Fourier inversion). Let G be a compact Lie group. The Fouriertransform f �→ FGf = f defines a surjective isometry L2(G) → L2(G). Theinverse Fourier transform is given by

(F−1G H)(x) =

∑[ξ]∈G

dim(ξ) Tr (ξ(x) H(ξ)), (10.11)

and we haveF−1

G ◦ FG = id and FG ◦ F−1G = id

on L2(G) and L2(G), respectively. Moreover, the Fourier transform FG is unitary,FG

∗ = F−1G , and for any H ∈ S(G) we have

(FG∗H)(x) = F−1

G (H∗)(x−1) =(F−1

G H)(x)

for all x ∈ G, where H∗(ξ) := H(ξ)∗ for all ξ ∈ Rep(G).

Proof. In Theorem 7.6.13 we have already shown that the Fourier transform isa surjective isometry L2(G) → L2(G). By Corollary 7.6.10 the inverse Fouriertransform is given by (10.11). Let us show the last part. Let f ∈ C∞(G) andH ∈ S(G). Then we have

(f,FG∗H)L2(G) = (FGf,H)L2(G) =

∑[ξ]∈G

dim(ξ) Tr(f(ξ) H(ξ)∗

)=

∑[ξ]∈G

dim(ξ) Tr(∫

G

f(x) ξ∗(x) dx H(ξ)∗)

=∫

G

f(x)

⎧⎨⎩ ∑[ξ]∈G

dim(ξ) Tr(ξ(x−1) H(ξ)∗

)⎫⎬⎭ dx

=∫

G

f(x) F−1G (H∗)(x−1) dx,

which implies FG∗H(x) = F−1

G (H∗)(x−1). Finally, the unitarity of the Fouriertransform follows from continuing the calculation:

(f,FG∗H)L2(G) =

∑[ξ]∈G

dim(ξ) Tr(∫

G

f(x) ξ∗(x) dx H(ξ)∗)

=∫

G

f(x)

⎧⎨⎩ ∑[ξ]∈G

dim(ξ) Tr (ξ∗(x) H(ξ)∗)

⎫⎬⎭ dx


=∫

G

f(x)

⎧⎨⎩ ∑[ξ]∈G

dim(ξ)Tr (ξ(x) H(ξ))

⎫⎬⎭ dx

=∫

G

f(x)(F−1

G H)(x) dx. �

Remark 10.3.24 (Sobolev space Hs(G)). On the Fourier transform side, in viewof Theorem 10.3.23, the Sobolev space Hs(G) can be characterised by

Hs(G) = {f ∈ D′(G) : 〈ξ〉sf(ξ) ∈ L2(G)}.

We also note that since G is compact, for s = 2k and k ∈ N, we have thatf ∈ H2k(G) if (−LG)kf ∈ L2(G). By Theorem 10.3.25, the Fourier transform FG

is a continuous bijection from H2k(G) to the space⎧⎨⎩F ∈M(G) :∑

[ξ]∈G

dim(ξ) 〈ξ〉2k||F (ξ)||2HS <∞

⎫⎬⎭.

We now analyse the Fourier transforms on C∞(G) and S(G) in preparationfor their extension to the spaces of distributions.

Theorem 10.3.25 (Fourier transform on C∞(G) and S(G)). The Fourier transformFG : C∞(G) → S(G) and its inverse F−1

G : S(G) → C∞(G) are continuous,satisfying F−1

G ◦ FG = id and FG ◦ F−1G = id on C∞(G) and S(G), respectively.

Proof. By (10.7) in Remark 10.3.11, writing C∞(G) =⋂

k∈N H2k(G), the smooth-ness f ∈ C∞(G) is equivalent to f ∈ H2k(G) for all k ∈ N, by Remark 10.3.24.This means that ∑

[ξ]∈G

dim(ξ) 〈ξ〉2k||f ||2HS <∞

for all k ∈ N. Hence FGf ∈ S(G). Consequently, fj → f in C∞(G) impliesthat fj → f in H2k(G) for all k ∈ N. Taking the Fourier transform we see thatq2k(FGfj − FGf) → 0 as j → ∞ for all k ∈ N, which implies that FGfj → FGf

in S(G). Inverting this argument implies the continuity of the inverse Fouriertransform F−1

G from S(G) to C∞(G). The last part of the theorem follows fromTheorem 10.3.23. �

Corollary 10.3.26 (S(G) is a Montel nuclear space). The space S(G) is a Montelspace and a nuclear space.

Proof. This follows from the same properties of C∞(G) because S(G) is homeo-morphic to C∞(G), a homeomorphism given by the Fourier transform. See Exer-cises B.3.35 and B.3.51. �


Definition 10.3.27 (Space S ′(G)). The space S ′(G) of slowly increasing or tempereddistributions on the unitary dual G is defined as the space of all H ∈ M(G) forwhich there exists some k ∈ N such that∑

[ξ]∈G

dim(ξ) 〈ξ〉−k||H(ξ)||HS <∞.

The convergence in S ′(G) is defined as follows. We will say that Hj ∈ S ′(G)converges to H ∈ S ′(G) in S ′(G) as j →∞, if there exists some k ∈ N such that∑

[ξ]∈G

dim(ξ) 〈ξ〉−k||Hj(ξ)−H(ξ)||HS → 0

as j →∞.

Lemma 10.3.28 (Trace and Hilbert–Schmidt norm). Let H be a Hilbert space andlet A,B ∈ S2(H) be Hilbert–Schmidt operators5. Then

|Tr(AB)| ≤ ||A||HS ||B||HS .

Proof. By the Cauchy–Schwartz inequality for the (Hilbert–Schmidt) inner prod-uct on S2(H) we have

|Tr(AB)| = |〈A,B∗〉HS | ≤ ||A||HS ||B||HS ,

proving the required estimate. �

Definition 10.3.29 (Duality 〈·, ·〉G). Let H ∈ S ′(G) and h ∈ S(G). We write

〈H,h〉G :=∑

[ξ]∈G

dim(ξ) Tr (H(ξ) h(ξ)). (10.12)

The sum is well defined in view of

〈H,h〉G ≤∑

[ξ]∈G

dim(ξ) |Tr (H(ξ) h(ξ))|

≤∑

[ξ]∈G

dim(ξ) ‖H(ξ)‖HS ‖h(ξ)‖HS

≤

⎛⎝ ∑[ξ]∈G

dim(ξ) 〈ξ〉−k ‖H(ξ)‖2HS

⎞⎠1/2⎛⎝ ∑[ξ]∈G

dim(ξ) 〈ξ〉k ‖h(ξ)‖2HS

⎞⎠1/2

<∞,

5Recall Definition B.5.43.


which is finite in view of Proposition 10.3.21 and Corollary 10.3.22. Here we usedLemma 10.3.28 and the Cauchy–Schwarz inequality (see Lemma 3.3.28) in theestimate. The bracket 〈·, ·〉G in (10.12) introduces the duality between S ′(G) andS(G) so that S ′(G) is the dual space to S(G).

Proposition 10.3.30 (Sequential density of S(G) in S ′(G)). The space S(G) issequentially dense in S ′(G).

Proof. We use the standard approximation in sequence spaces by cutting the se-quence to obtain its approximation. Thus, let H ∈ S ′(G). For ξ ∈ G we define

Hj(ξ) :={

H(ξ), if 〈ξ〉 ≤ j,0, if 〈ξ〉 > j.

Let k ∈ N be such that ∑[ξ]∈G

dim(ξ) 〈ξ〉−k ||H(ξ)||HS <∞. (10.13)

Clearly Hj ∈ S(G) for all j, and Hj → H in S ′(G) as j →∞, because∑[ξ]∈G

dim(ξ) 〈ξ〉−k||Hj(ξ)−H(ξ)||HS =∑

[ξ]∈G,〈ξ〉>j

dim(ξ) 〈ξ〉−k||H(ξ)||HS → 0

as j → 0, in view of the convergence of the series in (10.13) and the fact that theeigenvalues of the Laplacian on G are increasing to infinity. �

We now establish a relation of the duality brackets 〈·, ·〉G and 〈·, ·〉G with theFourier transform and its inverse. This will allow us to extend the actions of theFourier transforms to the spaces of distributions on G and G.

Proposition 10.3.31 (Dualities and Fourier transforms). Let ϕ ∈ C∞(G), h ∈S(G). Then we have the identities

〈FGϕ, h〉G =⟨ϕ, ι ◦ F−1

G h⟩

G

and ⟨F−1

G h, ϕ⟩

G= 〈h,FG(ι ◦ ϕ)〉G,

where (ι ◦ ϕ)(x) = ϕ(x−1).


〈FGϕ, h〉G =∑

[ξ]∈G

dim(ξ) Tr (ϕ(ξ) h(ξ))

=∑

[ξ]∈G

dim(ξ) Tr(∫

G

ϕ(x) ξ∗(x) dx h(ξ))


=∫

G

ϕ(x)

⎧⎨⎩ ∑[ξ]∈G

dim(ξ) Tr(ξ(x−1) h(ξ)

)⎫⎬⎭ dx

(10.11)=

∫G

ϕ(x) (F−1G h)(x−1) dx

=⟨ϕ, ι ◦ F−1

G h⟩

G.

Similarly, we have⟨F−1

G h, ϕ⟩

G=

∫G

(F−1G h)(x) ϕ(x) dx

=∫

G

⎧⎨⎩ ∑[ξ]∈G

dim(ξ) Tr (ξ(x) h(ξ))

⎫⎬⎭ ϕ(x) dx

=∑

[ξ]∈G

dim(ξ) Tr({∫

G

ξ(x−1) ϕ(x−1) dx

}h(ξ)

)

=∑

[ξ]∈G

dim(ξ) Tr({∫

G

ξ∗(x) (ι ◦ ϕ)(x) dx

}h(ξ)

)=

∑[ξ]∈G

dim(ξ) Tr (ι ◦ ϕ(ξ) h(ξ))

= 〈h,FG(ι ◦ ϕ)〉G,


Proposition 10.3.31 motivates the following definitions:

Definition 10.3.32 (Fourier transforms on D′(G) and S ′(G)). For u ∈ D′(G), wedefine FGu ≡ u ∈ S ′(G) by

〈FGu, h〉G :=⟨u, ι ◦ F−1

G h⟩

G, (10.14)

for all h ∈ S(G). For H ∈ S ′(G), we define F−1G H ∈ D′(G) by⟨

F−1G H,ϕ

⟩G

:= 〈H,FG(ι ◦ ϕ)〉G, (10.15)

for all ϕ ∈ C∞(G).

Theorem 10.3.33 (Well-defined and continuous). For u ∈ D′(G) and H ∈ S ′(G),their forward and inverse Fourier transforms FGu ∈ S ′(G) and F−1

G H ∈ D′(G)are well defined. Moreover, the mappings FG : D′(G)→ S ′(G) and F−1

G : S ′(G)→D′(G) are continuous, and

F−1G ◦ FG = id and FG ◦ F−1

G = id

on D′(G) and S ′(G), respectively.


Proof. For h ∈ S(G), its inverse Fourier transform satisfies F−1G h ∈ C∞(G) by

Theorem 10.3.25. Therefore, ι ◦ F−1G h ∈ C∞(G), so that the definition in (10.14)

makes sense. Moreover, the mappings F−1G : S(G) → C∞(G) and ι : C∞(G) →

C∞(G) are continuous, so that FGu is a continuous functional on S(G), implyingthat FGu ∈ S ′(G). Let us now show that FG : D′(G) → S ′(G) is continuous.Indeed, if uj → u in D′(G), then

⟨uj , ι ◦ F−1

G h⟩

G→

⟨u, ι ◦ F−1

G h⟩

Gas j →∞ for

every h ∈ S(G), implying that FGuj → FGu in S ′(G).The proof that F−1

G H ∈ D′(G) for every H ∈ S ′(G) is similar, as well as thecontinuity of F−1

G : S ′(G)→ D′(G), and are left as Exercise 10.3.34.To show that F−1

G ◦ FG = id on D′(G), we take u ∈ D′(G) and ϕ ∈ C∞(G),and calculate⟨

(F−1G ◦ FG)u, ϕ

⟩G

(10.15)= 〈FGu,FG(ι ◦ ϕ)〉G

(10.14)=

⟨u, ι ◦ F−1

G (FG(ι ◦ ϕ))⟩

G

Theorem 10.3.25= 〈u, ϕ〉G.

Similarly, for H ∈ S ′(G) and h ∈ S(G), we have⟨(FG ◦ F−1

G )H,h⟩

G

(10.14)=

⟨F−1

G H, ι ◦ F−1G h

⟩G

(10.15)=

⟨H,FG

(ι ◦ ι ◦ F−1

G h)⟩

G

Theorem 10.3.25= 〈H,h〉G,

completing the proof. �Exercise 10.3.34. Complete the proof of Theorem 10.3.33 by showing that F−1

G H ∈D′(G) for every H ∈ S ′(G) and that F−1

G : S ′(G)→ D′(G) is continuous.

Corollary 10.3.35 (Sequential density of C∞(G) in D′(G)). The space C∞(G) issequentially dense in D′(G).

Proof. The statement follows from Proposition 10.3.30 saying that the space S(G)is sequentially dense in S ′(G), and properties of the Fourier transform from The-orems 10.3.25 and 10.3.33. �

10.3.3 Spaces Lp(G)

Definition 10.3.36 (Spaces Lp(G)). For 1 ≤ p < ∞, we will write Lp(G) ≡�p

(G, dimp( 2

p− 12 )

)for the space of all H ∈ S ′(G) such that

||H||Lp(G) :=

⎛⎝ ∑[ξ]∈G

(dim(ξ))p( 2p− 1

2 ) ||H(ξ)||pHS

⎞⎠1/p

<∞.


For p =∞, we will write L∞(G) ≡ �∞(G, dim−1/2

)for the space of all H ∈ S ′(G)

such that||H||L∞(G) := sup

[ξ]∈G

(dim(ξ))−1/2 ||H(ξ)||HS <∞.

We will usually write Lp(G) but the notation �p(G, dimp( 2

p− 12 )

)can be also used

to emphasize that these spaces have a structure of weighted sequence spaces onthe discrete set G, with the weights given by the powers of the dimensions of therepresentations.

Exercise 10.3.37. Prove that spaces Lp(G) are Banach spaces for all 1 ≤ p ≤ ∞.

Remark 10.3.38. Two important cases of L2(G) = �2(G, dim1

)and L1(G) =

�1(G, dim3/2

)are defined by the norms

||H||L2(G) :=

⎛⎝ ∑[ξ]∈G

dim(ξ) ||H(ξ)||2HS

⎞⎠1/2

,

which is already familiar from (10.8), and by

||H||L1(G) :=∑

[ξ]∈G

(dim(ξ))3/2 ||H(ξ)||HS .

We now discuss several properties of spaces Lp(G). We first recall a result onthe interpolation of weighted spaces from [14, Theorem 5.5.1]:

Theorem 10.3.39 (Interpolation of weighted spaces). Let us write dμ0(x) =w0(x) dμ(x), dμ1(x) = w1(x) dμ(x), and write Lp(w) = Lp(w dμ) for the weightw. Suppose that 0 < p0, p1 <∞. Then

(Lp0(w0), Lp1(w1))θ,p = Lp(w),

where 0 < θ < 1, 1p = 1−θ

p0+ θ

p1, and w = w

p(1−θ)/p00 w

pθ/p11 .

From this we obtain:

Proposition 10.3.40 (Interpolation of Lp(G) spaces). Let 1 ≤ p0, p1 <∞. Then(Lp0(G), Lp1(G)

)θ,p

= Lp(G),

where 0 < θ < 1 and 1p = 1−θ

p0+ θ

p1.

Proof. The statement follows from Theorem 10.3.39 if we regard Lp(G) =�p

(G, dimp( 2

p− 12 )

)as a weighted sequence space over G with the weight given

by dim(ξ)p( 2p− 1

2 ). �


Lemma 10.3.41 (Hilbert–Schmidt norm of representations). If ξ ∈ Rep(G), then

||ξ(x)||HS =√

dim(ξ)

for every x ∈ G.

Proof. We have

||ξ(x)||HS = (Tr (ξ(x) ξ(x)∗))1/2 =(Tr Idim(ξ)

)1/2 =√

dim(ξ),

by the unitarity of the representation ξ. �Proposition 10.3.42 (Fourier transforms on L1(G) and L1(G)). The Fourier trans-form FG is a linear bounded operator from L1(G) to L∞(G) satisfying

||f ||L∞(G) ≤ ||f ||L1(G).

The inverse Fourier transform F−1G is a linear bounded operator from L1(G) to

L∞(G) satisfying||F−1

G H||L∞(G) ≤ ||H||L1(G).

Proof. Using f(ξ) =∫

Gf(x) ξ(x)∗ dx, by Lemma 10.3.41 we get

||f(ξ)||HS ≤∫

G

|f(x)| ||ξ(x)∗||HS dx ≤ (dim(ξ))1/2||f ||L1(G).

Therefore,

||f ||L∞(G) = sup[ξ]∈G

(dim(ξ))−1/2||f(ξ)||HS ≤ ||f ||L1(G).

On the other hand, using (F−1G H)(x) =

∑[ξ]∈G dim(ξ) Tr (ξ(x) H(ξ)), by Lemma

10.3.28 and Lemma 10.3.41 we have

|(F−1G H)(x)| ≤

∑[ξ]∈G

dim(ξ) ||ξ(x)||HS ||H(ξ)||HS

=∑

[ξ]∈G

(dim(ξ))3/2 ||H(ξ)||HS

= ||H||L1(G),

from which we get ||F−1G H||L∞(G) ≤ ||H||L1(G). �

Theorem 10.3.43 (Hausdorff–Young inequality). Let 1 ≤ p ≤ 2 and 1p + 1

q = 1.

Let f ∈ Lp(G) and H ∈ Lp(G). Then ||f ||Lq(G) ≤ ||f ||Lp(G) and ||F−1G H||Lq(G) ≤

||H||Lp(G).


Theorem 10.3.43 follows from the L1 → L∞ and L2 → L2 boundedness inProposition 10.3.42 and in Proposition 10.3.17, respectively, by the following inter-polation theorem in [14, Corollary 5.5.4] (which is also a consequence of Theorem10.3.39):

Theorem 10.3.44 (Stein–Weiss interpolation). Let 1 ≤ p0, p1, q0, q1 <∞ and let

T : Lp0(U,w0 dμ)→ Lq0(V, w0 dν), T : Lp1(U,w1 dμ)→ Lq1(V, w1 dν),

with norms M0 and M1, respectively. Then

T : Lp(U,w dμ)→ Lq(V, w dν)

with norm M≤M1−θ0 Mθ

1 , where 1p = 1−θ

p0+ θ

p1, 1

q = 1−θq0

+ θq1

, w=wp(1−θ)/p00 w

pθ/p11

and w = w0p(1−θ)/p0w1

pθ/p1 .

We now turn to the duality between spaces Lp(G):

Theorem 10.3.45 (Duality of Lp(G)). Let 1 ≤ p < ∞ and 1p + 1

q = 1. Then(Lp(G)

)′= Lq(G).

Proof. The duality is given by the bracket 〈·, ·〉G in Definition 10.3.29:

〈H,h〉G :=∑

[ξ]∈G

dim(ξ) Tr (H(ξ) h(ξ)).

Assume first 1 < p < ∞. Then, if H ∈ Lp(G) and h ∈ Lq(G), using Lemma10.3.28 we get∣∣〈H,h〉G

∣∣ ≤ ∑[ξ]∈G

dim(ξ) ‖H(ξ)‖HS ‖h(ξ)‖HS

=∑

[ξ]∈G

(dim(ξ))2p− 1

2 ‖H(ξ)‖HS (dim(ξ))2q− 1

2 ‖h(ξ)‖HS

≤

⎛⎝ ∑[ξ]∈G

(dim(ξ))p( 2p− 1

2 ) ‖H(ξ)‖pHS

⎞⎠1/p

×

⎛⎝ ∑[ξ]∈G

(dim(ξ))q( 2q− 1

2 ) ‖h(ξ)‖qHS

⎞⎠1/q

= ||H||Lp(G) ||h||Lq(G),


where we also used the discrete Holder inequality (Lemma 3.3.28). Let now p = 1.In this case we have∣∣〈H,h〉G

∣∣ ≤∑

[ξ]∈G

(dim(ξ))3/2 ‖H(ξ)‖HS (dim(ξ))−1/2 ‖h(ξ)‖HS

≤ ||H||L1(G) ||h||L∞(G).

We leave the other part of the proof as an exercise. �

Remark 10.3.46 (Sobolev spaces Lpk(G)). If we use difference operators �α from

Definition 10.7.1, we can also define Sobolev spaces Lpk(G), k ∈ N, on the unitary

dual G by

Lpk(G) =

{H ∈ Lp(G) : �αH ∈ Lp(G) for all |α| ≤ k

}.

10.4 Symbols of operators

Let G be a compact Lie group. Let us endow D(G) = C∞(G) with the usual testfunction topology (which is the uniform topology of C∞(G); we refer the readerto Section 10.12 for some additional information on the topics of distributionsand Schwartz kernels if more introduction is desirable). For a continuous linearoperator A : C∞(G) → C∞(G), let KA, LA, RA ∈ D′(G×G) denote respectivelythe Schwartz, left-convolution and right-convolution kernels, i.e.,

Af(x) =∫

G

KA(x, y) f(y) dy

=∫

G

LA(x, xy−1) f(y) dy

=∫

G

f(y) RA(x, y−1x) dy (10.16)

in the sense of distributions. To simplify the notation in the sequel, we will of-ten write integrals in the sense of distributions, with a standard distributionalinterpretation.

Proposition 10.4.1 (Relations between kernels). We have

RA(x, y) = LA(x, xyx−1), (10.17)

as well as

RA(x, y) = KA(x, xy−1) and LA(x, y) = KA(x, y−1x),

with the standard distributional interpretation.

10.4. Symbols of operators 551

Proof. Equality (10.17) follows directly from (10.16). The proof of the last twoequalities is just a change of variables. Indeed, (10.16) implies that KA(x, y) =RA(x, y−1x). Denoting v = y−1x, we have y = xv−1, so that KA(x, xv−1) =RA(x, v). Similarly, KA(x, y) = LA(x, xy−1) from (10.16) and the change w =xy−1 yield y = w−1x, and hence KA(x,w−1x) = LA(x,w). �

We also note that left-invariant operators on C∞(G) correspond to right-con-volutions f �→ f ∗k. Since we identify the Lie algebra g of G with the left-invariantvector fields on G, it will be most natural to study right-convolution kernels in thesequel. Let us explain this in more detail:Remark 10.4.2 (Left or right?). For g ∈ D′(G), define the respective left-convolu-tion and right-convolution operators l(f), r(f) : C∞(G)→ C∞(G) by

l(f)g := f ∗ g,

r(f)g := g ∗ f.

In this notation, the relation between left- and right-convolution kernels of theseconvolution operators in the notation of (10.16) becomes Ll(f)(x, y) = f(y) =Rr(f)(x, y). Also, if Y ∈ g, then Proposition 10.2.9 implies that DY l(f) = l(f)DY .Let the respective left and right regular representations of G be denoted by πL, πR :G→ U(L2(G)), i.e.,

πL(x)f(y) = f(x−1y),πR(x)f(y) = f(yx).

Operator A is

left-invariant if πL(x) A = A πL(x),right-invariant if πR(x) A = A πR(x),

for every x ∈ G. Notice that A is

left-invariant ⇐⇒ right-convolution,

right-invariant ⇐⇒ left-convolution.

Indeed, we have, for example,

[πR(x)l(f)g](z) = (f ∗ g)(zx)

=∫

G

f(y) g(y−1zx) dy

=∫

G

f(y) (πR(x)g)(y−1z) dy

= [f ∗ πR(x)g](z)= [l(f)πR(x)g](z),

so that πR(x)l(f) = l(f)πR(x), and similarly πL(x)r(f) = r(f)πL(x).


The Lie algebras are often (but not always) identified with left-invariantvector fields, which are right-convolutions, that is why our starting choice in thesequel are right-convolution kernels. We refer to Remark 10.4.13 for a furtherdiscussion of left and right.

10.4.1 Full symbols

We now define symbols of operators on G.

Definition 10.4.3 (Symbols of operators on G). Let ξ : G → U(Hξ) be an irre-ducible unitary representation. The symbol of a linear continuous operator A :C∞(G)→ C∞(G) at x ∈ G and ξ ∈ Rep(G) is defined as

σA(x, ξ) := rx(ξ) ∈ End(Hξ),

where rx(y) = RA(x, y) is the right-convolution kernel of A as in (10.16). Hence

σA(x, ξ) =∫

G

RA(x, y) ξ(y)∗ dy

in the sense of distributions, and by Corollary 10.2.10 the right-convolution kernelcan be regained from the symbol as well:

RA(x, y) =∑

[ξ]∈G

dim(ξ) Tr (ξ(y) σA(x, ξ)) , (10.18)

where this equality is interpreted distributionally. We now show that operator Acan be represented by its symbol:

Theorem 10.4.4 (Quantization of operators). Let σA be the symbol of a continuouslinear operator A : C∞(G)→ C∞(G). Then

Af(x) =∑

[ξ]∈G

dim(ξ) Tr(ξ(x) σA(x, ξ) f(ξ)

)(10.19)

for every f ∈ C∞(G) and x ∈ G.

Proof. Let us define a right-convolution operator Ax0 ∈ L(C∞(G)) by its kernelRA(x0, y) = rx0(y), i.e., by

Ax0f(x) :=∫

G

f(y) rx0(y−1x) dy = (f ∗ rx0)(x).

ThusσAx0

(x, ξ) = rx0(ξ) = σA(x0, ξ),


so that by (10.6) we have

Ax0f(x) =∑

[ξ]∈G

dim(ξ) Tr(ξ(x) Ax0f(ξ)

)=

∑[ξ]∈G

dim(ξ) Tr(ξ(x) σA(x0, ξ) f(ξ)

),

where we used that f ∗ rx0 = rx0 f by Proposition 10.2.7. This implies the result,because Af(x) = Axf(x), for each fixed x. �

Definition 10.4.3 and Theorem 10.4.4 justify the following notation:

Definition 10.4.5 (Pseudo-differential operators). For a symbol σA, the corre-sponding operator A defined by (10.19) will be also denoted by Op(σA). Theoperator defined by formula (10.19) will be called the pseudo-differential operatorwith symbol σA.

If we fix representations to be matrix representations we can express all theformulae above in matrix components. Thus, if ξ : G→ U(dim(ξ)) are irreducibleunitary matrix representations then

Af(x) =∑

[ξ]∈G

dim(ξ)dim(ξ)∑m,n=1

ξ(x)nm

⎛⎝dim(ξ)∑k=1

σA(x, ξ)mk f(ξ)kn

⎞⎠ ,

and as a consequence of (10.18) and Corollary 10.2.10 we also have formally:

RA(x, y) =∑

[ξ]∈G

dim(ξ)dim(ξ)∑m,n=1

ξ(y)nm σA(x, ξ)mn. (10.20)

Alternatively, setting Aξ(x)mn := (A(ξmn))(x), we have

σA(x, ξ)mn =dim(ξ)∑k=1

ξkm(x) (Aξkn)(x), (10.21)

1 ≤ m,n ≤ dim(ξ), which follows from the following theorem:

Theorem 10.4.6 (Formula for the symbol via representations). Let σA be the sym-bol of a continuous linear operator A : C∞(G)→ C∞(G). Then for all x ∈ G andξ ∈ Rep(G) we have

σA(x, ξ) = ξ(x)∗(Aξ)(x). (10.22)


Proof. Working with matrix representations ξ : G→ U(dim(ξ)), we have

dim(ξ)∑k=1

ξkm(x) (Aξkn)(x)

(10.19)=

∑k

ξkm(x)∑

[η]∈G

dim(η) Tr(η(x) σA(x, η) ξkn(η)

)=

∑k

ξkm(x)∑

[η]∈G

dim(η)∑i,j,l

η(x)ij σA(x, η)jl ξkn(η)li

=∑k,j

ξkm(x) ξ(x)kj σA(x, ξ)jn

Rem. 10.2.4= σA(x, ξ)mn,

where we take η = ξ if η ∈ [ξ] in the sum, so that ξkn(η)li = 〈ξkn, ηil〉L2 by (10.5),which equals 1

dim(ξ) if ξ = η, k = i and n = l, and zero otherwise. �

Remark 10.4.7 (Formula for symbol on Tn). Since in the case of the torus G = Tn

by Remark 10.2.4 representations of Tn are given by ek(x) = ei2πx·k, k ∈ Zn,formula (10.22) gives the formula for the toroidal symbol

σA(x, k) := σA(x, ek) = e−i2πx·k(A ei2πx·k)(x),

k ∈ Zn, as in Theorem 4.1.4.Remark 10.4.8 (Symbol of the Laplace operator). We note that by Theorem10.3.13 the symbol of the Laplace operator L = LG on G is σL(x, ξ) = −λ[ξ]Idim ξ,where Idim ξ is the identity mapping on Hξ and λ[ξ] are the eigenvalues of −L.

Remark 10.4.9 (Symbol as a mapping on G× G). The symbol of A ∈ L(C∞(G))is a mapping

σA : G× Rep(G)→⋃

ξ∈Rep(G)

End(Hξ),

where σA(x, ξ) ∈ End(Hξ) for every x ∈ G and ξ ∈ Rep(G). However, it can beviewed as a mapping on the space G× G. Indeed, let ξ, η ∈ Rep(G) be equivalentvia an intertwining isomorphism U ∈ Hom(ξ, η): i.e., such that there exists a linearunitary bijection U : Hξ → Hη such that U η(x) = ξ(x) U for every x ∈ G, thatis η(x) = U−1 ξ(x) U . Then by Remark 10.2.5 we have f(η) = U−1 f(ξ) U , andhence also

σA(x, η) = U−1 σA(x, ξ) U.

Therefore, taking any representation from the same class [ξ] ∈ G leads to the sameoperator A in view of the trace in formula (10.19). In this sense we may think thatsymbol σA is defined on G× G instead of G× Rep(G).


Remark 10.4.10 (Symbol of right-convolution). Notice that if A = (f �→ f ∗ a)then RA(x, y) = a(y) and hence

σA(x, ξ) = a(ξ),

and hence Af(ξ) = a(ξ) f(ξ) = σA(x, ξ) f(ξ).

Proposition 10.4.11 (Symbol of left-convolution). If B = (f �→ b ∗ f) is the left-convolution operator, then

LB(x, y) = b(y), RB(x, y) = LB(x, xyx−1) = b(xyx−1),

and the symbol of B is given by

σB(x, ξ) = ξ(x)∗ b(ξ) ξ(x).

Exercise 10.4.12. Prove Proposition 10.4.11. (Hint: use (10.4) and (10.17).)

Remark 10.4.13 (What if we started with left-convolution kernels?). What if wehad chosen right-invariant vector fields and corresponding left-convolution opera-tors as the starting point of the Fourier analysis? Let us define another “Fouriertransform” by

πf (ξ) :=∫

G

f(x) ξ(x) dx.

Then πf∗g = πf πg, and a continuous linear operator A : C∞(G) → C∞(G) canbe presented by

Af(x) =∑

[ξ]∈G


)=

∑[ξ]∈G

dim(ξ) Tr (ξ(x)∗ σA(x, ξ) πf (ξ)) ,

where

σA(x, ξ) = πy �→LA(x,y)(ξ)= ξ(x) (A(ξ∗))(x).

In the coming symbol considerations this left–right choice is encountered, e.g., asfollows:

σAB(x, ξ) ∼ σA(x, ξ) σB(x, ξ) + · · · if we use right-convolutions,σAB(x, ξ) ∼ σB(x, ξ) σA(x, ξ) + · · · if we use left-convolutions.

There is an explicit link between the left–right cases. We refer to Section 10.11 fora further discussion of these issues for the operator-valued symbols. At the sametime, we note that these choices already determine the need to work with rightactions on homogeneous spaces in Chapter 13, so that the homogeneous spacesthere are K\G instead of G/K, see Remark 13.2.5 for the discussion of this issue.


Remark 10.4.14. We have now associated a unique full symbol σA to each continu-ous linear operator A : C∞(G)→ C∞(G). Here σA(x, ξ) : Hξ → Hξ is a linear op-erator for each x ∈ G and each irreducible unitary representation ξ : G→ U(Hξ).The correspondence A �→ σA is linear in the sense that

σA+B(x, ξ) = σA(x, ξ) + σB(x, ξ) and σλA(x, ξ) = λσA(x, ξ),

where λ ∈ C. However, σAB(x, ξ) is not usually σA(x, ξ)σB(x, ξ) (unless B is aright-convolution operator, so that the symbol σB(x, ξ) = b(ξ) does not depend onthe variable x ∈ G). A composition formula will be established in Theorem 10.7.8below.

10.4.2 Conjugation properties of symbols

In the sequel, we will need conjugation properties of symbols which we will nowanalyse for this purpose.

Definition 10.4.15 (φ-pushforwards). Let φ : G → G be a diffeomorphism, f ∈C∞(G), A : C∞(G) → C∞(G) continuous and linear. Then the φ-pushforwardsfφ ∈ C∞(G) and Aφ : C∞(G)→ C∞(G) are defined by

fφ := f ◦ φ−1,

Aφf :=(A(fφ−1)

)φ

= A(f ◦ φ) ◦ φ−1.

Notice thatAφ◦ψ = (Aψ)φ .

Exercise 10.4.16. Using the local theory of pseudo-differential operators show thatA ∈ Ψμ(G) if and only if Aφ ∈ Ψμ(G).

Definition 10.4.17. For u ∈ G, let uL, uR : G→ G be defined by

uL(x) := ux and uR(x) := xu.

Then (uL)−1 = (u−1)L and (uR)−1 = (u−1)R. The inner automorphism φu : G→G defined in Proposition 10.2.6 by φu(x) := u−1xu satisfies

φu = u−1L ◦ uR = uR ◦ u−1

L .

Proposition 10.4.18. Let u ∈ G, B = AuL, C = AuR

and F = Aφu . Then we havethe following relations between symbols:

σB(x, ξ) = σA(u−1x, ξ),σC(x, ξ) = ξ(u)∗ σA(xu−1, ξ) ξ(u),σF (x, ξ) = ξ(u)∗ σA(uxu−1, ξ) ξ(u).


Especially, if A = (f �→ f ∗ a), i.e., σA(x, ξ) = a(ξ), then

σB(x, ξ) = a(ξ),σC(x, ξ) = ξ(u)∗ a(ξ) ξ(u)

= σF (x, ξ).

Proof. We notice that F = C(u−1)L, so it suffices to consider only operators B and

C. For the operator B = AuL, we get∫

G

f(z) RB(x, z−1x) f(z) dz = Bf(x)

= A(f ◦ uL)(u−1L (x))

=∫

G

f(uy) RA(u−1x, y−1u−1x) dy

=∫

G

f(z) RA(u−1x, z−1x) dz,

so RB(x, y) = RA(u−1x, y), yielding

σB(x, ξ) = σA(u−1x, ξ).

For the operator C = AuR, we have∫

G

f(z) RC(x, z−1x) dz = Cf(x)

= A(f ◦ uR)(u−1R (x))

=∫

G

f(yu) RA(xu−1, y−1xu−1) f(yu) dy

=∫

G

f(z) RA(xu−1, uz−1xu−1) dz,

so that RC(x, y) = RA(xu−1, uyu−1), yielding

σC(x, ξ) =∫

G

RC(x, y) ξ(y)∗ dy

=∫

G

RA(xu−1, uyu−1) ξ(y)∗ dy

=∫

G

RA(xu−1, z) ξ(u−1zu)∗ dz

=∫

G

RA(xu−1, z) ξ(u)∗ ξ(z)∗ ξ(u) dz

= ξ(u)∗ σA(xu−1, ξ) ξ(u)

and completing the proof. �


Let us now record how push-forwards by translation affect vector fields.

Lemma 10.4.19 (Push-forwards of vector fields). Let u ∈ G, Y ∈ g and let E =DY : C∞(G)→ C∞(G) be defined by

DY f(x) =ddt


. (10.23)

ThenEuR

= Eφu= Du−1Y u,

i.e.,DY (f ◦ uR)(xu−1) = DY (f ◦ φu)(uxu−1) = Du−1Y uf(x).

Proof. We have

EuRf(x) = E(f ◦ uR)(xu−1)

=ddt

(f ◦ uR)(xu−1 exp(tY ))∣∣t=0

=ddt

f(xu−1 exp(tY )u)∣∣t=0

=ddt

f(x exp(tu−1Y u)∣∣t=0

= Du−1Y uf(x).

Due to the left-invariance, we have EuL= E, so that

Eφu= (Eu−1

L)uR

= EuR= Du−1Y u.

For more transparency, we also calculate directly:

Eφuf(x) = E(f ◦ φu)(uxu−1)

=ddt

(f ◦ φu)(uxu−1 exp(tY ))∣∣t=0

=ddt

f(xu−1 exp(tY )u)∣∣t=0

=ddt

f(x exp(tu−1Y u))∣∣t=0

= Du−1Y uf(x),

yielding the same result. �Remark 10.4.20 (Symbol of iDY can be diagonalised). Notice first that the com-plex vector field iDY is symmetric:

(iDY f, g)L2(G) =∫

G

(iDY f)(x) g(x) dx

= −i∫

G

f(x) DY g(x) dx

= (f, iDY g)L2(G) .

10.5. Boundedness of operators on L2(G) 559

Hence it is always possible to choose a representative ξ ∈ Rep(G) from each [ξ] ∈

G such that σiDY(x, ξ) is a diagonal matrix

⎛⎜⎝λ1

. . .λdim(ξ)

⎞⎟⎠, with diagonal

entries λj ∈ R, which follows because symmetric matrices can be diagonalised byunitary matrices. Notice that then also the commutator of symbols satisfy

[σiDY, σA](x, ξ)mn = (λm − λn) σA(x, ξ)mn.

10.5 Boundedness of operators on L2(G)

In this section we will state some natural conditions on the symbol of an operatorA : C∞(G)→ C∞(G) to guarantee its boundedness on L2(G). Recall first that theHilbert–Schmidt inner product of matrices is defined as a special case of DefinitionB.5.43:

Definition 10.5.1 (Hilbert–Schmidt inner product). The Hilbert–Schmidt innerproduct of A,B ∈ Cm×n is

〈A,B〉HS := Tr(B∗A) =m∑

i=1

n∑j=1

BijAij ,

with the corresponding norm ‖A‖HS := 〈A,A〉1/2HS , and the operator norm

‖A‖op := sup{‖Ax‖�2 : x ∈ Cn×1, ‖x‖�2 ≤ 1

}= ‖A‖�2→�2 ,

where ‖x‖�2 = (∑n

j=1 |xj |2)1/2 is the usual Euclidean norm.

Let A,B ∈ Cn×n. Then by Theorem 12.6.1 proved in Section 12.6 we have

‖AB‖HS ≤ ‖A‖op ‖B‖HS .

Moreover, we also have

‖A‖op = sup{‖AX‖HS : X ∈ Cn×n, ‖X‖HS ≤ 1

}.

By this, taking the Fourier transform of the convolution and using Plancherel’sformula (Corollary 7.6.10), by Proposition 10.2.7 we get

Proposition 10.5.2 (Operator norm of convolutions). We have

‖g �→ f ∗ g‖L(L2(G)) = ‖g �→ g ∗ f‖L(L2(G)) = supξ∈Rep(G)

‖f(ξ)‖op. (10.24)

We also note that ‖f(ξ)‖op = ‖f(η)‖op if [ξ] = [η] ∈ G.


We now extend this property to operators that are not necessarily left- orright-invariant. First we introduce derivatives of higher order on the Lie group G:

Definition 10.5.3 (Operators ∂α on G). Let {Yj}dim(G)j=1 be a basis for the Lie

algebra of G, and let ∂j be the left-invariant vector fields corresponding to Yj ,∂j = DYj

, as in (10.23). For α ∈ Nn0 , let us denote ∂α = ∂α1

1 · · · ∂αnn . Sometimes

we denote these operators by ∂αx .

Remark 10.5.4 (Orderings). We note that unless G is commutative, operators ∂j

do not in general commute. Thus, when we talk about “all operators ∂α”, we meanthat we take these operators in all orderings. However, if we fix a certain orderingof Yj ’s, the commutator of a general ∂α with ∂α taken in this particular ordering isan operator of lower order (this can be easily seen either for the simple properties ofcommutators in Exercise D.1.5 or from the general composition Theorem 10.7.9).The commutator is again a combination of operators of the form ∂β with |β| ≤|α| − 1. Thus, since usually we require some property to hold for example “forall ∂α with |α| ≤ N”, we can rely iteratively on the fact that the assumption isalready satisfied for ∂β , thus making this ordering issue less important.

Theorem 10.5.5 (Boundedness of operators on L2(G)). Let G be a compact Liegroup of dimension n and let k be an integer such that k > n/2. Let σA be thesymbol of a linear continuous operator A : C∞(G) → C∞(G). Assume that thereis a constant C such that

‖∂αx σA(x, ξ)‖op ≤ C

for all x ∈ G, all ξ ∈ Rep(G), and all |α| ≤ k. Then A extends to a boundedoperator from L2(G) to L2(G).

Proof. Let Af(x) = (f ∗ rA(x))(x), where rA(x)(y) = RA(x, y) is the right-convolution kernel of A. Let Ayf(x) = (f ∗ rA(y))(x), so that Axf(x) = Af(x).Then

‖Af‖2L2(G) =∫

G

|Axf(x)|2 dx ≤∫

G

supy∈G

|Ayf(x)|2 dx,


supy∈G

|Ayf(x)|2 ≤ C∑|α|≤k

∫G

|∂αy Ayf(x)|2 dy.

Therefore, using the Fubini theorem to change the order of integration, we obtain

‖Af‖2L2(G) ≤ C∑|α|≤k

∫G

∫G

|∂αy Ayf(x)|2 dx dy

≤ C∑|α|≤k

supy∈G

∫G

|∂αy Ayf(x)|2 dx

10.6. Taylor expansion on Lie groups 561

= C∑|α|≤k

supy∈G

‖∂αy Ayf‖2L2(G)

≤ C∑|α|≤k

supy∈G

‖f �→ f ∗ ∂αy rA(y)‖2L(L2(G))‖f‖2L2(G)

(10.24)

≤ C∑|α|≤k

supy∈G

sup[ξ]∈G

‖∂αy σA(y, ξ)‖2op‖f‖2L2(G),

where the last inequality holds due to (10.24). This completes the proof. �

10.6 Taylor expansion on Lie groups

As Taylor polynomial expansions are useful in obtaining symbolic calculus onRn, we would like to have analogous expansions on a group G. Here, the Taylorexpansion formula on G will be obtained by embedding G into some Rm, usingthe Taylor expansion formula in Rm, and then restricting it back to G.

Let U ⊂ Rm be an open neighbourhood of some point �e ∈ Rm. The Nthorder Taylor polynomial PNf : Rm → C of f ∈ C∞(U) at �e is given by

PNf(�x) =∑

α∈Nm0 : |α|≤N

1α!

(�x− �e)α ∂αx f(�e).

Then the remainder ENf := f − PNf satisfies

ENf(�x) =∑

|α|=N+1

(�x− �e)α fα(�x)

for some functions fα ∈ C∞(U). In particular,

ENf(�x) = O(|�x− �e|N+1) as �x→ �e.

Let G be a compact Lie group; we would like to approximate a smooth function u :G→ C using a Taylor polynomial type expansion nearby the neutral element e ∈G. By Corollary 8.0.4 we may assume that G is a closed subgroup of GL(n, R) ⊂Rn×n, the group of real invertible (n×n)-matrices, and thus a closed submanifoldof the Euclidean space of dimension m = n2. This embedding of G into Rm will bedenoted by x �→ �x, and the image of G under this embedding will be still denotedby G. Also, if x ∈ G, we may still write x for �x to simplify the notation. LetU ⊂ Rm be a small enough open neighbourhood of G ⊂ Rm such that for each�x ∈ U there exists a unique nearest point p(�x) ∈ G (with respect to the Euclideandistance). For u ∈ C∞(G) we define f ∈ C∞(U) by

f(�x) := u(p(�x)).


The effect is that f is constant in the directions perpendicular to G. As above, wemay define the Euclidean Taylor polynomial PNf : Rm → C at e ∈ G ⊂ Rm. Letus define PNu : G→ C as the restriction,

PNu := PNf |G.

We call PNu ∈ C∞(G) a Taylor polynomial of u of order N at e ∈ G. Then forx ∈ G, we have

u(x)− PNu(x) =∑

|α|=N+1

(x− e)α uα(x)

for some functions uα ∈ C∞(G), where we set (x−e)α := (�x−�e)α. There should beno confusion with this notation because there is no substraction on the group G,so subtracting group elements means subtracting them when they are embeddedin a higher-dimensional linear space. Taylor polynomials on G are given by

PNu(x) =∑|α|≤N

1α!

(x− e)α ∂(α)x u(e),

where we set∂(α)

x u(e) := ∂αx f(�e). (10.25)

Remark 10.6.1. We note that in this way we can obtain different forms of theTaylor series. For example, it may depend on the embedding of G into GL(n, R),on the choice of the coordinates in Rn × Rn, etc.

Let us now consider the example of G = SU(2). Recall the quaternionicidentification

(x01 + x1i + x2j + x3k �→ (x0, x1, x2, x3)) : H→ R4,

to be discussed in detail in Section 11.4. Moreover, there is the identificationH ⊃ S3 ∼= SU(2), given by

�x = (x0, x1, x2, x3) �→(

x0 + ix3 x1 + ix2

−x1 + ix2 x0 − ix3

)=

(x11 x12

x21 x22

)= x.

Hence we identify (1, 0, 0, 0) ∈ R4 with the neutral element of SU(2).

Remark 10.6.2. Notice that the functions

q+(x) = x12 = x1 + ix2,

q−(x) = x21 = −x1 + ix2,

q0(x) = x11 − x22 = 2ix3

also vanish at the identity element of SU(2).


A function u ∈ C∞(S3) can be extended to f ∈ C∞(U) = C∞(R4 \ {0}) by

f(�x) := u(�x/‖�x‖).

Therefore, we obtain PNu ∈ C∞(S3),

PNu(�x) :=∑|α|≤N

1α!

(�x− �e)α∂α

x f(�e),

where �e = (1, 0, 0, 0). Expressing this in terms of x ∈ SU(2), we obtain Taylorpolynomials for x ∈ SU(2) in the form

PNu(x) =∑|α|≤N

1α!

(x− e)α∂(α)

x u(e),

where we write ∂(α)x u(e) := ∂α

x f(�e), and where

(x− e)α := (�x− �e)α

= (x0 − 1)α1xα21 xα3

2 xα43

=(

x11 + x22

2− 1

)α1(

x12 − x21

2

)α2(

x12 + x21

2i

)α3(

x11 − x22

2i

)α4

.

This gives an example of possible Taylor monomials on SU(2).

10.7 Symbolic calculus

In this section, we study global symbols of pseudo-differential operators on com-pact Lie groups, as defined in Definition 10.4.3. We also derive elements of thecalculus in quite general classes of symbols. For this, we first introduce differ-ence operators acting on symbols in the ξ-variable. These are analogues of the∂ξ-derivatives in Rn and of the difference operators �ξ on Tn, and are obtainedby the multiplication by “coordinate functions” on the Fourier transform side.

10.7.1 Difference operators

As explained in Section 10.6, smooth functions on a group G can be approxi-mated by Taylor polynomial type expansions. More precisely, there exist partialdifferential operators ∂

(α)x of order |α| on G such that for every u ∈ C∞(G) we

have

u(x) =∑|α|≤N

1α!

qα(x−1) ∂(α)x u(e) +

∑|α|=N+1

qα(x−1) uα(x)

∼∑α≥0

1α!

qα(x−1) ∂(α)x u(e) (10.26)


in a neighbourhood of e ∈ G, where uα ∈ C∞(G), and qα ∈ C∞(G) satisfyqα+β = qαqβ , and ∂

(α)x are as in (10.25). Moreover, here q0 ≡ 1, and qα(e) = 0 if

|α| ≥ 1.

Definition 10.7.1 (Difference operators �αξ ). Let us define difference operators�α

ξ

acting on Fourier coefficients by �αξ f(ξ) := qαf(ξ). Notice that �α+β

ξ = �αξ�

βξ .

Remark 10.7.2. The technical choice of writing qα(x−1) in (10.26) is dictated byour desire to make the asymptotic formulae in Theorems 10.7.8 and 10.7.10 looksimilar to the familiar Euclidean formulae in Rn, and by an obvious freedom in se-lecting among different forms of Taylor polynomials qα, see Remark 10.6.1. For ex-ample, on SU(2), if we work with operators Δ+,Δ−,Δ0 defined in (12.14)–(12.16),we can choose the form of the Taylor expansion (10.26) adapted to functionsq+, q−, q0. On SU(2), we can observe that q+(x−1) = −q−(x), q−(x−1) = −q+(x),q0(x−1) = −q0(x), so that for |α| = 1 the functions qα(x) and qα(x−1) are linearcombinations of q+, q−, q0. In terms of the quaternionic identification, these arefunctions from Remark 10.6.2. Taylor monomials (x− e)α from the previous sec-tion, when restricted to SU(2), can be expressed in terms of functions q+, q−, q0.For an argument of this type we refer to the proof of Lemma 12.4.5.Remark 10.7.3 (Differences reduce the order of symbols). In Theorem 12.3.6 wewill apply the differences on the symbols of specific differential operators on SU(2).In general, on a compact Lie group G, a difference operator of order |γ| appliedto a symbol of a partial differential operator of order N gives a symbol of orderN − |γ|. More precisely:

Proposition 10.7.4 (Differences for symbols of differential operators). Let

D =∑|α|≤N

cα(x) ∂αx (10.27)

be a partial differential operator with coefficients cα ∈ C∞(G), and ∂αx as in Def-

inition 10.5.3. For q ∈ C∞(G) such that q(e) = 0, we define difference operator�q acting on symbols by

�q f(ξ) := qf(ξ).

Then we obtain

�qσD(x, ξ) =∑|α|≤N

cα(x)∑β≤α

(α

β

)(−1)|β| (∂β

x q)(e) σ∂α−βx

(x, ξ), (10.28)

which is a symbol of a partial differential operator of order at most N − 1. Moreprecisely, if q has a zero of order M at e ∈ G then Op(�qσD) is of order N −M .

Proof. Let D in (10.27) be be a partial differential operator, where cα ∈ C∞(G)and ∂α

x : D(G)→ D(G) is left-invariant of order |α|. If |α| = 1 and φ, ψ ∈ C∞(G)then we have the Leibniz property

∂αx (φψ) = φ (∂α

x ψ) + (∂αx φ) ψ,


leading to

0 =∫

G

φ(x) ∂αx ψ(x) dx +

∫G

∂αx φ(x) ψ(x) dx.

More generally, for |α| ∈ N0, φ ∈ C∞(G) and f ∈ D′(G), we have for the distri-butional derivatives∫

G

φ(x) ∂αx f(x) dx = (−1)|α|

∫G

∂αx φ(x) f(x) dx,

with a standard distributional interpretation. Recall that the right-convolutionkernel RA ∈ D′(G×G) of a continuous linear operator A : D(G)→ D(G) satisfies

Aφ(x) =∫

G

φ(y) RA(x, y−1x) dy.

For instance, informally

φ(x) =∫

G

φ(y) δe(y−1x) dy =∫

G

φ(y) δx(y) dy,

where δp ∈ D′(G) is the Dirac delta distribution at p ∈ G. Notice that

∂αx φ(x) =

∫G

(−1)|α| (∂αy φ)(xy−1) δe(y) dy

=∫

G

φ(xy−1) ∂αy δe(y) dy.

The right-convolution kernel of the operator D from (10.27) is given by

RD(x, y) =∑|α|≤N

cα(x) ∂αy δe(y).

Let Dq : C∞(G)→ C∞(G) be defined by

σDq(x, ξ) := �qσD(x, ξ),

i.e.,RDq

(x, y) = q(y) RD(x, y).

Then Dq = Op(σDq ) is a differential operator:

Dqφ(x) =∫

G

φ(xy−1) q(y)∑|α|≤N

cα(x) ∂αy δe(y) dy

=∑|α|≤N

(−1)|α| cα(x)∫

G

∂αy

(φ(xy−1) q(y)

)δe(y) dy

=∑|α|≤N

(−1)|α| cα(x)∑β≤α

(α

β

)(−1)|α−β| (∂β

x q)(e) ∂α−βx φ(x).


Thus

�qσD(x, ξ) =∑|α|≤N

cα(x)∑β≤α

(α

β

)(−1)|β| (∂β

x q)(e) σ∂α−βx

(x, ξ).

Hence if q has a zero of order M at e ∈ G then Dq is of order N −M . �Exercise 10.7.5. Provide the distributional interpretation of all the steps in theproof of Proposition 10.7.4.

10.7.2 Commutator characterisation

Definition 10.7.6 (Operator classesAmk (M)). For a compact closed manifold M , let

Am0 (M) denote the set of those continuous linear operators A : C∞(M)→ C∞(M)

which are bounded from Hm(M) to L2(M). Recursively defineAmk+1(M)⊂Am

k (M)such that A ∈ Am

k (M) belongs to Amk+1(M) if and only if [A,D] = AD −DA ∈

Amk (M) for every smooth vector field D on M .

We now recall a variant of the commutator characterisation of pseudo-differ-ential operators given in Theorem 5.3.1 which assures that the behaviour of com-mutators in Sobolev spaces characterises pseudo-differential operators:

Theorem 10.7.7. A continuous linear operator A : C∞(M) → C∞(M) belongs toΨm(M) if and only if A ∈ ⋂∞

k=0Amk (M).

We note that in such a characterisation on a compact Lie group M = G,it suffices to consider vector fields of the form D = Mφ∂x, where Mφf := φf ismultiplication by φ ∈ C∞(G), and ∂x is left-invariant. Notice that

[A,Mφ∂x] = Mφ [A, ∂x] + [A,Mφ] ∂x,

where [A,Mφ]f = A(φf) − φAf . Hence we need to consider compositions MφA,AMφ, A ◦ ∂x and ∂x ◦A. First, we observe that

σMφA(x, ξ) = φ(x) σA(x, ξ), (10.29)σA◦∂x

(x, ξ) = σA(x, ξ) σ∂x(x, ξ), (10.30)

σ∂x◦A(x, ξ) = σ∂x(x, ξ) σA(x, ξ) + (∂xσA)(x, ξ), (10.31)

where σ∂x(x, ξ) is independent of x ∈ G. Here (10.29) and (10.30) are straightfor-

ward and (10.31) follows by the Leibniz formula:

∂x ◦Af(x) = ∂x

∑[ξ]∈G


)=

∑[ξ]∈G

dim(ξ) Tr((∂xξ)(x) σA(x, ξ) f(ξ)

)+

∑[ξ]∈G

dim(ξ) Tr(ξ(x) ∂xσA(x, ξ) f(ξ)

),


where we used that σ∂x(x, ξ) = ξ(x)∗(∂xξ)(x) by Theorem 10.4.6 to obtain (10.31).

Next we claim that we have the formula

σAMφ(x, ξ) ∼

∑α≥0

1α!�α

ξ σA(x, ξ) ∂(α)x φ(x),

where ∂(α)x are certain partial differential operators of order |α|. This follows from

the general composition formula in Theorem 10.7.8.

10.7.3 Calculus

Here we discuss elements of the symbolic calculus of operators. First we recall thefundamental quantity 〈ξ〉 from Definition 10.3.18 that will allow us to introducethe orders of operators. We note that this scale 〈ξ〉 on G is determined by theeigenvalues of the Laplace operator L on G. We now formulate the result oncompositions:

Theorem 10.7.8 (Composition formula I). Let m1,m2 ∈ R and ρ > δ ≥ 0. LetA,B : C∞(G)→ C∞(G) be continuous and linear, their symbols satisfy∥∥�α

ξ σA(x, ξ)∥∥

op≤ Cα 〈ξ〉m1−ρ|α|,∥∥∂β

xσB(x, ξ)∥∥

op≤ Cβ 〈ξ〉m2+δ|β|,

for all multi-indices α and β, uniformly in x ∈ G and [ξ] ∈ G. Then


1α!

(�αξ σA)(x, ξ) ∂(α)

x σB(x, ξ), (10.32)

where the asymptotic expansion means that for every N ∈ N we have∥∥∥∥∥∥σAB(x, ξ)−∑|α|<N

1α!

(�αξ σA)(x, ξ) ∂(α)

x σB(x, ξ)

∥∥∥∥∥∥op

≤ CN 〈ξ〉m1+m2−(ρ−δ)N.

Proof. First,

ABf(x) =∫

G

(Bf)(xz) RA(x, z−1) dz

=∫

G

∫G

f(xy−1) RB(xz, yz) dy RA(x, z−1) dz,


where we use the standard distributional interpretation of integrals. Hence

σAB(x, ξ)

=∫

G

RAB(x, y) ξ(y)∗ dy

=∫

G

∫G

RA(x, z−1) ξ(z−1)∗ RB(xz, yz) ξ(yz)∗ dz dy

=∑|α|<N

1α!

∫G

∫G

RA(x, z−1) qα(z−1) ξ(z−1)∗

×∂(α)x RB(x, yz) ξ(yz)∗ dz dy

+∑|α|=N

∫G

∫G

RA(x, z−1) qα(z−1) ξ(z−1)∗ uα(x, yz) ξ∗(yz) dz dy

=∑|α|<N

1α!

(�αξ σA)(x, ξ) ∂(α)

x σB(x, ξ) +∑|α|=N

(�αξ σA)(x, ξ) uα(x, ξ).

Now the statement follows because we have

‖uα(x, ξ)‖op ≤ C〈ξ〉m1+δN

since uα(x, y) is the remainder in the Taylor expansion of RB(x, y) in x only andso it satisfies similar estimates to those of σB with respect to ξ. This completesthe proof. �

A similar proof yields another version of the composition formula:

Theorem 10.7.9 (Composition formula II). Let m1,m2 ∈ R and ρ > δ ≥ 0. LetA,B : C∞(G)→ C∞(G) be continuous and linear, their symbols satisfying∥∥∂β

x�αξ σA(x, ξ)

∥∥op

≤ Cα 〈ξ〉m1−ρ|α|+δ|β|,∥∥∂βx�α

ξ σB(x, ξ)∥∥

op≤ Cβ 〈ξ〉m2−ρ|α|+δ|β|,

for all multi-indices α and β, uniformly in x ∈ G and [ξ] ∈ G. Then


1α!

(�αξ σA)(x, ξ) ∂(α)

x σB(x, ξ),

where the asymptotic expansion means that for every N ∈ N we have∥∥∥∥∥∥�γξ ∂β

x

⎛⎝σAB(x, ξ)−∑|α|<N

1α!

(�αξ σA)(x, ξ) ∂(α)

x σB(x, ξ)

⎞⎠∥∥∥∥∥∥op

≤ CN 〈ξ〉m1+m2−(ρ−δ)N−ρ|γ|+δ|β|.


Let us complement Theorems 10.7.8 and 10.7.9 with a formula for the adjointoperator:

Theorem 10.7.10 (Adjoint). Let m ∈ R and ρ > δ ≥ 0. Let A : C∞(G)→ C∞(G)be continuous and linear, with symbol σA satisfying∥∥�α

ξ ∂βxσA(x, ξ)

∥∥op≤ Cα 〈ξ〉m−ρ|α|+δ|β|,

for all multi-indices α, uniformly in x ∈ G and [ξ] ∈ G. Then the symbol of A∗ is

σA∗(x, ξ) ∼∑α≥0

1α!�α

ξ ∂(α)x σA(x, ξ)∗,

where the asymptotic expansion means that for every N ∈ N we have∥∥∥∥∥∥�γξ ∂β

x

⎛⎝σA(x, ξ)−∑|α|<N

1α!�α

ξ ∂(α)x σA(x, ξ)∗

⎞⎠∥∥∥∥∥∥op

≤ CN 〈ξ〉m−(ρ−δ)N−ρ|γ|+δ|β|.

Proof. First we observe that writing

A∗g(y) =∫

G

g(x) RA∗(y, x−1y) dx,

we get the relationRA∗(y, x−1y) = RA(x, y−1x)

between kernels, which means that

RA∗(x, v) = RA(xv−1, v−1).

From this we find

σA∗(x, ξ) =∫

G

RA∗(x, v) ξ(v)∗ dv

=∫

G

RA(xv−1, v−1) ξ(v)∗ dv

=∑|α|<N

1α!

∫G

qα(v) ∂(α)x RA(x, v−1) ξ(v)∗ dv +RN (x, ξ)

=∑|α|<N

1α!�α

ξ ∂(α)x σA(x, ξ)∗ +RN (x, ξ),

where the last formula for the asymptotic expansion follows in view of

σA(x, ξ)∗ =(∫

G

RA(x, v) ξ∗(v) dv

)∗=

∫G

RA(x, v−1) ξ∗(v) dv,

and estimate for the remainder RN (x, ξ) follows by an argument similar to thatin the proof of Theorem 10.7.8. �


Above we considered the symbol of the adjoint. Since KAt(x, y) = KA∗(x, y),Theorem 10.7.10 provides a follow-up:

Corollary 10.7.11 (Transpose). Let A :C∞(G)→C∞(G) be as in Theorem 10.7.10.Then the symbol of the transpose At is

σAt(x, ξ) ∼∑α≥0

1α!�α

ξ ∂(α)x σA(x, ξ)∗,

where the right-convolution kernel of A : C∞(G) → C∞(G) is defined byRA(x, y) := RA(x, y), and where the asymptotic expansion is interpreted as inTheorem 10.7.10. �

We postpone the discussion of the asymptotic expansion of the parametrixfor elliptic operators in Theorem 10.9.10 to after we introduce symbol classes.

10.7.4 Leibniz formula

For studying products of pseudo-differential symbols, it would be beneficial tohave a Leibniz-like formula for the “derivatives with respect to the dual variableξ ∈ G”. Classically for smooth functions σA, σB : R → C, the Leibniz formula isthe familiar

(σAσB)′(ξ) = σ′A(ξ) σB(ξ) + σA(ξ) σ′B(ξ).

In the context of pseudo-differential calculus on the torus T, we have T ∼= Z; forfunctions σA, σB : Z→ C a useful Leibniz-like formula reads

�ξ(σAσB) = (�ξσA) σB + σA (�ξσB) + (�ξσA) (�ξσB) ,

where the difference operator is defined by �ξσ(ξ) = σ(ξ + 1)− σ(ξ).Let G be a compact Lie group. Given q ∈ C∞(G) and f ∈ D′(G), let

�q f(ξ) := qf(ξ). (10.33)

Let σA(ξ) = a(ξ) and σB(ξ) = b(ξ). Then

�q(σAσB)(ξ) = q(b ∗ a)(ξ)

= b ∗ (qa)(ξ) + Rb(ξ)

= (�qσA(ξ))σB(ξ) + Rb(ξ),

where the remainder operator R : D′(G)→ D′(G) is given by

Rb = q(b ∗ a)− b ∗ (qa),

that isRb(x) =

∫G

b(y)(q(x)− q(y−1x)

)a(y−1x) dy.

10.8. Boundedness on Sobolev spaces Hs(G) 571

Thus by the Taylor expansion from Section 10.6 we have

Rb(x) ∼∫

G

b(y)

⎛⎝ ∑|α|≥0

c(q, α)(x) y(α)

⎞⎠ a(y−1x) dy,

where

c(q, α)(x) :=1α!

∂(α)z

(q(x)− q(z−1x)

)∣∣∣z=e

.

As usual, according to (10.33), the difference �c(q,α) is

�c(q,α)f(ξ) = c(q, α)f(ξ). (10.34)

Notice that c(q, 0)(x) = q(x) − q(x) = 0, so that we get an asymptotic Leibnizformula:

Theorem 10.7.12 (Asymptotic Leibniz formula). For symbols σA and σB, we have

�q(σAσB)(ξ) ∼ (�qσA(ξ))σB(ξ) +∑|α|>0

�c(q,α)

(σA(ξ)

(�α

ξ σB(ξ)))

.

10.8 Boundedness on Sobolev spaces Hs(G)

In this section we show conditions on the symbol for operators to be boundedon Sobolev spaces Hs(G). The Sobolev space Hs(G) of order s ∈ R can be de-fined via a smooth partition of unity of the closed manifold G, and there areother definitions as well, in particular in terms of the Laplace operator on G, seeDefinition 10.3.9. We also recall Definition 10.3.18 of the quantity 〈ξ〉 on the Liegroup G, measuring the order of operators compared to that of the Laplacian L:if −Lξvw(x) = λ[ξ]ξ

vw(x), then

〈ξ〉 := (1 + λ[ξ])1/2.

according to (10.9). Now we can formulate the result on the Sobolev space bound-edness:

Theorem 10.8.1 (Boundedness of operators on Sobolev spaces). Let G be a compactLie group. Let A be a continuous linear operator from C∞(G) to C∞(G) and letσA be its symbol. Assume that there are constants μ,Cα ∈ R such that

‖∂αx σA(x, ξ)‖op ≤ Cα 〈ξ〉μ

holds for all x ∈ G, ξ ∈ Rep(G) and all multi-indices α. Then A extends to abounded operator from Hs(G) to Hs−μ(G), for all s ∈ R.


Remark 10.8.2. Notice that we may easily show a special case of this result withs = μ. Namely, if σA is as in Theorem 10.8.1, then∥∥∂α

x

(σA(x, ξ)〈ξ〉−μ

)∥∥op≤ Cα

for every multi-index α. Here σA(x, ξ)〈ξ〉−μ = σA◦Ξ−μ(x, ξ), and thus Theo-rem 10.5.5 implies that A◦Ξ−μ is bounded on L2(G), so that A∈L(Hμ(G),L2(G)).

Proof of Theorem 10.8.1. Observing the continuous mapping Ξs :Hs(G)→L2(G),we have to prove that operator Ξs−μ◦A◦Ξ−s is bounded from L2(G) to L2(G). Letus denote B = A ◦Ξ−s, so that the symbol of B satisfies σB(x, ξ) = 〈ξ〉−s

σA(x, ξ)for all x ∈ G and ξ ∈ Rep(G). Since Ξs−μ ∈ Ψs−μ(G), by (10.24) and Lemma10.9.3 its symbol satisfies

‖�αξ σΞs−μ(x, ξ)‖op ≤ C ′α 〈ξ〉s−μ−|α|

. (10.35)

Now we can observe that the asymptotic formula in Theorem 10.7.8 works for thecomposition Ξs−μ ◦B in view of (10.35), and we obtain

∂βxσΞs−μ◦B(x, ξ) ∼

∑α≥0

1α!

(�α

ξ σΞs−μ(x, ξ))〈ξ〉−s ∂(α)

x ∂βxσA(x, ξ).

It follows that ∥∥∂βxσΞs−μ◦B(x, ξ)

∥∥op≤ C ′′β ,

so that Ξs−μ ◦ B is bounded on L2(G) by Theorem 10.5.5. This completes theproof. �Remark 10.8.3 (Functional analytic argument). The boundedness of operators onthe Sobolev spaces follows also by a purely functional analytic argument if Gis compact. Indeed, because the space C∞(G) is nuclear (by adopting ExerciseB.3.51), the tensor product space C∞(G) ⊗π �∞(G) can be endowed with theFrechet topology given by the seminorms supx∈G,[ξ]∈G |∂α

x a(x, ξ)|. Hence, as soonas the operator satisfies supx∈G,[ξ]∈G |∂α

x a(x, ξ)| < ∞ for finitely many α, it isbounded on the corresponding space Hs(G).

Exercise 10.8.4. Work out the details of this argument.

10.9 Symbol classes on compact Lie groups

The goal of this section is to describe the pseudo-differential symbol inequalitieson compact Lie groups that yield Hormander’s classes Ψm(G). On the way tocharacterising the usual Hormander classes Ψm(G) in Theorem 10.9.6, we needsome properties concerning symbols of pseudo-differential operators that we willestablish in the next sections.

10.9. Symbol classes on compact Lie groups 573

10.9.1 Some properties of symbols of Ψm(G)

Given an operator from Hormander’s class Ψm(G), we can derive some informationabout its full symbol as defined in Definition 10.4.3. A precise characterisation ofthe class Ψm(G) will be given in Theorem 10.9.6.

Lemma 10.9.1. Let A ∈ Ψm(G). Then there exists a constant C <∞ such that

‖σA(x, ξ)‖op ≤ C〈ξ〉m

for all x ∈ G and ξ ∈ Rep(G). Also, if u ∈ G and if B is an operator with symbolσB(x, ξ) = σA(u, ξ), then B ∈ Ψm(G).

Proof. First, B ∈ Ψm(G) follows from the local theory of pseudo-differential op-erators, by studying

Bf(x) =∫

G

KA(u, ux−1y) f(y) dy.

Hence the right-convolution operator B is bounded from Hs(G) to Hs−m(G),implying ‖σA(u, ξ)‖ ≤ C〈ξ〉m. �Exercise 10.9.2. Provide the details for the proof of Lemma 10.9.1.

Lemma 10.9.3. Let A ∈ Ψm(G). Then Op(�αξ ∂β

xσA) ∈ Ψm−|α|(G) for all α, β.

Proof. First, given A ∈ Ψm(G), let us define σB(x, ξ) = �αξ ∂β

xσA(x, ξ). We mustshow that B ∈ Ψm−|α|(G). If here |β| = 0, we obtain

Bf(x) =∫

G

f(xy−1) qα(y) RA(x, y) dy =∫

G

qα(y−1x) KA(x, y) f(y) dy.

Moving to local coordinates, we need to study

Bf(x) =∫

Rn

φ(x, y) KA(x, y) f(y) dy,

where A ∈ Ψm(Rn × Rn) with φ ∈ C∞(Rn×Rn), the kernel KA being compactlysupported. Let us calculate the symbol of B:

σB(x, ξ) =∫

Rn

e2πi(y−x)·ξ φ(x, y) KA(x, y) dy

∼∑γ≥0

1γ!

∂γz φ(x, z)|z=x

∫G

e2πi(y−x)·ξ (y − x)γ KA(x, y) dy

=∑γ≥0

1γ!

∂γz φ(x, z)|y=x Dγ

ξ σA(x, ξ).

This shows that B ∈ Ψm(Rn × Rn). We obtain Op(�αξ σA) ∈ Ψm−|α|(G) if A ∈

Ψm(G).


Next we show that B = Op(∂βxσA) ∈ Ψm(G). We may assume that |β| = 1.

Left-invariant vector field ∂βx is a linear combination of terms of the type c(x)Dx,

where c ∈ C∞(G) and Dx is right-invariant. By the previous considerations on B,we may remove c(x) here, and consider only C = Op(DxσA). Since RA(x, y) =KA(x, xy−1), we get

DxRA(x, y) = (Dx + Dz)KA(x, z)|z=xy−1 ,

leading to

Cf(x) =∫

G

f(xy−1) DxRA(x, y) dy =∫

G

f(y) (Dx + Dy)KA(x, y) dy.

Thus, we study local operators of the form

Cf(x) =∫

Rn

f(y)(φ(x, y)∂β

x + ψ(x, y)∂βy

)KA(x, y) dy,

where the kernel of A ∈ Ψm(Rn × Rn) has compact support, φ, ψ ∈ C∞(Rn×Rn),and φ(x, x) = ψ(x, x) for every x ∈ Rn. Let C = D + E, where

Df(x) =∫

Rn

f(y) φ(x, y)(∂β

x + ∂βy

)KA(x, y) dy,

Ef(x) =∫

Rn

f(y) (ψ(x, y)− φ(x, y)) ∂βy KA(x, y) dy.

By the above considerations about B, we may assume that φ(x, y) ≡ 1 here, andobtain σD(x, ξ) = ∂β

xσA(x, ξ). Thus D ∈ Ψm(Rn × Rn). Moreover,

Ef(x) ∼∑γ≥0

1γ!

∂γz (ψ(x, z)− φ(x, z))|z=x

∫Rn

f(y) (y − x)γ ∂βy KA(x, y) dy,

yielding

σE(x, ξ) ∼∑γ≥0

cγ(x) ∂γξ

(ξβ σA(x, ξ)

)for some functions cγ ∈ C∞(Rn) for which c0(x) ≡ 0. Since |β| = 1, this showsthat E ∈ Ψm(Rn × Rn). Thus Op(∂β

xσA) ∈ Ψm(G) if A ∈ Ψm(G). �Lemma 10.9.4. Let A ∈ Ψm(G) and let D : C∞(G)→ C∞(G) be a smooth vectorfield. Then Op(σAσD) ∈ Ψm+1(G) and Op([σA, σD]) ∈ Ψm(G).

Proof. For simplicity, we may assume that D = Mφ∂x, where ∂x is left-invariantand φ ∈ C∞(G). Now

σA(x, ξ) σD(x, ξ) = φ(x) σA(x, ξ) σ∂x(ξ) = σMφA◦∂x

(x, ξ),


and it follows from the local theory that MφA◦∂x ∈ Ψm+1(G). Thus Op(σAσD) ∈Ψm+1(G). Next,

σD(x, ξ) σA(x, ξ) = φ(x) σ∂x(ξ) σA(x, ξ)(10.31)

= φ(x) (σ∂x◦A(x, ξ)− (∂xσA)(x, ξ))= σMφ◦∂x◦A(x, ξ)− φ(x) (∂xσA)(x, ξ).

From this we see that

Op([σA, σD]) = MφA ◦ ∂x −Mφ∂x ◦A + Mφ Op(∂xσA)= Mφ[A, ∂x] + Mφ Op(∂xσA).

Here Op(∂xσA) ∈ Ψm(G) by Lemma 10.9.3. Hence Op([σA, σD]) belongs to Ψm(G)as a sum of operators from Ψm(G). �

10.9.2 Symbol classes Σm(G)

Combined with asymptotic expansion (10.32) for composing operators, commu-tator characterisation Theorem 10.7.7 motivates defining the following symbolclasses

Σm(G) =∞⋂

k=0

Σmk (G)

that we will show to characterise Hormander’s classes Ψm(G). The classes Σmk (G)

are defined iteratively in the following way:

Definition 10.9.5 (Symbol classes Σm(G)). Let m ∈ R. We denote σA ∈ Σm0 (G) if

sing supp (y �→ RA(x, y)) ⊂ {e} (10.36)

and if‖�α

ξ ∂βxσA(x, ξ)‖op ≤ CAαβm 〈ξ〉m−|α|, (10.37)

for all x ∈ G, all multi-indices α, β, and all ξ ∈ Rep(G), where 〈ξ〉 is defined in(10.9). Then we say that σA ∈ Σm

k+1(G) if and only if

σA ∈ Σmk (G), (10.38)

σ∂j σA − σAσ∂j ∈ Σmk (G), (10.39)

(�γξ σA) σ∂j ∈ Σm+1−|γ|

k (G), (10.40)

for all |γ| > 0 and 1 ≤ j ≤ dim(G). Let

Σm(G) :=∞⋂

k=0

Σmk (G).

We write A ∈ Op Σm(G) if and only if σA ∈ Σm(G).


Theorem 10.9.6 (Equality Op Σm(G) = Ψm(G)). Let G be a compact Lie groupand let m ∈ R. Then A ∈ Ψm(G) if and only if σA ∈ Σm(G).

Proof. First, applying Theorem 10.7.8 to σA ∈ Σmk+1(G), we notice that [A,D] ∈

Op Σmk (G) for any smooth vector field D : C∞(G) → C∞(G). Consequently,

if here A ∈ Op Σm(G) then also [A,D] ∈ Op Σm(G). By Remark 10.8.2 weobtain OpΣm(G) ⊂ L(Hm(G), L2(G)). Consequently, Theorem 10.7.7 impliesOp Σm(G) ⊂ Ψm(G).

Conversely, we have to show that Ψm(G) ⊂ Op Σm(G). This follows byLemma 10.9.1, and Lemmas 10.9.3 and 10.9.4. More precisely, let A ∈ Ψm(G).Then we have

Op(�αξ ∂β

xσA) ∈ Ψm−|α|(G),

Op([σ∂j

, σA])∈ Ψm(G),

Op((�γ

ξ σA)σ∂j

)∈ Ψm+1−|γ|(G).

Moreover, ‖σA(x, ξ)‖ ≤ C〈ξ〉m by Lemma 10.9.1, and the singular support y �→RA(x, y) is contained in {e} ⊂ G. This completes the proof. �Corollary 10.9.7. The set Σm(G) is invariant under x-freezings, x-translations andξ-conjugations. More precisely, if (x, ξ) �→ σA(x, ξ) belongs to Σm(G) and u ∈ Gthen also the following symbols belong to Σm(G):

(x, ξ) �→ σA(u, ξ), (10.41)(x, ξ) �→ σA(ux, ξ), (10.42)(x, ξ) �→ σA(xu, ξ), (10.43)(x, ξ) �→ ξ(u)∗ σA(x, ξ) ξ(u). (10.44)

Proof. The symbol classes Σm(G) are defined by conditions (10.36)–(10.40), whichare checked for points x ∈ G fixed (with constants uniform in x). Therefore itfollows that Σm(G) is invariant under the x-freezing (10.41), and under the leftand right x-translations (10.42),(10.43). The x-freezing property (10.41) wouldhave followed also from Lemma 10.9.1 and Theorem 10.9.6. From the general localtheory of pseudo-differential operators it follows that A ∈ Ψm(G) if and only ifthe φ-pullback Aφ belongs to the same class Ψm(G), where Aφf = A(f ◦φ) ◦φ−1.This, combined with the x-translation invariance and Proposition 10.4.18, impliesthe conjugation invariance in (10.44). �

From Theorem 10.9.6 and Lemma 10.9.3 we also obtain:

Corollary 10.9.8. If σA ∈ Σm(G) then �αξ ∂β

xσA ∈ Σm−|α|(G).

Finally, we formulate a simple relation between convergence of symbols andoperators. It is a straightforward consequence of Theorems 10.5.5 and 10.8.1 forL2(G) and Hs(G) cases, respectively. In Corollary 12.4.11 we will give an improve-ment of this result on the group SU(2).


Corollary 10.9.9 (Convergence of symbols and operators). Let σ ∈ Σ0(G), andassume that a sequence σk ∈ Σ0(G) satisfies inequalities (10.37) uniformly in k.Let N ∈ N be such that N > dim(G)/2, and assume that for all |β| ≤ N we havethe convergence

∂βxσk(x, ξ)→ ∂β

xσ(x, ξ) as k →∞ (10.45)

in the operator norm, uniformly over all x ∈ G and ξ ∈ Rep(G). Then Opσk →Opσ strongly on L2(G).

Moreover, if the convergence (10.45) holds for all multi-indices β, thenOpσk → Opσ strongly on Hs(G) for any s ∈ R.

From the asymptotic expansion for the composition of pseudo-differentialoperators in Section 10.7.3, we get an expansion for a parametrix of an ellipticoperator:

Theorem 10.9.10 (Parametrix). Let σAj∈ Σm−j(G), and let

σA(x, ξ) ∼∞∑

j=0

σAj(x, ξ).

Assume that A is elliptic in the sense that σA0(x, ξ) = σB0(x, ξ)−1 is an invertiblematrix for every x and ξ, and that B0 = Op(σB0) ∈ Ψ−m(G). Then there existsσB ∈ Σ−m(G) such that I −BA and I −AB are smoothing operators. Moreover,

σB(x, ξ) ∼∞∑

k=0

σBk(x, ξ),

where the operators Bk ∈ Σ−m−k(G) are defined recursively by

σBN(x, ξ) := −σB0(x, ξ)

N−1∑k=0

N−k∑j=0

∑|γ|=N−j−k

1γ!

[�γ

ξ σBk(x, ξ)

]∂(γ)

x σAj(x, ξ).

(10.46)

Proof. If σI ∼ σBA holds for some σB ∼∑∞

k=0 σBk, then by Theorem 10.7.8 we

have

Idim(ξ) = σI(x, ξ) ∼ σBA(x, ξ)

∼∑γ≥0

1γ!

[�γ

ξ σB(x, ξ)]∂(γ)

x σA(x, ξ)

∼∑γ≥0

1γ!

[�γ

ξ

∞∑k=0

σBk(x, ξ)

]∂(γ)

x

∞∑j=0

σAj(x, ξ).


From this, we want to find σBk. Now Idim(ξ) = σB0(x, ξ) σA0(x, ξ), and for |γ| ≥ 1

we can demand that

0 =∑

|γ|=N−j−k

1γ!

[�γ

ξ σBk(x, ξ)

]∂(γ)

x σAj(x, ξ).

Then (10.46) provides the solution to these equations, and the reader may verifyalso that σBN

∈ Σ−m−N (G). Thus σB ∼ ∑∞k=0 σBk

. Finally, notice that σI ∼σBA. �Exercise 10.9.11. Show that σBN

∈ Σ−m−N (G) in Theorem 10.9.10.

Exercise 10.9.12. In the notation of Theorem 10.9.10, let us write

σC0(x, ξ) := σB0(x, ξ) = σA0(x, ξ)−1,

and let σC ∼∑∞

N=0 σCN, where

σCN(x, ξ) = −σC0(x, ξ)

N−1∑k=0

N−k∑j=0

∑|γ|=N−j−k

1γ!

[�γ

ξ σAj(x, ξ)

]∂(γ)

x σCk(x, ξ).

Check that also C is a parametrix of A.

10.10 Full symbols on compact manifolds

In this section we discuss how the introduced constructions are mapped by globaldiffeomorphisms.

Let Φ : G → M be a diffeomorphism from a compact Lie group G toa smooth manifold M . Such diffeomorphisms can be obtained for large classesof compact manifolds by the Poincare conjecture type results. For example, ifdim G = dim M = 3 it is now known that such Φ exists for any closed simply-connected6 manifold, and we can take G ∼= S3 ∼= SU(2). We will now explain howthe diffeomorphism Φ induces the quantization of operators on M from that onG.

Let us endow M with the natural Lie group structure induced by Φ, i.e., withthe group multiplication ((x, y) �→ x · y) : M ×M →M defined by

x · y := Φ(Φ−1(x) Φ−1(y)

).

The spaces C∞(G) and C∞(M) are isomorphic via mappings

Φ∗ : C∞(G)→ C∞(M), f �→ fΦ = f ◦ Φ−1,

Φ∗ : C∞(M)→ C∞(G), g �→ gΦ−1 = g ◦ Φ,

6For the definition of simply-connectedness see Definition 8.3.18.

10.11. Operator-valued symbols 579

and the Haar integral on M is given by∫M

g dμM ≡∫

M

g dx :=∫

G

g ◦ Φ dμG,

where μG is the Haar measure on G, because∫M

g(x · y) dx =∫

M

g(Φ(Φ−1(x) Φ−1(y))) dx

=∫

G

(g ◦ Φ)(Φ−1(x) Φ−1(y)) d(Φ−1(x))

=∫

G

(g ◦ Φ)(z) dz

=∫

M

g(x) dx.

Moreover, Φ∗ : C∞(G) → C∞(M) extends to a linear unitary bijection Φ∗ :L2(μG)→ L2(μM ):∫

M

g(x) h(x) dx =∫

G

(g ◦ Φ) (h ◦ Φ) dμG.

Notice also that there is an isomorphism

Φ∗ : Rep(G)→ Rep(M), ξ �→ Φ∗(ξ) = ξ ◦ Φ

of irreducible unitary representations. Thus G ∼= M in this sense. This immediatelyimplies that the whole construction of symbols of pseudo-differential operators onM is equivalent to that on G.

In Section 12.5 we will give an example of this identification in the case ofSU(2) ∼= S3.

10.11 Operator-valued symbols

In this section we discuss another notion of a symbol which we call operator-valuedsymbols of operators.

We recall left- and right-convolution operators from Remark 10.4.2, whichwill now play an important role:

Definition 10.11.1 (Convolution operators l(f) and r(f)). For f ∈ D′(G), wedefine the respective left-convolution and right-convolution operators l(f), r(f) :C∞(G)→ C∞(G) by

l(f)g := f ∗ g,

r(f)g := g ∗ f.


Exercise 10.11.2. To check that we indeed have l(f), r(f) : C∞(G)→ C∞(G) onecan use a distributional interpretation similar to the one in Section 1.4.2. Workout the details of this.

Remark 10.11.3. For f ∈ L2(G), in the literature, operator l(f) is sometimes calleda “global” Fourier transform of f . Such terminology can be found in, e.g., W.F.Stinespring [120], where it is denoted by π(f), and for related integration theory ofoperators by I.E. Segal, see [107] and [108]. However, since we are dealing with thequantization of operators on G that have both left- and right-convolution kernels,we want to keep track of both left- and right-convolution operators. At the sametime, it will turn out that the proceeding theory of full symbols is better adaptedto the operator r(f) because our starting point was right-convolution kernels.

As usual, the left and right regular representations of G are denoted byπL, πR : G→ U(L2(G)), respectively, i.e.,

πL(x)f(y) = f(x−1y),πR(x)f(y) = f(yx).

Exercise 10.11.4. Verify that these representations are indeed unitary. For exam-ple, check that πR(x)−1 = πR(x−1) = πR(x)∗, and similarly for πL(x).

Keeping in mind the right and left Peter–Weyl Theorems (Theorem 7.5.14and Remark 7.5.14), the Fourier inversion formulae may be viewed in the followingform:

Proposition 10.11.5 (“Fourier inversion formulae”). Let f ∈ C∞(G). Then wehave

r(f) =∫

G

f(y) πR(y)∗ dy and l(f) =∫

G

f(y) πL(y) dy. (10.47)

Conversely, for every x ∈ G we have

f(x) = Tr (r(f) πR(x)) and f(x) = Tr (l(f) πL(x)∗), (10.48)

where Tr is the trace functional, see Definition B.5.38; notice that Tr(AB) =Tr(BA). These formulae have an almost everywhere extension to L2(μG).

Proof. We will prove the case of right-convolutions since for left-convolutions it issimilar. First, we can write

(r(f)g)(x) = (g ∗ f)(x)

=∫

G

g(xy−1)f(y) dy

=∫

G

f(y)(πR(y−1)g)(x) dy,


from which we obtain (10.47) in view of the unitarity πR(y−1) = πR(y)∗. Theproof of (10.48) is somewhat lengthier if we provide the details. First we observethat (10.47) implies that

r(f)πR(x) =∫

G

f(y) πR(y)∗ πR(x) dy =∫

G

f(y) πR(y−1x) dy. (10.49)

By the Peter–Weyl Theorem (Theorem 7.5.14) whose notation we will use in thisproof, we know that {dim(φ)φij}φ,i,j is an orthonormal basis in L2(μG), so thatby Definition B.5.38 of the trace we have

Tr(r(f) πR(x)) =∑

[φ]∈G

dim(φ)∑i,j=1

dim(φ)(r(f)πR(x)φij , φij) (10.50)

(10.49)=

∑[φ]∈G

dim(φ)∑i,j=1

dim(φ)∫

G

∫G

f(y)φij(zy−1x)φij(z) dy dz

=∑

[φ]∈G

dim(φ)∑i,j=1

dim(φ)∫

G

∫G

f(xw−1)φij(zw)φij(z) dw dz,

where in the last equality we changed the variables w := y−1x. Using φ(zw) =φ(z)φ(w), we obtain

φ(zw)ij =dim(φ)∑

k=1

φik(z)φkj(w) (10.51)

and

φ(y−1x)jj =dim(φ)∑

k=1

φji(y−1)φij(x) =dim(φ)∑

k=1

φij(y)φij(x), (10.52)

to be used later as well. We also notice that∫G

φik(z)φij(z) dz = 〈φik, φij〉L2 =1

dim(φ)δkj (10.53)

in view of Lemma 7.5.12. Plugging (10.51) and (10.53) into (10.50) we get

Tr(r(f) πR(x)) =∑

[φ]∈G

dim(φ)∑i,j=1

∫G

f(xw−1) φjj(w) dw

=∑

[φ]∈G

dim(φ)∑j=1

dim(φ)∫

G

f(xw−1) φjj(w) dw


y:=xw−1

=∑

[φ]∈G

dim(φ)∑j=1

dim(φ)∫

G

f(y) φjj(y−1x) dy

(10.52)=

∑[φ]∈G

dim(φ)∑j=1

dim(φ) 〈f, φij〉L2 φij(x)

= f(x), (10.54)

with the last equality in view of Corollary 7.6.7. �

Corollary 10.11.6 (Character description). For every f ∈ L2(G) we have

f =∑

[φ]∈G

dim(φ) f ∗ χφ,

where χφ is the character of φ. Thus, the projection of f ∈ L2(G) to Hφ is givenby f �→ f ∗ χφ.

Proof. Writing the expression in the line of (10.54) as a trace, we see that

∑[φ]∈G

dim(φ)∑j=1

dim(φ)∫

G

f(xw−1)φjj(w) dw

=∑

[φ]∈G

dim(φ)(f ∗ Trφ)(x)

=∑

[φ]∈G

dim(φ)(f ∗ χφ)(x).

Consequently (10.54) implies the statement. �

Exercise 10.11.7. Provide details for the proof of Proposition 10.11.5 in the left-convolution case.

Definition 10.11.8 (Right and left kernels RA(x), LA(x)). Let A : C∞(G) →C∞(G) be a linear continuous operator, and let RA(x, y), LA(x, y) ∈ D(G)⊗D′(G)be its right-convolution and left-convolution kernels, respectively7. For every x ∈ Gwe define RA(x), LA(x) ∈ D′(G) by

[RA(x)](y) := RA(x, y) and [LA(x)](y) := LA(x, y).

In this notation we can write

(Af)(x) = [f ∗RA(x)](x) = [r(RA(x))f ](x), (10.55)

7As defined in (10.16) in Section 10.4, and shown in Lemma 10.12.5 in Section 10.12.


and similarly(Af)(x) = [LA(x) ∗ f ](x) = [l(LA(x))f ](x). (10.56)

This motivates the following definition.

Definition 10.11.9 (Operator-valued symbols). Let A : C∞(G) → C∞(G) be alinear continuous operator. We define its right operator-valued symbol rA : G →L(C∞(G)) by

x �→ rA(x) := r(RA(x)).

Similarly, we define its left operator-valued symbol lA : G �→ L(C∞(G)) by

x �→ lA(x) := l(LA(x)).

We observe that (10.55) and (10.56) imply the equality

[rA(x)f ](x) = Af(x) = [lA(x)]f(x). (10.57)

Lemma 10.11.10. For every f ∈ C∞(G) and every x ∈ G we have

r(rA(x)f) = rA(x)r(f) and l(lA(x)f) = lA(x)l(f).

Proof. We prove only the “right” case since the “left” one is similar. We have asimple calculation

r(rA(x)f)g = g ∗ (rA(x)f)= g ∗ (f ∗RA(x))

Prop. 7.7.3= (g ∗ f) ∗RA(x)= r(RA(x))(g ∗ f)= rA(x)r(f)g,

completing the proof. �Exercise 10.11.11. Prove the left cases of Lemma 10.11.10 as well as of the followingtheorem:

Theorem 10.11.12 (Operator-valued quantization). Let A : C∞(G) → C∞(G) bea linear continuous operator. Then we have

Af(x) = Tr(rA(x) r(f) πR(x))= Tr(lA(x) l(f) πL(x)∗), (10.58)

for all f ∈ C∞(G) and x ∈ G.

Proof. We can write

Af(x)(10.57)

= (rA(x)f)(x)(10.48)

= Tr (r[rA(x)f ] πR(x))Lemma 10.11.10= Tr (rA(x) r(f) πR(x)),

completing the proof for the right case. The left one is similar. �


Remark 10.11.13 (Symbol as a family of convolution operators). In view of The-orem 10.11.12 the operator-valued symbols lA and rA can be regarded as a familyof convolution operators obtained from A by “freezing” it at points x ∈ G. Wenote that if A ∈ L(D(G)) is a left-invariant operator, i.e., AπL(x) = πL(x)Afor every x ∈ G, then its right operator-valued symbol is the constant mappingx �→ rA(x) ≡ A (A is a right-convolution operator).

Remark 10.11.14 (Operator-valued symbols and operators). The quantizationA �→ rA, lA is an injective linear mapping. Conversely, starting from an operator-valued symbol we can define the corresponding operator. Indeed, let ρ : G →L(D′(G)) be a mapping such that ρ(x) = r(s(x)), for some s ∈ D(G) ⊗ D′(G).Then we can define Op(ρ) ∈ L(D(G)) by

(Op(ρ)f)(x) = (ρ(x)f)(x),

for which s is the right-convolution kernel and ρ is the right operator-valued sym-bol.

Corollary 10.11.15 (Decomposition in matrix blocks). According to the Peter–Weyl Theorem, all operators in (10.58) have direct sum decompositions with corre-sponding finite-dimensional square matrix blocks rA(x, ξ) := rA(x)|Hξ , lA(x, ξ) :=lA(x)|Hξ , πR(x, ξ) := πR(x)|Hξ , πL(x, ξ) := πL(x)|Hξ , where ξ ∈ Rep(G) is arepresentation ξ : G→ U(Hξ). In this notation (10.58) implies

Af(x) =∑ξ∈G

Tr (πR(x, ξ) rA(x, ξ) r(f)|Hξ) (10.59)

=∑ξ∈G

Tr (πL(x, ξ)∗ lA(x, ξ) l(f)|Hξ) .

The meaning of operators r(f)|Hξ and l(f)|Hξ can be clarified as follow: if ξ ∈Rep(G), then

l(u)ξ(x) = u(ξ)ξ(x) and r(u)ξ(x) = ξ(x)u(ξ).

Let us show the last formula for l(u). Indeed, we have

l(u)ξ(x) =∫

G

u(y)ξ(y−1x) dy =(∫

G

u(y)ξ(y)∗ dy

)ξ(x) = u(ξ)ξ(x),

and the calculation for r(u) is similar, where we can commute ξ(x) and the integralsince ξ(x) is finite-dimensional.

We now establish a relation between different quantizations. Namely, forevery x ∈ G the mapping rA(x) : C∞(G) → C∞(G) from Definition 10.11.9 islinear and continuous, so that we can find its symbol σrA(x)(y, ξ) according toDefinition 10.4.3.


Theorem 10.11.16 (Relation between quantizations). Let A : C∞(G) → C∞(G)be a linear continuous operator and let rA, lA : G → L(C∞(G)) be its right andleft operator-valued symbols, respectively. Then for all x, y ∈ G and ξ ∈ Rep(G)we have

σrA(x)(y, ξ) = σA(x, ξ) (10.60)

andσlA(x)(y, ξ) = ξ(y)∗ξ(x)σA(x, ξ)ξ(x)∗ξ(y). (10.61)

The operator-valued symbols take the form

[rA(x)f ](y) =∑

[ξ]∈G

dim(ξ) Tr(ξ(y) σA(x, ξ) f(ξ)

)(10.62)

and

[lA(x)f ](y) =∑

[ξ]∈G

dim(ξ) Tr(ξ(x) σA(x, ξ) ξ(x)∗ ξ(y) f(ξ)

). (10.63)

Finally, we also have

Af(x) = Tr (Op(σB) r(f) πR(x)), (10.64)

where σB(y, ξ) := σA(x, ξ).

Remark 10.11.17. Formulae (10.62) and (10.63) when evaluated at y = x giveus Af(x), thus recovering equality (10.57). Formula (10.60) shows that the oper-ator rA(x) when quantized in terms of symbols from Definition 10.4.3 becomesa multiplier, with its symbol independent of y. Formula (10.61) is slightly morecomplicated as a consequence of the fact that the symbols from Definition 10.4.3are better adapted to right-convolutions, see Remark 10.4.2. In the “left” caseadditional conjugations in (10.61) appear in view of Proposition 10.4.11.

Proof of Theorem 10.11.16. We start with the right operator-valued symbol rA(x).By Theorem 10.4.6 its symbol can be found as

σrA(x)(y, ξ) = ξ(y)∗[rA(x)ξ](y).

Thus, we calculate

[rA(x)ξ](y) = [r(RA(x))ξ](y)= [ξ ∗RA(x)](y)

=∫

G

ξ(yz−1)RA(x)(z)dz

= ξ(y)∫

G

ξ(z)∗RA(x, z)dz

= ξ(y)σA(x, ξ),


yielding (10.60). Here, as usual, we used the finite dimensionality of representationsto commute ξ(y) with the integral. This implies (10.62) by Theorem 10.4.4:

[rA(x)f ](y) =∑

[ξ]∈G

dim(ξ) Tr(ξ(y) σrA(x)(y, ξ) f(ξ)

)=

∑[ξ]∈G

dim(ξ) Tr(ξ(y) σA(x, ξ) f(ξ)

).

Moreover, (10.64) follows from (10.60) by Theorem 10.11.12, where B = rA(x).Similarly, in view of

σlA(x)(y, ξ) = ξ(y)∗[lA(x)ξ](y)

we calculate

[lA(x)ξ](y) = [l(LA(x))ξ](y)= [LA(x) ∗ ξ](y)

=∫

G

LA(x)(yz−1)ξ(z)dz

=∫

G

LA(x, yz−1)ξ(z)dz.

Using formula (10.17), i.e., RA(x, w) = LA(x, xwx−1) with w = x−1yz−1x, givesz = xw−1x−1y and

LA(x, yz−1) = RA(x, x−1yz−1x).

Continuing the calculation for [lA(x)ξ](y) above, we get

[lA(x)ξ](y) =∫

G

LA(x, yz−1)ξ(z)dz

=∫

G

RA(x, x−1yz−1x)ξ(z)dz

=∫

G

RA(x, w)ξ(xw−1x−1y)dz

= ξ(x)(∫

G

RA(x,w)ξ(w)∗ dw

)ξ(x−1)ξ(y)

= ξ(x)σA(x, ξ)ξ(x)∗ξ(y),

yielding (10.61). This implies (10.63) by Theorem 10.4.4. �

In Theorem 10.5.5 we presented conditions on the L2(G)-boundedness ofoperators in terms of symbols σ(x, ξ). We now briefly discuss the same questionin terms of the operator-valued symbols. We will do this in terms of the rightoperator-valued symbols because the left case is the same, see Exercise 10.11.21.


Definition 10.11.18 (Derivations of operator-valued symbols). Let operator A :C∞(G) → C∞(G) be linear and continuous, with right operator-valued symbolrA : G→ L(C∞(G)). Let p(D) ∈ L(C∞(G)) be a partial differential operator. Foreach x ∈ G we define

(p(D)rA)(x) = p(D)rA(x) := r(RB(x)),

where RB(x)(y) = RB(x, y) and

RB = (p(D)⊗ id)RA.

Operator B defined by Bf(x) = [r(RB(x))f ](x) is then also a linear continuousoperator from C∞(G)) to C∞(G)) because p(D) ∈ L(C∞(G)), id ∈ L(D′(G)).

Theorem 10.11.19 (Boundedness on L2(G)). Let G be a compact Lie group ofdimension n. Let A be a linear continuous operator from C∞(G) to C∞(G) andassume that its right operator-valued symbol rA satisfies rA ∈ Ck(G,L(L2(G)))with k > n/2. Then A extends to a bounded linear operator from L2(G) to L2(G).

Proof. By (10.57) we have

‖Af‖2L2(G) =∫

G

|(rA(x)f)(x)|2 dx

≤∫

G

supy∈G

|(rA(y)f)(x)|2 dx,


supy∈G

|(rA(y)f)(x)|2 ≤ Ck

∑|α|≤k

∫G

|((∂αy rA)(y)f)(x)|2 dy.

Therefore using the Fubini theorem to change the order of integration, we obtain

‖Af‖2L2(G) ≤ C∑|α|≤k

∫G

∫G

|((∂αy rA)(y)f)(x)|2 dx dy

≤ C∑|α|≤k

supy∈G

∫G

|((∂αrA)(y)f)(x)|2 dx

= C∑|α|≤k

supy∈G

‖(∂αrA)(y)f‖2L2(G)

≤ C∑|α|≤k

supy∈G

‖(∂αrA)(y)‖2L(L2(G))‖f‖2L2(G).

The proof is complete, because G is compact and rA ∈ Ck(G,L(L2(G))). �


Theorem 10.11.19 yields the following Sobolev boundedness result:

Corollary 10.11.20. Let G be a compact Lie group and let A be a linear continuousoperator from C∞(G) to C∞(G) such that its right operator-valued symbol rA

satisfies rA ∈ C∞(G,L(Hm(G),H0(G))), m ∈ R. Then A extends to a boundedlinear operator from Hm(G) to H0(G).

Exercise 10.11.21. Show that we can replace rA by the left operator-valued symbollA in Theorem 10.11.19 and Corollary 10.11.20 for them to remain true. Work outdetails of the proofs.

Remark 10.11.22 (M.E. Taylor’s characterisation). Let g be the Lie algebra of acompact Lie group G, and let n = dim(G) = dim(g). By the exponential mappingexp : g → G, a neighbourhood of the neutral element e ∈ G can be identifiedwith a neighbourhood of 0 ∈ g. Let Xm = Sm

1# ⊂ Sm1,0(R

n × Rn) consist of thex-invariant symbols (x, ξ) �→ p(ξ) in Sm

1,0(Rn × Rn) with the usual Frechet space

topology (given by the optimal constants in symbol estimates as seminorms). Adistribution k ∈ D′(G) with a sufficiently small support is said to belong to spaceXm if

sing supp(k) ⊂ {e} and k ∈ Xm ⊂ C∞(g′),

where the Fourier transform k is the usual Fourier transform on g ∼= Rn, and thedual space satisfies g′ ∼= Rn (and we are using the exponential coordinates for k(y)when y ≈ e ∈ G). If k ∈ Xm then the convolution operator

u �→ k ∗ u, k ∗ u(x) =∫

G

k(xy−1) u(y) dy,

is said to belong to space OPXm, which is endowed with the natural Frechet spacestructure obtained from Xm. Formally, let k(x, y) = kx(y) be the left-convolutionkernel of a linear operator K : C∞(G)→ C∞(G), i.e.,

Ku(x) =∫

G

kx(xy−1) u(y) dy.

In [128], M.E. Taylor showed that K ∈ Ψm(G) if and only if the mapping

(x �→ (u �→ kx ∗ u)) : G→ OPXm

is smooth; here naturally u �→ kx ∗ u must belong to OPXm for each x ∈ G.This approach was pursued in [128] as well as in [136, 139] in the setting of theleft-convolution kernels, and relied on the exponential mapping and on the pseudo-differential operator on the Lie algebra (see Remark 10.11.22). In this approachmany arguments may be restricted to a suitable neighbourhood of the identityelement. However, the notion of the symbol σA from Definition 10.4.3 appearsto be more practical as it allows a finite-dimension realisation of the symbol andworks globally on the group.


Remark 10.11.23 (Operator-valued calculus). It is possible to construct the calcu-lus of operator-valued symbols, including compositions, adjoints, transposes, andthe inverse. Exponential coordinates of the Lie algebra versions of it can be foundin [128, 139]. However, using our calculus in Section 10.7 together with Corollary10.11.15 one can obtain this directly on the group without referring to the expo-nential coordinates in the neighbourhood of the origin. We leave this as an exercisefor an interested reader.

10.11.1 Example on the torus Tn

We will show here how the construction of this section applies for the operator-valued quantization of operator on the torus Tn. The starting point for this ex-ample is the operator-valued quantization formula (10.58) in Theorem 10.11.12:

Af(x) = Tr(rA(x) r(f) πR(x))= Tr(lA(x) l(f) πL(x)∗), (10.65)

where G is the unitary dual of G, πL : G → L(L2(G)) is the left regular repre-sentation of G on L2(G), rA, lA : G → L(D(G)) are the right and left operator-valued symbol of A, and r(f) and l(f) are the right and left convolution operators.According to the Peter–Weyl Theorem, these operators also have direct sum de-compositions with corresponding finite-dimensional square matrix blocks whichappear in (10.59).

Let us inspect and summarise what this means in the special case of then-torus

G = Tn.

The 1-torus T = R/Z can be identified with the multiplicative group of the unitcircle in the plane. Let U(1) be the group of 1× 1 unitary matrices; U(1) can beidentified with the unit circle, again. For each ξ ∈ Zn, define eξ : Tn → U(1) by

eξ(x) = ei2πx·ξ;

up to isomorphism, an irreducible unitary representation of G = Tn is some eξ.Thus G can be identified with the integer lattice Zn, and {eξ : ξ ∈ Zn} is anorthonormal basis for L2(G), where the Haar measure on G = Tn is obtainedfrom the Lebesgue measure of Rn. Let us analyse this in terms of the left regularrepresentation πL : G→ L(L2(G)) which is now

(πL(x)f)(y) = f(y − x) (for almost every y ∈ Tn).

Especially, we notice

(πL(x)∗eξ)(y) = (πL(−x)eξ)(y) = eξ(y + x) = ei2πx·ξ eξ(y). (10.66)


Let u ∈ C∞(G). Then l(u) ∈ L(L2(G)) is the convolution operator given by

(l(u)f)(x) = (u ∗ f)(x) =∫

G

u(x− y) f(y) dy =∫

G

u(y) f(x− y) dy.

Especially, we havel(u)eξ = u(ξ) eξ, (10.67)

where u(ξ) ∈ C is the usual Fourier coefficient of u : Tn → C. Let A be a continuouslinear operator from C∞(G) to C∞(G). Recall that here G = Tn and G = Zn; letus define σA : G× G→ C by

σA(x, ξ) = e−i2πx·ξ (σAeξ)(x),

which is the toroidal symbol of A as well as the 1× 1 matrix symbol, see Remark10.4.7. The left operator-valued symbol of A is the convolution operator-valuedmapping lA : G→ L(C∞(G)) that by (10.57) satisfies

(lA(x)u)(x) = (Au)(x).

Thereby and by Theorem 10.11.16 we have

(lA(x)u)(y) =∑ξ∈Zn

eξ(y) σA(x, ξ) u(ξ). (10.68)

In particular, (10.66) and (10.68) imply that

πL(x)∗lA(x)eξ = πL(x)∗eξσA(x, ξ)eξ(ξ) = ei2πx·ξσA(x, ξ)eξ. (10.69)

The trace of a linear operator B : L2(G)→ L2(G) is

Tr(B) =∑ξ∈Zn

〈Beξ, eξ〉L2(G),

where 〈f, g〉L2(G) =∫

Gf(y) g(y) dy is the inner product of L2(G), see Definition

B.5.38. Let us now explore formula (10.65) in the case of G = Tn:

Tr (πL(x)∗ lA(x) l(u)) =∑ξ∈Zn

〈πL(x)∗ lA(x) l(u) eξ, eξ〉L2(G)

(10.67)=

∑ξ∈Zn

〈πL(x)∗ lA(x) u(ξ) eξ, eξ〉L2(G)

(10.69)=

∑ξ∈Zn

〈 ei2πx·ξ σA(x, ξ) u(ξ) eξ, eξ〉L2(G)

=∑ξ∈Zn

ei2πx·ξ σA(x, ξ) u(ξ) 〈eξ, eξ〉L2(G)

=∑ξ∈Zn

ei2πx·ξ σA(x, ξ) u(ξ)

= (Au)(x),

10.12. Appendix: integral kernels 591

where the last equality is the toroidal quantization of the operator A, given inTheorem 4.1.3. This is the case on the commutative group G = Tn, where spacesHξ = span(eξ) are all one-dimensional. On non-commutative compact Lie groups,even using the quantization (10.59), πL(x, ξ), σA(x, ξ) and l(u)|Hξ are no longernumbers (more precisely: no longer 1× 1 matrices), but they are dim(ξ)× dim(ξ)matrices, where the dimension dim(ξ) ∈ Z+ depends on the corresponding repre-sentation ξ ∈ G and is usually greater than 1.

Exercise 10.11.24. Work out the above example for the “right” representation πR

and right operator-valued symbols rA. In particular show that on the torus “left”and “right” coincide.

10.12 Appendix: integral kernels

Here we provide a short appendix with more technical explanations about integralkernels of operators on the compact Lie group G.

Definition 10.12.1 (Duality betweenD(G) andD′(G)). LetD(G) be the set C∞(G)equipped with the usual Frechet space topology defined by seminorms pα(f) =maxx∈G |∂αf(x)|, with ∂α as in Definition 10.5.3. Thus, the convergence on D(G)is just the uniform convergence of functions and all their derivatives: fk → f inC∞(G) (or in D(G)) if ∂αfk(x) → ∂αf(x) for all x ∈ G, due to the compactnessof G.

Let D′(G) = L(D(G), C) be its dual, i.e., the set of distributions with D(G)as the test function space. We equip the space of distributions with the weak∗-topology. The duality D′(G)×D(G)→ C is denoted by

〈f, φ〉 := f(φ),

where φ ∈ D(G) and f ∈ D′(G), and an embedding D(G) ↪→ D′(G), ψ �→ fψ = ψis given by

〈ψ, φ〉 =∫

G

ψ(x) φ(x) dy

(in the same way as on Rn in Remark 1.3.7).

Definition 10.12.2 (Transpose and adjoint). The transpose of A ∈ L(D(G)) isAt ∈ L(D′(G)) defined by the equality

〈Atf, φ〉 = 〈f,Aφ〉,

and the adjoint A∗ ∈ L(D′(G)) is given by the equality

(A∗f, φ) = (f,Aφ),

where (f, φ) = 〈f, φ〉, φ(x) := φ(x) is the complex conjugate.


Exercise 10.12.3. For f ∈ D′(G) and for the left-convolution operator l(f)φ :=f ∗ φ show that l(f)∗ = l(f) where f(x) = f(x−1), and l(f)t = l(f) wheref(x) = f(x−1).

Remark 10.12.4 (Schwartz kernel). Let D(G) ⊗D′(G) denote the complete locallyconvex tensor product of the nuclear spaces D(G) and D′(G) (see Section B.3.1).Then K ∈ D(G) ⊗D′(G) defines a linear operator A ∈ L(D(G)) by

〈f,Aφ〉 := 〈K, φ⊗ f〉. (10.70)

In fact, the Schwartz Kernel Theorem states that L(D(G)) and D(G) ⊗D′(G) areisomorphic: for every A ∈ L(D(G)) there exists a unique KA ∈ D(G) ⊗D′(G) suchthat the duality (10.70) is satisfied with K = KA, which is called the Schwartzkernel of A. Duality (10.70) gives us also the interpretation for

(Aφ)(x) =∫

G

KA(x, y) φ(y) dy.

For a more general Schwartz kernel theorem see Theorem B.3.55 and DefinitionB.3.56.

Lemma 10.12.5. Let A ∈ L(D(G)), and let

LA(x, y) := KA(x, y−1x)

in the sense of distributions. Then LA ∈ D(G) ⊗D′(G).

Proof. Notice that D(G) ⊗D(G) ∼= D(G×G). Let us define the multiplication

m : D(G)⊗D(G)→ D(G), m(f ⊗ g)(x) := f(x)g(x),

the co-multiplication

Δ : D(G)→ D(G) ⊗D(G), (Δf)(x, y) := f(xy),

and the antipode

S : D(G)→ D(G), (Sf)(x) := f(x−1).

These mappings are a part of the (nuclear Frechet) Hopf algebra structure of D(G),see Chapter 9 (as well as, e.g., [1] or [126]). The mappings are all continuous andlinear.

The convolution of operators A,B ∈ L(D(G)) is said to be the operator

A ∗B := m(A⊗B)Δ ∈ L(D(G));

it is easy to calculate the Schwartz kernel

KA∗B(x, y) =∫

G

KA(x, yz−1) KB(x, z) dz,

10.12. Appendix: integral kernels 593

orKA∗B = (m⊗Δt)(id⊗ τ ⊗ id)(KA ⊗KB),

whereτ : D′(G) ⊗D(G)→ D(G) ⊗D′(G), τ(f ⊗ φ) := φ⊗ f,

andΔt : D′(G) ⊗D′(G)→ D′(G)

is the transpose of the co-multiplication Δ, Δt(f ⊗ g) = f ∗ g (i.e., Δt extendsthe convolution of distributions). Now (A ∗ S)S ∈ L(D(G)), hence K(A∗S)S ∈D(G) ⊗D′(G) by the Schwartz kernel theorem, and

K(A∗S)S(x, y) = KA∗S(x, y−1)

=∫

G

KA(x, y−1z−1) KS(x, z) dz

= KA(x, y−1x)= LA(x, y). �

Remark 10.12.6. Any distribution s ∈ D(G) ⊗D′(G) can be considered as a map-ping

s : G→ D′(G), x �→ s(x),

where s(x)(y) := s(x, y). If D ∈ L(D(G)) and M ∈ L(D′(G)) then (D ⊗M)s ∈D(G) ⊗D′(G). For instance, D could be a partial differential operator, and M amultiplication.

Exercise 10.12.7. Prove Lemma 10.12.5 for the right-convolution kernel RA(x, y).

Chapter 11

Fourier Analysis on SU(2)

In this chapter we develop elements of Fourier analysis on the group SU(2) in aform suitable for the consequent development of the theory of pseudo-differentialoperators on SU(2) in Chapter 12. Certain results from this chapter can be found in[148], which, together with [154] we can recommend for further reading, includingfor some instances of explicitly calculated Clebsch–Gordan coefficients. However,on this occasion, with pseudo-differential operators in mind and the form of theanalysis necessary for us and adopted to Chapter 10, we present an independentexposition of SU(2) with considerably more direct proofs and different argumentscompared to, e.g., [148].

11.1 Preliminaries: groups U(1), SO(2), and SO(3)

First, we discuss a simpler model of commutative groups U(1) ∼= SO(2), to treatrotation group SO(3) in a similar manner later. Following that, we study thespecial unitary group SU(2). For the definitions of SO(n) and SU(n) we refer toRemarks 6.2.9 and 6.2.10, respectively. We start by discussing Lie algebras of U(1),SO(2), and SO(3).

Definition 11.1.1 (Group U(1)). Let g be the real vector space of 1-by-1-matrices

X = X(x) =(i2πx

), (x ∈ R).

Of course, one may treat X(x) as a pure imaginary number, but we insist on thematrix interpretation to anticipate future developments. Then

exp(X(x)) =(ei2πx

)is a matrix belonging to the Lie group G = U(1), the unitary matrix group ofdimension 1. Of course, U(1) ∼= {z ∈ C : |z| = 1} ∼= T1 = R/Z.

596 Chapter 11. Fourier Analysis on SU(2)

Definition 11.1.2 (Group SO(2)). Let g be the real vector space of matrices

X = X(x) =(

0 −xx 0

), (x ∈ R).

Now

exp(X(x)) =(

cos(x) − sin(x)sin(x) cos(x)

)is a matrix belonging to the Lie group G = SO(2). If |x| < π/2 and g = (gij)2i,j=1 =exp(X(x)) then

x = arcsin(g21) = arcsin ((g21 − g12)/2) .

Definition 11.1.3 (Group SO(3)). Let the real vector space g consist of matricesof the form

X(x) =

⎛⎝ 0 −x3 x2

x3 0 −x1

−x2 x1 0

⎞⎠ ,

where x = (xj)3j=1 ∈ R3. Thus x �→ X(x) identifies R3 with g, and we equip g

with the Euclidean norm ‖X(x)‖g := ‖x‖R3 =√

x21 + x2

2 + x23. Actually, g is a Lie

algebra with the Lie product (A,B) �→ [A,B] := AB −BA, since

[X(x), X(y)] = X(

⎛⎝x2y3 − x3y2

x3y1 − x1y3

x1y2 − x2y1

⎞⎠).

It turns out that g is the Lie algebra of the group G = SO(3). Let

X1 := X((1, 0, 0)), X2 := X((0, 1, 0)), X3 = X((0, 0, 1)).

Then[X1, X2] = X3, [X2, X3] = X1 and [X3, X1] = X2.

If x ∈ R3 and t := ‖x‖R3 then we have the Rodrigues representation formula

exp(X(x)) = I + X(x)sin(t)

t+ X(x)2

1− cos(t)t2

which is equal to⎛⎝ 1 + (x21 − t2) 1−cos t

t2 −x3sin t

t + x1x21−cos t

t2 x2sin t

t + x1x31−cos t

t2

x3sin t

t + x1x21−cos t

t2 1 + (x22 − t2) 1−cos t

t2 −x1sin t

t + x2x31−cos t

t2

−x2sin t

t + x1x31−cos t

t2 x1sin t

t + x2x31−cos t

t2 1 + (x23 − t2) 1−cos t

t2

⎞⎠ .

If here ‖x‖R3 < π and g = (gij)3i,j=1 = exp(X(x)), we obtain the formula

x = (xj)3j=1 =t

2 sin(t)

⎛⎝g32 − g23

g13 − g31

g21 − g12

⎞⎠ ,

where sin(t) =√

1− cos2(t) with cos(t) = (g11 + g22 + g33 − 1)/2.

11.1. Preliminaries: groups U(1), SO(2), and SO(3) 597

11.1.1 Euler angles on SO(3)

Euler angles are useful as local coordinates on SO(3). First we note that rotationsof R3 by an angle t ∈ R around the xj-axis, j = 1, 2, 3, respectively, are expressedby the matrices ωj(t) = exp(tXj) given by⎛⎝1 0 0

0 cos t − sin t0 sin t cos t

⎞⎠ ,

⎛⎝ cos t 0 sin t0 1 0

− sin t 0 cos t

⎞⎠ ,

⎛⎝cos t − sin t 0sin t cos t 00 0 1

⎞⎠ .

We represent rotations by Euler angles φ, θ, ψ ∈ R. Any g ∈ SO(3) is of the form

g = ω(φ, θ, ψ) := ω3(φ) ω2(θ) ω3(ψ),

where −π < φ,ψ ≤ π and 0 ≤ θ ≤ π. If 0 < θ1, θ2 < π then ω(φ1, θ1, ψ1) =ω(φ2, θ2, ψ2) if and only if θ1 = θ2 and φ1 ≡ φ2 (mod 2π) and ψ1 ≡ ψ2 (mod 2π);thus we conclude that the Euler angles provide local coordinates for the manifoldSO(3) nearby a point ω(φ, θ, ψ) whenever θ ≡ 0 (mod π).

Let g = ω(φ, θ, ψ) be the Euler angle representation of g ∈ SO(3), where−π < φ,ψ ≤ π, 0 ≤ θ ≤ π, so that ω(φ, θ, ψ) is⎛⎝cos φ cos θ cos ψ − sin φ sin ψ − cos φ cos θ sin ψ − sin φ cos ψ cos φ sin θ

sinφ cos θ cos ψ + cos φ sin ψ − sin φ cos θ sinψ + cos φ cos ψ sinφ sin θ− sin θ cos ψ sin θ sin ψ cos θ

⎞⎠ .

The group SO(3) acts transitively on the space S2, as ω(φ, θ, ψ) moves the northpole e3 = (0, 0, 1)T ∈ S2 to the point

ω(φ, θ, ψ)e3 =

⎛⎝cos φ sin θsinφ sin θ

cos θ

⎞⎠ .

If 0 < θ < π and −π/2 < φ, ψ < π/2, then the Euler angles and the exponentialcoordinates are related by

ω(φ, θ, ψ) = exp(X(x)),

where x ∈ R3, 0 < t := ‖x‖R3 < π,

cos t = (cos(φ + ψ)(1 + cos θ) + cos θ − 1)/2,

and

x =t

2 sin t

⎛⎝ sin θ(sinψ − sinφ)sin θ(cos ψ + cos φ)

(1 + cos θ) sin(φ + ψ)

⎞⎠ .


11.1.2 Partial derivatives on SO(3)

For an element g ∈ SO(3) we now introduce the notation for its components.Thus, if g denotes elements of the group SO(3), we can view it as the identitymapping from SO(3) to SO(3). In its matrix components we can write this asg = (gij)3i,j=1 : SO(3) → SO(3). In particular, if g ∈ SO(3), we can denote itsmatrix components by gij , 1 ≤ i, j ≤ 3, so that we obtain functions (g �→ gij) :SO(3) → R which are smooth on G: gij ∈ C∞(G); with this identification andnotation, g = (gij)3i,j=1 : SO(3)→ SO(3) is the identity mapping. Define

∂1 = ∂(1,0,0), ∂2 = ∂(0,1,0), ∂3 = ∂(0,0,1).

Notice that(∂kf)(g) =

ddt

f(ωk(t)g)|t=0.

In particular, we have∂kg = ω′k(0)g.

Hereω′1(0) = X1, ω′2(0) = X2, , ω′3(0) = X3,

so that ∂1g, ∂2g and ∂3g are respectively⎛⎝ 0 0 0−g31 −g32 −g33

g21 g22 g23

⎞⎠ ,

⎛⎝ g31 g32 g33

0 0 0−g11 −g12 −g13

⎞⎠ ,

⎛⎝−g21 −g22 −g23

g11 g12 g13

0 0 0

⎞⎠ .

11.1.3 Invariant integration on SO(3)

We recall that on a compact group G there exists a unique translation-invariantregular Borel probability measure, called the Haar measure μG (see Remark 7.4.4);customarily L2(G) refers to L2(G, μG). Integrations on G are (unless otherwisementioned) with respect to μG, so we may write∫

G

f(x) dx

instead of∫

Gf dμG =

∫G

f(x) dμG(x).Using the Euler angle coordinates on G = SO(3), we define an orthogonal

projection PS2 ∈ L(L2(SO(3))) by

(PS2f)(ω(φ, θ, ψ)) =12π

∫ π

−π

f(ω(φ, θ, ψ)) dψ

for almost all g = ω(φ, θ, ψ). With the natural interpretation we have PS2f ∈L2(S2), and if f ∈ C∞(SO(3)) then PS2f ∈ C∞(S2). Thereby∫

SO(3)

f(x) dx =∫

S2PS2f dσ,

11.2. General properties of SU(2) 599

where the measure dσ on the sphere is the normalised angular part of the Lebesguemeasure of R3. This yields the Haar integral on SO(3):

f �→∫

SO(3)

f(x) dx =1

8π2

∫ π

−π

∫ π

0

∫ π

−π

f(ω(φ, θ, ψ)) sin(θ) dφ dθ dψ. (11.1)

Exercise 11.1.4. The reader can check rigorously now that (11.1) indeed definesthe Haar integral on SO(3).

11.2 General properties of SU(2)

From now on, we shall study the group of two-dimensional special unitary matrices,denoted by SU(2). In other words, we study

SU(2) ={u ∈ C2×2 : det(u) = 1 and u∗u = I

},

where I =(

1 00 1

)∈ C2×2 is the identity matrix of dimension 2. Indeed, SU(2) is

a matrix group, since for u, v ∈ SU(2), we have

det(uv) = det(u) det(v) = 1,

(uv)∗(uv) = v∗u∗uv = I,

det(u∗) = det(u) = 1,(u∗)∗u∗ = (uu∗)∗ = I∗ = I,

u−1 = u∗ ∈ SU(2).

Lemma 11.2.1 (Elements of SU(2)). The matrix u ∈ C2×2 belongs to SU(2) if andonly if it is of the form

u =(

α β

−β α

),

where α, β ∈ C are such that |α|2 + |β|2 = 1. Moreover, SU(2) is a compact group.

Proof. If u ∈ C2×2 is as above, we have det(u) = |α|2 + |β|2 = 1 and

u∗u =(

α −β

β α

) (α β

−β α

)=

(|α|2 + |β|2 0

0 |β|2 + |α|2)

= I

so that u ∈ SU(2). Now suppose

u =(

u11 u12

u21 u22

)∈ C2×2

belongs to SU(2). Then u−1 = u∗ and 1 = det(u) = u11u22 − u12u21. Specifically,

u−1 =1

det(u)

(u22 −u12

−u21 u11

)=

(u22 −u12

−u21 u11

)


and

u−1 = u∗ =(

u11 u21

u12 u22

).

This yields the desired form for u, with α = u11, β = u12.From the proof so far, we see that the mapping (u �→ (u11, u12)) : SU(2)→ C2

provides a homeomorphism from SU(2) to the Euclidean unit sphere of C2. ThusSU(2) is compact. �

11.3 Euler angle parametrisation of SU(2)

The unitary group U(1) = {u ∈ C : u∗u = 1} is often parametrised by the anglet ∈ R, i.e., u(t) = eit ∈ U(1); of course, the parameter range 0 ≤ t < 2π is sufficienthere. These angles provide convenient expressions for group operations:

u(t0) u(t1) = u(t0 + t1) and u(t)−1 = u(−t). (11.2)

In analogy to this, elements of SU(2) can be endowed with so-called Euler angles,our next topic.

Notice that the Euclidean unit sphere of C2 is naturally identified with theEuclidean unit sphere S3 of R4. An easy (non-unique) way to parametrise thepoints (u11, u12) of the unit sphere of C2 by r, s, t ∈ R is

u11 = cos(r) eis, u12 = i sin(r) eit,

resulting in

u =(

cos(r) eis i sin(r) eit

i sin(r) e−it cos(r) e−is

). (11.3)

Putting r = 0 in (11.3) we obtain a one-parametric subgroup of matrices(eis 00 e−is

),

and with s = 0 = t we get another one-parametric subgroup of matrices(cos(r) i sin(r)

i sin(r) cos(r)

).

Multiplying elements of these one-parametric subgroups, let us define

u(2s, 2r, 2t) :=(

eis 00 e−is

) (cos(r) i sin(r)

i sin(r) cos(r)

) (eit 00 e−it

)=

(cos(r) ei(s+t) i sin(r) ei(s−t)

i sin(r) e−i(s−t) cos(r) e−i(s+t)

).

Clearly, by Lemma 11.2.1 any matrix u ∈ SU(2) is of this form. This leads us tothe Euler angle parametrisation of SU(2):

11.3. Euler angle parametrisation of SU(2) 601

Definition 11.3.1 (Euler’s angles on SU(2)). Euler’s angles (φ, θ, ψ) from the pa-rameter ranges

0 ≤ φ < 2π, 0 ≤ θ ≤ π, −2π ≤ ψ < 2π (11.4)

correspond to the group element

u(φ, θ, ψ) =(

cos( θ2 ) ei(φ+ψ)/2 i sin( θ

2 ) ei(φ−ψ)/2

i sin( θ2 ) e−i(φ−ψ)/2 cos( θ

2 ) e−i(φ+ψ)/2

)∈ SU(2).

Exercise 11.3.2. Check that the parameter ranges for φ, θ, ψ in (11.4) are sufficient.Show that the Euler angle parametrisation is almost injective in the sense that if(φ1, θ1, ψ1) = (φ2, θ2, ψ2) with 0 ≤ φj < 2π, 0 < θj < π, −2π ≤ ψ < 2π, thenu(φ1, θ1, ψ1) = u(φ2, θ2, ψ2).

The angle parametrisation of U(1) behaves well with respect to the groupoperations in (11.2). Unfortunately, the situation with the Euler angles of SU(2)is complicated. Nevertheless, let us study this problem.

Exercise 11.3.3. Let u(φ, θ, ψ) =(

a bc d

). Verify that

2aa = 1 + cos(θ),2ab = i eiφ sin(θ),−2ab = i eiψ sin(θ).

Notice how these formulae allow one to recover Euler angles φ, θ, ψ from the matrixu(φ, θ, ψ).

Let us examine the multiplication

u(φ, θ, ψ) = u(φ0, θ0, ψ0) u(φ1, θ1, ψ1).

Abbreviating 2rj := θj and s := ψ0 + φ1, we have

u(φ, θ, ψ) = u(φ0, 0, 0) v u(0, 0, ψ1),

where

v = u(0, θ0, ψ0) u(φ1, θ1, 0)

=(

cos r0 eiψ0/2 i sin r0 e−iψ0/2

i sin r0 eiψ0/2 cos r0 e−iψ0/2

) (cos r1 eiφ1/2 i sin r1 eiφ1/2

i sin r1 e−iφ1/2 cos r1 e−iφ1/2

)=

(cos(r0) cos(r1)eis − sin(r0) sin(r1)e−is . . .

i sin(r0) cos(r1)eis + i cos(r0) sin(r1)e−is . . .

).


Notice that there is no need for calculating the second column of the matrix v.

Applying Exercise 11.3.3 to u = u(φ0, 0, 0) v u(0, 0, ψ1) =(

a bc d

), we get

1 + cos(θ) = 2aa

= 2 cos(r0)2 cos(r1)2 + 2 sin(r0)2 sin(r1)2

−4 cos(2s) cos(r0) sin(r0) cos(r1) sin(r1)

=24(1 + cos(θ0))(1 + cos(θ1)) +

24(1− cos(θ0))(1− cos(θ1))

− cos(2s) sin(θ0) sin(θ1)= 1 + cos(θ0) cos(θ1)− cos(ψ0 + φ1) sin(θ0) sin(θ1),

ieiφ sin(θ)= 2ab

= 2ieiφ0[ei2s cos(r0)2 cos(r1) sin(r1)− e−i2s sin(r0)2 sin(r1) cos(r1)

+ cos(r0) sin(r0) cos(r1)2 − sin(r0) cos(r0) sin(r1)2]

= ieiφ0 [cos(2s) cos(2r0) sin(2r1) + i sin(2s) sin(2r1)+ sin(2r0) cos(2r1)]

= ieiφ0 [cos(ψ0 + φ1) cos(θ0) sin(θ1) + i sin(ψ0 + φ1) sin(θ1)+ sin(θ0) cos(θ1)] ,

ieiψ sin(θ)= −2ab

= 2ieiψ1[ei2s cos(r0) sin(r0) cos(r1)2 − e−i2s sin(r0) cos(r0) sin(r1)2

+cos(r0)2 cos(r1) sin(r1)− sin(r0)2 sin(r1) cos(r1)]

= ieiψ1 [cos(2s) sin(2r0) cos(2r1) + i sin(2s) sin(2r0)+ cos(2r0) sin(2r1)]

= ieiψ1 [cos(ψ0 + φ1) sin(θ0) cos(θ1) + i sin(ψ0 + φ1) sin(θ0)+ cos(θ0) sin(θ1)] .

Let us collect the outcomes:

Proposition 11.3.4. Let u(φ, θ, ψ) = u0 u1, where uj = u(φj , θj , ψj). Then

cos(θ) = cos(θ0) cos(θ1)− cos(ψ0 + φ1) sin(θ0) sin(θ1),

eiφ = eiφ0cos(ψ0 + φ1) cos(θ0) sin(θ1) + i sin(ψ0 + φ1) sin(θ1) + sin(θ0) cos(θ1)

sin(θ),

eiψ = eiψ1cos(ψ0 + φ1) sin(θ0) cos(θ1) + i sin(ψ0 + φ1) sin(θ0) + cos(θ0) sin(θ1)

sin(θ).

11.4. Quaternions 603

Exercise 11.3.5. Let u(φ1, θ1, ψ1) = u(φ0, θ0, ψ0)−1, where

0 ≤ φk < 2π, 0 ≤ θk ≤ π and − 2π ≤ ψk < 2π.

Express (φ1, θ1, ψ1) in terms of (φ0, θ0, ψ0).

11.4 Quaternions

The quaternion space H is the associative R-algebra with a vector space basis{1, i, j,k}, where 1 ∈ H is the unit and

i2 = j2 = k2 = −1 = ijk.

The mappingx = (xm)3m=0 �→ x01 + x1i + x2j + x3k

identifies R4 with H, and the quaternion inner product is given by

(x, y) �→ 〈x, y〉H := x0y0 + x1y1 + x2y2 + x3y3,

the corresponding norm being

x �→ ‖x‖H := 〈x, x〉1/2H . (11.5)

For all x, y ∈ H we have‖xy‖H = ‖x‖H ‖y‖H;

if ‖x‖H = 1 then both y �→ xy and y �→ yx are linear isometries H → H. Inparticular, the unit sphere S3 ⊂ R4 ∼= H is a multiplicative group.

Exercise 11.4.1. Show that i, j,k satisfy the following multiplication rules:

i = jk = −kj, j = ki = −ik and k = ij = −ji.

11.4.1 Quaternions and SU(2)

A bijective homomorphism S3 → SU(2) is defined by

x �→ u(x) =(

x0 + ix3 x1 + ix2

−x1 + ix2 x0 − ix3

)=:

(α β

−β α

), (11.6)

with detu(x) = ||x||2H = |x|2 = 1. Thus from Lemma 11.2.1 we get:

Proposition 11.4.2 (S3 ∼= SU(2)). We have S3 ∼= SU(2).


The reader may ask why we reject a perhaps more obvious candidate for ahomomorphism, namely

x �→(

x0 + ix1 x2 + ix3

−x2 + ix3 x0 − ix1

);

the reason is that our choice fits perfectly with the choice of traditional Eulerangles, where the x3-axis is the “fundamental one”. On S3 ⊂ R4 ∼= H, the Eulerangle representation is

x =

⎛⎜⎜⎝x0

x1

x2

x3

⎞⎟⎟⎠ =

⎛⎜⎜⎝cos φ+ψ

2 cos θ2

− sin φ−ψ2 sin θ

2

cos φ−ψ2 sin θ

2

sin φ+ψ2 cos θ

2

⎞⎟⎟⎠ ,

where the ranges of parameters φ, θ, ψ are the same as in (11.4), i.e.,

0 ≤ φ < 2π, 0 ≤ θ ≤ π, −2π ≤ ψ < 2π.

Notice that the Euler angle representation is unique if and only if 0 < θ < π, butthis is not a true problem for us, since θ ∈ {0, π} corresponds to a set of lowerdimension on S3.Remark 11.4.3. Recall that for u = u(φ, θ, ψ) ∈ SU(2), the Euler angle represen-tation is given by

u =(

ei(φ+ψ)/2 cos θ2 ei(φ−ψ)/2i sin θ

2

e−i(φ−ψ)/2i sin θ2 e−i(φ+ψ)/2 cos θ

2

)=

(eiφ/2 0

0 e−iφ/2

) (cos(θ/2) i sin(θ/2)i sin(θ/2) cos(θ/2)

) (eiψ/2 0

0 e−iψ/2

),

where 0 ≤ φ < 2π, 0 ≤ θ ≤ π and −2π ≤ ψ < 2π.

11.4.2 Quaternions and SO(3)

Let P : H→ H be the orthogonal projection

P (x) := x1i + x2j + x3k ∼= (xm)3m=1.

The subspace P (H) ⊂ H is naturally identified with R3, and the cross product ofR3 is a “shadow” of the quaternion product:

P (P (x)P (y)) =xy − yx

2∼= (xm)3m=1 × (ym)3m=1.

Then S2 ⊂ R3 ∼= P (H) ⊂ H consists of points x with cos((φ + ψ)/2) = 0, i.e.,

x =

⎛⎜⎜⎝0

± cos φ sin θ2

± sin φ sin θ2

± cos θ2

⎞⎟⎟⎠ =

⎛⎜⎜⎝0

cos φ sin θ

sinφ sin θ

cos θ

⎞⎟⎟⎠ ,

11.4. Quaternions 605

where 0 ≤ φ < 2π and 0 ≤ θ, θ ≤ π. of S2. The quaternion conjugate of x ∈ H is

x∗ := x01− x1i− x2j− x3k.

It is easy to see that u(x∗) = u(x)∗. Now (x �→ x∗) : H → H is a linear isometrysuch that (x∗)∗ = x, (xy)∗ = y∗x∗ and x∗x = xx∗ = ‖x‖2H1. Consequently, if‖x‖H = 1 then

(y �→ xyx∗) : H→ H

is a linear orientation-preserving isometry mapping P (H) → P (H) bijectively;hence y �→ xyx∗ corresponds to a rotation Rx ∈ SO(3), and x �→ Rx is a grouphomomorphism S3 → SO(3). This surjective homomorphism is given by

x ∼= u(x) = u(φ, θ, ψ)

�→

⎛⎝Re(α2 + β2) − Im(α2 − β2) 2 Im(αβ)Im(α2 + β2) Re(α2 − β2) −2 Re(αβ)

2 Im(αβ) 2 Re(αβ) |α|2 − |β|2

⎞⎠ = ω(φ, θ, ψ), (11.7)

i.e., u(φ, θ, ψ) �→ ω(φ, θ, ψ), and its kernel is {1,−1} ∈ S3. Thus we obtain

Theorem 11.4.4 (SO(3) ∼= S3/{±1}). We have

SO(3) ∼= S3/{±1},and the space C∞(SO(3)) consists of smooth functions f ∈ C∞(S3) satisfyingf(−x) = f(x).

11.4.3 Invariant integration on SU(2)

Let us consider the doubly covering homomorphism

(u(φ, θ, ψ) �→ ω(φ, θ, ψ)) : SU(2)→ SO(3),

presented in formula (11.7) in Section 11.4.2. From this, recalling the Haar integral(11.1) of SO(3), we obtain the Haar integral for SU(2):

f �→∫

SU(2)

f(x) dx =1

16π2

∫ 2π

−2π

∫ π

0

∫ π

−π

f(u(φ, θ, ψ)) sin(θ) dφ dθ dψ.

Exercise 11.4.5. Deduce the Haar integral of SU(2) by considering the group asthe unit sphere in R4 and expressing the Lebesgue measure of R4 in “polar coor-dinates”.

11.4.4 Symplectic groups

Here we briefly review the notion of symplectic groups and show that the groupSU(2) can be also viewed as the first symplectic group Sp(1). In general, thesubgroups

O(n) < GL(n, R), U(n) < GL(n, C) and Sp(n) < GL(n, H)


have a lot in common, namely all of them are subgroups of unitary elements of therespective general linear groups with respect to their inner products, Euclidean,Hermitian, and symplectic, respectively.

For any n ∈ N, the symplectic group Sp(n) is the group of automorphisms ofHn preserving the norm. More precisely, for x, y ∈ Hn we can generalise the innerproduct (11.5) from H to Hn by setting

〈x, y〉Hn :=n∑

j=1

xjy∗j ,

so that the norm in Hn becomes ||x||Hn = 〈x, x〉Hn . The inner product 〈x, y〉Hn issometimes called the symplectic inner product.

Definition 11.4.6 (Symplectic groups Sp(n)). The symplectic group is defined asthe set

Sp(n) = {A ∈ GL(n, H) : ||Ax||Hn = ||x||Hn for all x ∈ Hn}.

As an immediate corollary we obtain:

Corollary 11.4.7. We have

Sp(1) ∼= {x ∈ H : ||x||H = 1} ∼= S3 ∼= SU(2).

Exercise 11.4.8. Show that if A ∈ Sp(n) then 〈Ax, Ay〉Hn = 〈x, y〉Hn for all x, y ∈Hn.

Remark 11.4.9 (Symplectic matrices). The inner-product preserving identificationH = C2 = R4 extends to the inner-product preserving identification Hn = C2n =R4n. In particular, norms on Hn and C2n coincide. Therefore, the symplectic groupSp(n) can be identified with a subgroup of GL(2n, C) preserving the Euclideannorm with respect to matrix multiplication. Thus, Sp(n) can be identified with

the subgroup of U(2n) of matrices of the form(

α β

−β α

), where α, β ∈ End(Cn).

Such matrices are called symplectic matrices.Remark 11.4.10 (Complex symplectic groups). The complex symplectic group isdefined by

Sp(n, C) = {A ∈ GL(2n, C) : AT JA = J},

where AT = A∗ is the transpose matrix of A and J =(

0 −In

In 0

), where In is

the identity matrix in GL(n, C). The matrix J has the property J2 = I2n so it canbe viewed as an extension of the complex identity i2 = −1 to higher dimensions.The product in GL(2n, C) defined by a · b := aT Jb is invariant under the actionof Sp(n, C).

Exercise 11.4.11. Show that Sp(n) = Sp(n, C) ∩ U(2n). Also show that Sp(n, C)is not compact but that Sp(n) is.

11.5. Lie algebra and differential operators on SU(2) 607

11.5 Lie algebra and differential operators on SU(2)

Let f ∈ C∞(SU(2)) and u : R → SU(2) be a smooth function. Let φ(t), θ(t), ψ(t)be the Euler angles of u(t). If 0 < φ(0) < 2π, 0 < θ(0) < π and −2π < ψ(0) < 2πthen

ddt

f(u)∣∣∣∣t=0

=(

φ′(0)∂

∂φ+ θ′(0)

∂

∂θ+ ψ′(0)

∂

∂ψ

)f(u(φ, θ, ψ))

∣∣∣∣t=0

. (11.8)

This is the derivative of f at u(0) ∈ SU(2) in the direction u′(0), as expressed inthe Euler angles. Of particular interest are those differential operators that arisenaturally from the Lie algebra.

Let us consider one-parametric subgroups ω1, ω2, ω3 : R→ SU(2), where

ω1(t) =(

cos(t/2) i sin(t/2)i sin(t/2) cos(t/2)

),

ω2(t) =(

cos(t/2) − sin(t/2)sin(t/2) cos(t/2)

),

ω3(t) =(

eit/2 00 e−it/2

).

Let us also make a special choice wj := ωj(π/2), i.e.,

w1 = 2−1/2

(1 ii 1

),

w2 = 2−1/2

(1 −11 1

),

w3 = 21/2

(1 + i 0

0 1− i

).

Exercise 11.5.1. Find the Euler angles of ωj(t).

Definition 11.5.2 (Basis of su(2)). Let Yj := ω′j(0), i.e.,

Y1 =12

(0 ii 0

), Y2 =

12

(0 −11 0

), Y3 =

12

(i 00 −i

).

Matrices Y1, Y2, Y3 constitute a basis for the real vector space su(2), the Lie algebraof SU(2).

Remark 11.5.3 (Pauli matrices). We note that the matrices 2i Yj , j = 1, 2, 3, are

known as Pauli (spin) matrices in physics. In our case, the coefficient i2 appears

in front of these Pauli matrices because we obtained Yj ’s as elements of the Liealgebra su(2) (leading to the coefficient i), and because of the use of Euler angles(leading to the coefficient 1

2 ). It can be also noted that k = span{Y3} and p =span{Y1, Y2} form a Cartan pair of the Lie algebra su(2), but we will not pursuethis topic here.


Exercise 11.5.4. Check the commutation relations

[Y1, Y2] = Y3, [Y2, Y3] = Y1, [Y3, Y1] = Y2.

State explicitly a Lie algebra isomorphism su(2)→ so(3).

Exercise 11.5.5. For x, y ∈ R3 let us define X, Y ∈ su(2) by

X =3∑

j=1

xjYj and Y =3∑

j=1

yjYj .

What has [X, Y ] to do with the vector cross product x× y ∈ R3?

Exercise 11.5.6. Let Y (y) = y1Y1 + y2Y2 + y3Y3. Show that

u = exp(Y (y)) = I cos(‖y‖) + Y (y)sin(‖y‖)‖y‖ ,

and (provided that ‖y‖ is small enough) that

y =‖y‖

sin(‖y‖)

⎛⎝−i(u12 + u21)u21 − u12

i(u22 − u11)

⎞⎠ .

Recall that an inner automorphism of a group G is a group isomorphismof the form (x �→ g−1xg) : G → G for some g ∈ G. Let us study specific innerautomorphisms of SU(2) conjugating the one-parametric subgroups ω1, ω2, ω3 toeach other.

Proposition 11.5.7 (Conjugating one-parametric subgroups to each other). Letwj = ωj(π/2) and t ∈ R. Then

w1 ω2(t) w−11 = ω3(t),

w2 ω3(t) w−12 = ω1(t),

w3 ω1(t) w−13 = ω2(t).

The differential versions of these formulae are

w1 Y2 w−11 = Y3,

w2 Y3 w−12 = Y1,

w3 Y1 w−13 = Y2.


w−11 ω3(t) w1 =

12

(1 −i−i 1

) (eit/2 0

0 e−it/2

) (1 ii 1

)=

12

(eit/2 + e−it/2 ieit/2 − ie−it/2

−ieit/2 + ie−it/2 eit/2 + e−it/2

)=

(cos(t/2) − sin(t/2)sin(t/2) cos(t/2)

)= ω2(t).


We also have

w2 ω3(t) w−12 =

12

(1 −11 1

) (eit/2 0

0 e−it/2

) (1 1−1 1

)=

12

(eit/2 + e−it/2 eit/2 − ie−it/2

eit/2 − ie−it/2 eit/2 + e−it/2

)=

(cos(t/2) i sin(t/2)i sin(t/2) cos(t/2)

)= ω1(t).

Finally, we have

w3 ω1(t) w−13

=12

(1 + i 0

0 1− i

) (cos(t/2) i sin(t/2)i sin(t/2) cos(t/2)

) (1− i 0

0 1 + i

)=

12

(1 + i 0

0 1− i

) ((1− i) cos(t/2) (i− 1)i sin(t/2)(i + 1) sin(t/2) (1 + i) cos(t/2)

)=

(cos(t/2) − sin(t/2)sin(t/2) cos(t/2)

)= ω2(t).

The differential version follows immediately by differentiating the above formulaewith respect to t at t = 0. �

Definition 11.5.8 (Left-invariant differential operators). To a vector Y ∈ su(2) weassociate the left-invariant differential operator DY : C∞(SU(2)) → C∞(SU(2))defined by

DY f(u) =ddt

f(u exp(tY ))∣∣∣∣t=0

.

In the sequel, we write Dj := DYj .

Proposition 11.5.9 (Derivatives D1, D2, D3 in Euler angles). Expressed in Eulerangles,

D1 = cos(ψ)∂

∂θ+

sin(ψ)sin(θ)

∂

∂φ− cos(θ)

sin(θ)sin(ψ)

∂

∂ψ,

D2 = − sin(ψ)∂

∂θ+

cos(ψ)sin(θ)

∂

∂φ− cos(θ)

sin(θ)cos(ψ)

∂

∂ψ,

D3 =∂

∂ψ,

provided that sin(θ) = 0, i.e., 0 < θ < π.


Proof. The simplest case is

D3f(u) =ddt

f(u(φ, θ, ψ) ω3(t))∣∣∣∣t=0

=ddt

f(u(φ, θ, ψ + t))∣∣∣∣t=0

=∂

∂ψf(u).

Next we shall deal with D1. According to (11.8), we need to calculate φ′(0), θ′(0),ψ′(0), where u(φ, θ, ψ) = u(φ0, θ0, ψ0) ω1(t). Let us exploit Proposition 11.3.4 withu(φ1, θ1, ψ1) = ω1(t) = u(0, t, 0), yielding

cos(θ) = cos(θ0) cos(t)− cos(ψ0) sin(θ0) sin(t),

ei(φ−φ0) sin(θ) = cos(ψ0) cos(θ0) sin(t) + i sin(ψ0) sin(t) + sin(θ0) cos(t),

eiψ sin(θ) = cos(ψ0) sin(θ0) cos(t) + i sin(ψ0) sin(θ0) + cos(θ0) sin(t).

Differentiating these equalities with respect to t at t = 0, we obtain

−θ′(0) sin(θ0) = − cos(ψ0) sin(θ0),iφ′(0) sin(θ0) + θ′(0) cos(θ0) = cos(ψ0) cos(θ0) + i sin(ψ0),

eiψ0 [iψ′(0) sin(θ0) + θ′(0) cos(θ0)] = cos(θ0).

Thereby θ′(0) = cos(θ0), φ′(0) = sin(ψ0)/ sin(θ0), and

iψ′(0) sin(θ0) =[e−iψ0 − cos(ψ0)

]cos(θ0) = −i sin(ψ0) cos(θ0),

yielding ψ′(0) = − sin(ψ0) cos(θ0)/ sin(θ0). This proves the expression for D1.Lastly, the D2 formula must be deduced. Again, we use equation (11.8), wherethis time u(φ, θ, ψ) = u(φ0, θ0, ψ0) ω2(t). Thus let

u(φ1, θ1, ψ1) = ω2(t) = u(π/2, t,−π/2),

and apply Proposition 11.3.4: noticing that sin(ψ0 + π/2) = cos(ψ0) and cos(ψ0 +π/2) = − sin(ψ0), we get

cos(θ) = cos(θ0) cos(t) + sin(ψ0) sin(θ0) sin(t),ei(φ−φ0) sin(θ) = − sin(ψ0) cos(θ0) sin(t) + i cos(ψ0) sin(t)

+ sin(θ0) cos(t),ei(ψ+π/2) sin(θ) = − sin(ψ0) sin(θ0) cos(t) + i cos(ψ0) sin(θ0)

+ cos(θ0) sin(t).

Differentiating these equalities with respect to t at t = 0, we get

−θ′(0) sin(θ0) = sin(ψ0) sin(θ0),iφ′(0) sin(θ0) + θ′(0) cos(θ0) = − sin(ψ0) cos(θ0) + i cos(ψ0),

ieiψ0 [iψ′(0) sin(θ0) + θ′(0) cos(θ0)] = cos(θ0).

In this case, θ′(0) = − sin(ψ0), φ′(0) = cos(ψ0)/ sin(θ0), and

iψ′(0) sin(θ0) =[sin(ψ0)− ie−iψ0

]cos(θ0) = −i cos(ψ0) cos(θ0),

so that ψ′(0) = − cos(ψ0) cos(θ0)/ sin(θ0). This completes the proof. �


Endowing SU(2) with the Riemannian metric inherited from R4 ⊂ S3 ∼=SU(2), we could show that {Y1, Y2, Y3} is orthogonal, so that operators D1, D2, D3

form a good choice for generating the first-order left-invariant partial differentialoperators. However, in order to simplify notation in the sequel, we will work withslightly different operators.

Definition 11.5.10 (Creation, annihilation, and neutral operators). Let us defineleft-invariant first-order partial differential operators ∂+, ∂−, ∂0 : C∞(SU(2)) →C∞(SU(2)), called creation, annihilation, and neutral operators, respectively, by⎧⎪⎨⎪⎩

∂+ := iD1 −D2,

∂− := iD1 + D2,

∂0 := iD3,

, i.e.,

⎧⎪⎨⎪⎩D1 = −i

2 (∂− + ∂+) ,

D2 = 12 (∂− − ∂+) ,

D3 = −i∂0.

The reader may ask why we work with operators ∂+, ∂−, ∂0 instead of vector fieldsD1, D2, D3. The reason is that many calculations become considerably shorter.Indeed, already expressions in Euler’s angles in Corollary 11.5.13 are quite simplerthan those in Proposition 11.5.9, as well as expressions in Theorem 11.9.3 aresimpler than those in Proposition 11.9.2. The other reason is that the symbols of∂+, ∂−, ∂0 will turn out to have fewer non-zero elements, see Theorem 12.2.1. Theterminology of “creation”, “annihilation”, and “neutral” operators is explained inRemark 12.2.3.

Remark 11.5.11 (Laplacian). The Laplacian L satisfies L = D21 +D2

2 +D23 and we

have[L, Dj ] = 0

for every j ∈ {1, 2, 3}. Notice that it can be expressed as

L = −∂20 − (∂+∂− + ∂−∂+)/2.

Exercise 11.5.12. Show that the operators ∂+, ∂−, ∂0 satisfy commutator relations

[∂0, ∂+] = ∂+, [∂−, ∂0] = ∂−, [∂+, ∂−] = 2∂0.

The operators ∂+, ∂−, ∂0 coincide with operators H+, H−, H0 in Vilenkin [148,p. 140].

By Proposition 11.5.9, we also have

Corollary 11.5.13 (Operators ∂+, ∂−, ∂0 in Euler angles).

∂+ = e−iψ

(i

∂

∂θ− 1

sin(θ)∂

∂φ+

cos(θ)sin(θ)

∂

∂ψ

),

∂− = eiψ

(i

∂

∂θ+

1sin(θ)

∂

∂φ− cos(θ)

sin(θ)∂

∂ψ

),

∂0 = i∂

∂ψ.


Exercise 11.5.14. For u =(

a(u) b(u)c(u) d(u)

)∈ SU(2), let us write

DY u :=(

DY a DY bDY c DY d

).

Without resorting to Euler angles, show that

D1u =i2

(b ad c

), D2u =

12

(b −ad −c

), D3u =

i2

(a −bc −d

),

∂+u =(−b 0−d 0

), ∂−u =

(0 −a0 −c

), ∂0u =

12

(−a b−c d

).

11.6 Irreducible unitary representations of SU(2)

In this section we shall determine irreducible unitary representations of SU(2) upto unitary equivalence. Our first task is to find a natural representation to startwith. Matrices act naturally on vectors by matrix multiplication, and this is indeeda good idea: let us identify z = (z1, z2) ∈ C2 with matrix z =

(z1 z2

)∈ C1×2,

and consider

T : SU(2)→ GL(C[z1, z2]), (T (u)f)(z) := f(zu),

where GL(C[z1, z2]) is the space of invertible linear mappings on the complexvector space C[z1, z2] consisting of two-variable polynomials f : C2 → C. ClearlyT is a representation of SU(2) on C[z1, z2], necessarily reducible as C[z1, z2] isinfinite-dimensional. So, decomposing T is the next step.

It is clear that the orders of polynomials f, T (u)f ∈ C[z1, z2] are the same,so that, e.g., polynomials f ∈ C[z1, z2] of order less than k ∈ N0 form a T -invariant subspace. But these subspaces can clearly be decomposed further. Foreach l ∈ 1

2N0, let Vl be the subspace of C[z1, z2] containing the homogeneouspolynomials of order 2l ∈ N0, i.e.,

Vl =

{f ∈ C[z1, z2] : f(z1, z2) =

2l∑k=0

akzk1z2l−k

2 , {ak}2lk=0 ⊂ C

}.

LetTl : SU(2)→ GL(Vl), (Tl(u)f)(z) = f(zu),

denote the restriction of T on the T -invariant subspace Vl of dimension (2l + 1) ∈Z+. Our objective is to show that Tl is irreducible, unitary with respect to anatural inner product of Vl, and that (up to unitary equivalence) there are noother irreducible unitary representations for SU(2). By regarding f ∈ Vl as anatural function on SU(2), we shall endow Vl with the L2-inner product of thegroup.

A natural basis for the vector space Vl is {plk : k ∈ {0, 1, . . . , 2l}}, where

plk(z) = zk1z2l−k

2 . (11.9)

11.6. Irreducible unitary representations of SU(2) 613

Theorem 11.6.1. As defined above, Tl is irreducible. Moreover, plk is an eigen-function of Tl(u(φ, 0, 0)) with eigenvalue eiφ(k−l/2).

Proof. By Schur’s Lemma (Corollary 6.3.25), irreducibility of Tl follows, if we canshow that its intertwining operators are necessarily scalar. So let A ∈ End(Vl) bean intertwining operator for Tl, i.e., such that for every u ∈ SU(2) it holds that

ATl(u) = Tl(u)A.

Due to the decomposition

u(s, r, t) = u(s, 0, 0)u(0, r, 0)u(0, 0, t),

checking the cases u = u(0, 0, s) and u = u(0, r, 0) suffices. First,

Tl(u(2s, 0, 0))plk(z) = plk(zu(2s, 0, 0)) = plk(z1eis, z2e−is) = eis(2k−l) plk(z),

so that plk is an eigenvector of Tl(u(2s, 0, 0)) with eigenvalue eis(2k−l). The samething is true for the vector Aplk ∈ Vl:

Tl(u(2s, 0, 0))Aplk(z) = ATl(u(2s, 0, 0))plk(z)

= A(z �→ eis(2k−l)plk(z)

)= eis(2k−l)Aplk(z).

Let Aplk =∑

j aj plj ∈ Vl. Then

eis(2k−l)Aplk = Tl(u(2s, 0, 0))∑

j

aj plj

=∑

j

aj Tl(u(2s, 0, 0)) plj =∑

j

aj eis(2j−l) plj ,

which is possible only if Aplk = ak plk. This yields especially

Tl(u(0, 2r, 0))Apl0(z) = Tl(u(0, 2r, 0)) a0pl0(z) = a0 pl0(zu(0, 2r, 0))= a0 (z1 sin(r) + cos(r)z2)l

= a0

∑k

(l

k

)sin(r)k cos(r)l−kplk(z).

On the other hand, this coincides with

ATl(u(0, 2r, 0))pl0(z) = A (z �→ pl0(zu(0, 2r, 0)))

= A(z �→ (z1 sin(r) + cos(r)z2)l

)= A

∑k

(l

k

)sin(r)k cos(r)l−k plk(z)

=∑

k

(l

k

)sin(r)k cos(r)l−k akplk(z).

Choosing r so that sin(r) = 0 = cos(r), we see that ak = a0 for each k. ThusA = a0I is scalar. �


Next we shall find out that, up to equivalence, representations Tl provide acomplete set of irreducible representations for SU(2).

Lemma 11.6.2 (Decomposition of elements of SU(2)). For g ∈ SU(2) there is adecomposition g = uhu−1, where u, h ∈ SU(2) such that h is diagonal.

Proof. Let g =(

a b

−b a

)∈ SU(2). If b = 0 then the claim is trivial with the

choice h := g and u = I, so we assume b = 0. The characteristic polynomial of gis Pg : C→ C, where

Pg(z) = det(zI − g)= (z − a)(z − a)− b(−b)

|a|2+|b|2=1= z2 − 2Re(a)z + 1.

Now |a| = 1, because b = 0, and therefore Pg has two distinct roots, i.e., g hastwo different eigenvalues z1, z2 ∈ C. Let

h =(

z1 00 z2

),

and let the column vectors of v ∈ C2×2 be the corresponding normalised eigen-vectors of g. Now g = vhv−1. The eigenvectors corresponding to the differenteigenvalues are orthogonal. Thus v ∈ U(2), and u := λv ∈ SU(2) with a suitablechoice of λ ∈ C, |λ| = 1. Now we have g = uhu−1. �

Theorem 11.6.3 (Completeness of the representation series). Let T∞ : SU(2) →GL(V ) be an irreducible representation. Then T∞ is equivalent to Tl, wheredim(V ) = 2l + 1.

Proof. Let m ∈ 12N0 ∪ {∞}. Let χm : SU(2) → C be the character of Tm. By

Theorem 7.8.6 it is enough to show that 〈χ∞, χl〉L2(SU(2)) = 0 for some l ∈ 12N0.

Notice thatχm(uhu−1) = χm(h).

By Lemma 11.6.2, we may identify χm with fm : R→ C defined by

fm(t) := χm(h(t)), where h(t) =(

eit 00 e−it

).

Thereby

〈χ∞, χl〉L2(SU(2)) =∫

SU(2)

χ∞(g) χl(g) dμSU(2)(g)

=12π

∫ 2π

0

f∞(t) fl(t) dt.

11.6. Irreducible unitary representations of SU(2) 615

Recall that {plk : 0 ≤ k ≤ 2l} is a basis for vector space Vl, and

Tl(h(t))plk(z) = plk(z h(t)) = plk(z1eit, z2e−it) = eit(2k−l)plk(z).

Thus we get

fl(t) = Tr (Tl(h(t))) =2l∑

k=0

eit(2k−l) =l∑

k=−l

eitk.

Thereby span{fl : l ∈ N0/2} is dense in the space of 2π-periodic continuous evenfunctions. Because f∞ : R→ C is a 2π-periodic continuous even function,∫ 2π

0

f∞(t) fl(t) dt = 0

for some l ∈ N0/2. This completes the proof. �

11.6.1 Representations of SO(3)

We now briefly review representations of SO(3) and related spherical harmonics.We will only sketch this topic since it is analogous to constructing representationsof SU(2). However, to provide an additional insight into the topic we indicate therelation to spherical harmonics.

The group SO(3) is a subgroup of the group GL(3, R) of three-dimensionalmatrices acting on R3 by matrix multiplication. In analogy to the spaces Vl in thecase of SU(2), we introduce the space Pl to be the complex subspace of the spaceC[x1, x2, x3] containing all the homogeneous polynomials of order l ∈ N0 in threevariables. The group SO(3) acts on Pl in the same way as in the case of SU(2),namely we have

Tl : SO(3)→ GL(Pl), (Tl(u)f)(x) = f(xu).

Exercise 11.6.4. Show that dimPl = 12 (l + 1)(l + 2).

The problem with the spaces Pl is that they do not yield irreducible repre-sentation spaces for l ≥ 2. For example, the space generated by the polynomialx2

1 + x22 + x2

3 is a non-trivial SO(3)-invariant subspace of P2. Thus, we have todecompose the spaces Pl further which is done in the following exercises.

Exercise 11.6.5 (Spherical harmonics). Let L = ∂2

∂x21

+ ∂2

∂x22

+ ∂2

∂x23

be the Laplacianon R3. A polynomial f ∈ Pl is called a harmonic polynomial of order l if Lf = 0,and we set Hl := {f ∈ Pl : Lf = 0}. Show that dimHl = 2l + 1. Restrictions ofharmonic polynomials to the sphere S2 are called spherical harmonics of order l.Consequently, since harmonic polynomials are homogeneous, the dimension of thespace of spherical harmonics of order l is also 2l + 1.

Exercise 11.6.6. Show that L is SO(3)-invariant. Consequently, show that Hl isan SO(3)-invariant subspace of Pl.


Exercise 11.6.7. Show that the mapping

Tl : SO(3)→ GL(Hl), (Tl(u)f)(x) = f(xu)

is an irreducible representation of SO(3). Moreover, show that an irreducible rep-resentation T∞ : SO(3) → GL(V ) is equivalent to Tl, where dim(V ) = 2l + 1 forl ∈ N0.

11.7 Matrix elements of representations of SU(2)

Recalling polynomials in (11.9), we see that the collection

{qlk : k ∈ {−l,−l + 1, . . . ,+l − 1,+l}}

is an orthonormal basis for the representation space Vl, where

qlk(z) =zl−kl zl+k

2√(l − k)!(l + k)!

.

Let us compute the matrix elements tlmn(φ, θ, ψ) of Tl(u(φ, θ, ψ)) with respect tothis basis.

Theorem 11.7.1 (Matrix elements of T l). Let u ∈ SU(2) be given by

u = u(φ, θ, ψ) =(

a bc d

)=

(ei(φ+ψ)/2 cos θ

2 ei(φ−ψ)/2i sin θ2


2

).

Then

tlmn(u) =(

ddz1

)l−m (d

dz2

)l+m (z1a + z2c)l−n(z1b + z2d)l+n√(l −m)!(l + m)!(l − n)!(l + n)!

, (11.10)

where one can see that the right-hand side does not depend on z. In Euler angles

tlmn(φ, θ, ψ) = e−i(mφ+nψ) P lmn(cos(θ)),

where

P lmn(x) = cl

mn

(1− x)(n−m)/2

(1 + x)(m+n)/2

(ddx

)l−m [(1− x)l−n(1 + x)l+n

]with

clmn = 2−l (−1)l−n in−m√

(l − n)! (l + n)!

√(l + m)!(l −m)!

.

11.7. Matrix elements of representations of SU(2) 617

Moreover, we have

tlmn(u) =

√(l −m)!(l + m)!(l − n)!(l + n)!

min{l−n,l−m}∑i=max{0,n−m}

(11.11)

× (l − n)!(l + n)!i!(l − n− i)!(l −m− i)!(n + m + i)!

aibl−m−icl−n−idn+m+i.

Definition 11.7.2 (Representations tl). The matrix (tlmn)m,n with indices m,nsuch that −l ≤ m,n ≤ l and l − m, l − n ∈ Z will be denoted by tl. The usualconvention in all the formulae is that 0! = 1. We note that although l,m, n can behalf-integers, the spacing between them is such that the differences between any ofthem are always integers!

Remark 11.7.3. Let 0 ≤ θ ≤ π and x = cos(θ). Notice that

(1− x)1/2 = 21/2 sin(θ/2),(1 + x)1/2 = 21/2 cos(θ/2).

In [148] the classical orthogonal polynomials of Legendre and Jacobi are connectedto functions P l

mn, which consequently could be called, e.g., generalised Legendre–Jacobi functions.

Proof of Theorem 11.7.1. First, notice that if vectors qm form an orthonormalbasis of a finite-dimensional inner product space V , then we can write u =∑

m (u, qm)V qm for all u ∈ V . Especially, Tqn =∑

m Tmn qm for a linear op-erator T : V → V , where the numbers Tmn = (Tqn, qm)V are the matrix elementsof T with respect to the chosen basis. Thus for the linear operator Tl(u) : Vl → Vl,the matrix coefficients tlmn(u) satisfy

Tl(u) qln(z) =∑m

tlmn(u) qlm(z)

=∑m

tlmn(u)zl−m1 zl+m

2√(l −m)!(l + m)!

.

Since Tl(u)qln(z) = qln(zu), we have

tlmn(u) =

√(l −m)!(l + m)!

(l −m)!(l + m)!

(d

dz1

)l−m (d

dz2

)l+m

Tl(u)qln(z)

=(

ddz1

)l−m (d

dz2

)l+m (z1a + z2c)l−n(z1b + z2d)l+n√(l −m)!(l + m)!(l − n)!(l + n)!

,

proving (11.10). Here we can use that the total number of derivatives is 2l andalso that (z1a + z2c)l−n(z1b + z2d)l+n is a homogeneous polynomial of degree


2l in (z1, z2) so that the value of the derivative corresponds to the coefficient ofzl−m1 zl+m

2 in this polynomial, and the right-hand side does not depend on z. Sowe can calculate(

ddz1

)l−m (d

dz2

)l+m [(z1a + z2c)l−n(z1b + z2d)l+n

]=

(d

dz1

)l−m (d

dz2

)l+m l−n∑i=0

l+n∑j=0

(l − n

i

)(l + n

j

)× aibjcl−n−idl+n−jzi+j

1 z2l−i−j2

=(

ddy

)l−m

(l + m)!l−n∑i=0

l+n∑j=0

(l − n

i

)(l + n

j

)× aibjcl−n−idl+n−jyi+j

∣∣y=0

= (l + m)!(

ddy

)l−m [(ay + c)l−n(by + d)l+n

]y=0

.

Recalling ad− bc = det(u) = 1, we notice that a(by + d)− b(ay + c) = 1, inspiringa change of variables: let x be such that a(by + d) = (x + 1)/2, i.e., b(ay + c) =(x− 1)/2. Hence dx/dy = 2ab, so that

tlmn =1√

(l −m)!(l + m)!(l − n)!(l + n)!(l + m)!

×(

ddy

)l−m [(ay + c)l−n(by + d)l+n

]y=0

=1√

(l −m)!(l + m)!(l − n)!(l + n)!(l + m)!

× (2ab)l−m

(ddx

)l−m[(

x− 12b

)l−n (x + 12a

)l+n]

x=2ad−1

=1√

(l −m)!(l + m)!(l − n)!(l + n)!(l + m)!

× 2−l−m bn−m

am+n

(ddx

)l−m [(x− 1)l−n(x + 1)l+n

]x=2ad−1

.

Here x = 2ad − 1 = 2 cos(θ/2)2 − 1 = cos(θ), i.e., cos(θ/2)2 = (1 + x)/2 andsin(θ/2)2 = (1− x)/2, a = ei(φ+ψ)/2 cos(θ/2) and b = ei(φ−ψ)/2i sin(θ/2). Thus

tlmn =

√(l + m)!

(l −m)!(l − n)!(l + n)!2−l−m

(ei(φ−ψ)/2

)n−m(ei(φ+ψ)/2

)n+m in−m

× sin(θ/2)n−m

cos(θ/2)m+n

(ddx

)l−m [(x− 1)l−n(x + 1)l+n

]x=cos θ

11.7. Matrix elements of representations of SU(2) 619

=

√(l + m)!

(l −m)!(l − n)!(l + n)!2−l−m e−i(mφ+nψ) in−m

×(

1−x2

)(n−m)/2(1+x2

)(m+n)/2

(ddx

)l−m [(x− 1)l−n(x + 1)l+n

]x=cos θ

=

√(l + m)!

(l −m)!(l − n)!(l + n)!2−l e−i(mφ+nψ) in−m

× (1− x)(n−m)/2

(1 + x)(m+n)/2

(ddx

)l−m [(x− 1)l−n(x + 1)l+n

]x=cos θ

.

Let us finally establish formula (11.11). Again, we note that the total number ofderivatives in (11.10) is 2l and also that (z1a + z2c)l−n(z1b + z2d)l+n is a homo-geneous polynomial of degree 2l in (z1, z2). Therefore,

tlmn(u) = (l −m)!(l + m)!α,

where α is the coefficient of zl−m1 zl+m

2 in the polynomial (z1a + z2c)l−n(z1b +z2d)l+n. Using the binomial formula, we have

α =l−n∑i=0

l+n∑j=0

i+j=l−m

(l − n

i

)(l + n

j

)aibjcl−n−idl+n−j . (11.12)

The restriction j ≤ l + n implies l−m = i + j ≤ i + l + n, so that i ≥ n−m. Onthe other hand, the restriction j ≥ 0 implies l−m = i + j ≥ i. Thus, substitutingj = l − m − i in (11.12) and letting i vary in the range max{0, n − m} ≤ i ≤min{l−n, l−m}, we obtain formula (11.11). This completes the proof of Theorem11.7.1. �Exercise 11.7.4. Prove that

(−1)l+m(−1)l+n = (−1)m−n (11.13)

if l ∈ 12N0 = {k : 2k ∈ N0} and m,n ∈ {−l,−l + 1, . . . ,+l − 1,+l}.

Corollary 11.7.5. For each θ, we have the identities

P l−m,−n(cos(θ)) = P l

mn(cos(θ)), (11.14)

P lnm(cos(θ)) = P l

mn(cos(θ)). (11.15)

Proof. By Theorem 11.7.1, notice that P lmn(cos(θ)) = tlmn(u) if

u =(

a bb a

), where

{a = cos(θ/2),b = i sin(θ/2).


Then Theorem 11.7.1 yields

tlmn(u) =(

ddz1

)l−m (d

dz2

)l+m (z1a + z2b)l−n (z1b + z2a)l+n√(l −m)!(l + m)!(l − n)!(l + n)!

,

and the (±m,±n)-symmetry in this formula implies (11.14). Notice also that

tlmn(u∗) =(

ddz1

)l−m (d

dz2

)l+m (z1a− z2b)l−n(−z1b + z2a)l+n√(l −m)!(l + m)!(l − n)!(l + n)!

=(

ddx1

)l−m ( −ddx2

)l+m (x1a + x2b)l−n(−x1b− x2a)l+n√(l −m)!(l + m)!(l − n)!(l + n)!

= (− 1)l+m(−1)l+n tlmn(u)(11.13)

= (− 1)m−n tlmn(u).

This leads to

P lnm(cos(θ)) = tlnm(u)

= tlmn(u∗)

= (− 1)m−n tlmn(u)

= (− 1)m−n P lmn(cos(θ))

Theorem 11.7.1= P lmn(cos(θ)),

proving (11.15). �

11.8 Multiplication formulae for representations

of SU(2)

On a compact group G, a function f : G→ C is called a trigonometric polynomialif its translates span the finite-dimensional vector space

span{(x �→ f(y−1x)) : G→ C | y ∈ G

},

see Section 7.6 for details. Trigonometric polynomials can be expressed as a linearcombination of matrix elements of irreducible unitary representations. Thus atrigonometric polynomial is continuous, and on a Lie group even C∞-smooth.Moreover, trigonometric polynomials form an algebra when endowed with theusual multiplication. On SU(2), actually,

tl′

m′n′ tlmn =l+l′∑

k=|l−l′|C

ll′(l+k)m′m(m′+m) C

ll′(l+k)n′n(n′+n) tl+k

(m′+m)(n′+n),

11.8. Multiplication formulae for representations of SU(2) 621

where Cll′(l+k)m′m(m′+m) are so-called Clebsch-Gordan coefficients, for which there are

explicit formulae [148]. We are going to derive basic multiplication formulae ontrigonometric polynomials tlmn : SU(2)→ C; for general multiplication of trigono-metric polynomials, one can use this computation iteratively.

Theorem 11.8.1. Let(t−− t−+

t+− t++

):= t1/2 =

(t1/2−1/2,−1/2 t

1/2−1/2,+1/2

t1/2+1/2,−1/2 t

1/2+1/2,+1/2

)and x± := x± 1/2 for x ∈ R. Then

tlmnt−− =

√(l −m + 1)(l − n + 1)

2l + 1tl

+

m−n− +

√(l + m)(l + n)

2l + 1tl

−m−n− ,

tlmnt++ =

√(l + m + 1)(l + n + 1)

2l + 1tl

+

m+n+ +

√(l −m)(l − n)

2l + 1tl

−m+n+ ,

tlmnt−+ =

√(l −m + 1)(l + n + 1)

2l + 1tl

+

m−n+ −√

(l + m)(l − n)2l + 1

tl−

m−n+ ,

tlmnt+− =

√(l + m + 1)(l − n + 1)

2l + 1tl

+

m+n− −√

(l −m)(l + n)2l + 1

tl−

m+n− .

Remark 11.8.2. Notice the pattern of the ± signs above!

Proof. It is enough to consider multiplication formulae for functions P lmn, because

by Theorem 11.7.1,

tlmn(φ, θ, ψ) = e−i(φm+ψn) P lmn(cos(θ)).

Moreover, by Corollary 11.7.5, the tlmnt−− formula implies the tlmnt++ formula,and the tlmnt−+ formula implies the tlmnt+− formula. Indeed, evaluating P belowat x = cos θ, we have

P lmn P

1/21/2,1/2 = P l

−m,−n P1/2−1/2,−1/2

=

√(l + m + 1)(l + n + 1)

2l + 1P l+

−m+,−n+ +

√(l −m)(l − n)

2l + 1P l−−m+,−n+

=

√(l + m + 1)(l + n + 1)

2l + 1P l+

m+,n+ +

√(l −m)(l − n)

2l + 1P l−

m+,n+

and

P lmn P

1/21/2,−1/2 = P l

−m,−n P1/2−1/2,1/2

=

√(l + m + 1)(l − n + 1)

2l + 1P l+

−m+,−n− +

√(l −m)(l + n)

2l + 1P l−−m+,−n−

=

√(l + m + 1)(l − n + 1)

2l + 1P l+

m+,n− +

√(l −m)(l + n)

2l + 1P l−

m+,n− .


Let us now prove the tlmnt−− formula. By Theorem 11.7.1, we have

P l+

m−n−(x) = cl+

m−n−(1− x)(n−m)/2

(1 + x)(m+n−1)/2

(ddx

)l−m+1 [(1− x)l−n+1(1 + x)l+n

],

= cl+

m−n−(1− x)(n−m)/2

(1 + x)(m+n−1)/2

(ddx

)l−m

× ddx

[(1− x)l−n+1(1 + x)l+n

],

where

ddx

[(1− x)l−n+1(1 + x)l+n

]= (1− x)l−n(1 + x)l+n−1 [−(l − n + 1)(1 + x) + (l + n)(1− x)]

= (1− x)l−n(1 + x)l+n−1 [−(2l + 1)(1 + x) + 2(l + n)]

= −(2l + 1) (1− x)l−n(1 + x)l+n + 2(l + n) (1− x)l−n(1 + x)l+n−1,

yielding

P l+

m−n−(x) = (2l + 1) cl+

m−n−P l

mn(x)clmn

P1/2−1/2,−1/2(x)

c1/2−1/2,−1/2

+2(l + n) cl+

m−n−P l−

m−n−(x)cl−m−n−

= (2l + 1)P l

mn(x) P1/2−1/2,−1/2(x)√

(l −m + 1)(l − n + 1)

−√

(l + m)(l + n)P l−

m−n−(x)√(l −m + 1)(l − n + 1)

.

Thereby

(2l + 1) P lmn(x) P

1/2−1/2,−1/2(x)

=√

(l −m + 1)(l − n + 1)P l+

m−n−(x) +√

(l + m)(l + n)P l−m−n−(x),

proving the tlmnt−− formula. The tlmnt−+ case is similar. Indeed, by Theorem11.7.1, we have

P l+

m−n+(x) = cl+

m−n+(1− x)(n−m+1)/2

(1 + x)(m+n)/2

(ddx

)l−m+1 [(1− x)l−n(1 + x)l+n+1

],

= cl+

m−n+(1− x)(n−m+1)/2

(1 + x)(m+n)/2

(ddx

)l−m ddx

[(1− x)l−n(1 + x)l+n+1

],

11.8. Multiplication formulae for representations of SU(2) 623

where

ddx

[(1− x)l−n(1 + x)l+n+1

]= (1− x)l−n−1(1 + x)l+n [−(l − n)(1 + x) + (l + n + 1)(1− x)]

= (1− x)l−n−1(1 + x)l+n [(2l + 1)(1− x)− 2(l − n)]

= (2l + 1) (1− x)l−n(1 + x)l+n − 2(l − n) (1− x)l−n−1(1 + x)l+n,

yielding

P l+

m−n+(x) = (2l + 1) cl+

m−n+P l

mn(x)clmn

P1/2−1/2,+1/2(x)

c1/2−1/2,+1/2

−2(l − n) cl+

m−n+

P l−m−n+(x)cl−m−n+

= (2l + 1)P l

mn(x) P1/2−1/2,+1/2(x)√

(l −m + 1)(l + n + 1)

+√

(l + m)(l − n)P l−

m−n+(x)√(l −m + 1)(l + n + 1)

.

Thereby

(2l + 1) P lmn(x) P

1/2−1/2,+1/2(x)

=√

(l −m + 1)(l + n + 1)P l+

m−n+(x)−√

(l + m)(l − n)P l−m−n+(x),

proving the tlmnt−+ formula. �Exercise 11.8.3. Calculate formulae for tlmnt++ and tlmnt+− in Theorem 11.8.1directly from definitions of tlmn, t++ and t+−.

Exercise 11.8.4. There are other forms of multiplication formulae that can be de-rived. For example, by using symmetries as in Corollary 11.7.5 derive the followingformulae:

tlmnt++ =

√(l −m + 1)(l − n + 1)

2l + 1tl

+

m−n− +

√(l + m)(l + n)

2l + 1tl

−m−n− ,

tlmnt+− =

√(l −m + 1)(l + n + 1)

2l + 1tl

+

m−n+ −√

(l + m)(l − n)2l + 1

tl−

m−n+ .


11.9 Laplacian and derivatives of representations

on SU(2)

Our aim now is to derive the formula for the Laplacian in terms of Euler anglesand determine its eigenvalues and eigenfunctions. We also want to find formulaefor derivatives of representations. First, recall the invariant differential operatorsD1, D2, D3 and their expressions in Euler angles in view of Proposition 11.5.9:

D1 := cos(ψ)∂

∂θ+

sin(ψ)sin(θ)

∂

∂φ− cos(θ)

sin(θ)sin(ψ)

∂

∂ψ,

D2 := − sin(ψ)∂

∂θ+

cos(ψ)sin(θ)

∂

∂φ− cos(θ)

sin(θ)cos(ψ)

∂

∂ψ,

D3 :=∂

∂ψ.

The Laplacian L is given by L = D21 + D2

2 + D23 and we have [L, Dj ] = 0, see

Remark 11.5.11. Therefore, we see that D23 = ∂2/∂ψ2, as well as calculate

D21 = cos ψ

(cos ψ

∂2

∂θ2+− cos θ sinψ

sin(θ)2∂

∂φ+

sin ψ

sin θ

∂2

∂θ ∂φ

−− sin(θ)2 − cos(θ)2

sin(θ)2sinψ

∂

∂ψ− cos θ

sin θsinψ

∂2

∂θ ∂ψ

)+

sin ψ

sin θ

(cos ψ

∂2

∂θ ∂φ+

sin ψ

sin θ

∂2

∂φ2− cos θ

sin θsinψ

∂2

∂φ ∂ψ

)−cos θ

sin θsinψ

(− sinψ

∂

∂θ+ cos ψ

∂2

∂θ ∂ψ

+cos ψ

sin θ

∂

∂φ+

sin ψ

sin θ

∂2

∂φ ∂ψ

−cos θ

sin θcos ψ

∂

∂ψ− cos θ

sin θsinψ

∂2

∂ψ2

),

so that after cancellations we get

D21 = cos(ψ)2

∂2

∂θ2− 2

cos ψ cos θ sin ψ

sin(θ)2∂

∂φ+

2 cos ψ sin ψ

sin θ

∂2

∂θ ∂φ

+(1 + cos(θ)2) cos ψ sin ψ

sin(θ)2∂

∂ψ− 2

cos ψ cos θ

sin θsinψ

∂2

∂θ ∂ψ

+sin(ψ)2

sin(θ)2∂2

∂φ2− 2

cos θ

sin(θ)2sin(ψ)2

∂2

∂φ ∂ψ

+cos(θ)sin(θ)

sin(ψ)2∂

∂θ

+cos(θ)2

sin(θ)2sin(ψ)2

∂2

∂ψ2.

11.9. Laplacian and derivatives of representations on SU(2) 625

Similarly, we have

D22 = − sinψ

(− sin ψ

∂2

∂θ2

+− cos θ cos ψ

sin(θ)2∂

∂φ+

cos ψ

sin θ

∂2

∂θ ∂φ

−− sin(θ)2 − cos(θ)2

sin(θ)2cos ψ

∂

∂ψ− cos θ

sin θcos ψ

∂2

∂θ ∂ψ

)+

cos ψ

sin θ

(− sinψ

∂2

∂θ ∂φ+

cos ψ

sin θ

∂2

∂φ2− cos θ

sin θcos ψ

∂2

∂φ ∂ψ

)−cos θ

sin θcos ψ

(− cos ψ

∂

∂θ− sin ψ

∂2

∂θ ∂ψ

+− sinψ

sin θ

∂

∂φ+

cos ψ

sin θ

∂2

∂φ ∂ψ

+cos θ

sin θsinψ

∂

∂ψ− cos θ

sin θcos ψ

∂2

∂ψ2

),

and after cancellations we get

D22 = sin(ψ)2

∂2

∂θ2+ 2

cos θ cos ψ sin ψ

sin(θ)2∂

∂φ− 2

cos ψ sinψ

sin θ

∂2

∂θ ∂φ

−1 + cos(θ)2

sin(θ)2cos ψ sin ψ

∂

∂ψ+ 2

cos θ

sin θcos ψ sinψ

∂2

∂θ ∂ψ

+cos(ψ)2

sin(θ)2∂2

∂φ2− 2

cos θ

sin(θ)2cos(ψ)2

∂2

∂φ ∂ψ

+cos θ

sin θcos(ψ)2

∂

∂θ

+cos(θ)2

sin(θ)2cos(ψ)2

∂2

∂ψ2.

Hence we obtain

Proposition 11.9.1 (Laplacian on SU(2)). In terms of Euler angles, the Laplacianon SU(2) is given by

L = D21 + D2

2 + D23

=∂2

∂θ2+

1sin(θ)2

∂2

∂φ2− 2

cos θ

sin(θ)2∂2

∂φ ∂ψ+

1sin(θ)2

∂2

∂ψ2

+cos θ

sin θ

∂

∂θ.

We now determine derivatives of representations.


Proposition 11.9.2 (Derivatives D1, D2, D3 of tlmn). We have

D1tlmn =

√(l − n)(l + n + 1)

−2itlm,n+1 +

√(l + n)(l − n + 1)

−2itlm,n−1,

D2tlmn =

√(l − n)(l + n + 1)

2tlm,n+1 −

√(l + n)(l − n + 1)

2tlm,n−1,

D3tlmn = −in tlmn.

Proof. Recall that

Djf(u) =ddt

f(u ωj(t))|t=0,

where f : SU(2) → C. Recall also that tlmn is a matrix element of the irreducibleunitary representation Tl : SU(2)→ End(Vl), acting by

Tl(u)v(z) = v(zu),

where v ∈ Vl is a homogeneous polynomial v : C2 → C of order 2l ∈ N0. Matrixelements tlmn : SU(2)→ C of Tl with respect to the basis

{qlk | k ∈ {−l,−l + 1, . . . ,+l − 1,+l}}

satisfyqln(zu) = Tl(u)qln(z) =

∑m

tlmn(u) qlm(z),

where

qlm(z) =zl−m1 zl+m

2√(l −m)! (l + m)!

.

Especially,

Djqln(zu) := Dj (u �→ qln(zu)) =∑m

(Djt

lmn

)(u) qlm(z).

First,

D3qln(zu) =ddt

qln(z u ω3(t))∣∣∣∣t=0

=ddt

((zu)1 eit/2

)l−n ((zu)2 e−it/2

)l+n√(l − n)! (l + n)!

∣∣∣∣∣t=0

=ddt

e−itn∣∣t=0

(zu)l−n1 (zu)l+n

2√(l − n)! (l + n)!

= −in qln(zu).

TherebyD3t

lmn(u) = −in tlmn(u).

11.9. Laplacian and derivatives of representations on SU(2) 627

Next,

D2qln(zu)

=ddt

qln(z u ω2(t))∣∣∣∣t=0

=ddt

((zu)1 cos t

2 + (zu)2 sin t2

)l−n (−(zu)1 sin t

2 + (zu)2 cos t2

)l+n√(l − n)! (l + n)!

∣∣∣∣∣t=0

=(l − n)(zu)2(zu)l−n−1

1 (zu)l+n2 − (zu)l−n

1 (l + n)(zu)1(zu)l+n−12

2√

(l − n)! (l + n)!

=

√(l − n)(l + n + 1)ql,n+1(zu)−

√(l + n)(l − n + 1)ql,n−1(zu)

2.

Thus D2tlmn(u) equals√

(l − n)(l + n + 1)tlm,n+1(u)−√

(l + n)(l − n + 1)tlm,n−1(u)2

.

Finally,

D1qln(zu)

=ddt

qln(z u ω1(t))∣∣∣∣t=0

=ddt

((zu)1 cos t

2 + (zu)2i sin t2

)l−n ((zu)1i sin t

2 + (zu)2 cos t2

)l+n√(l − n)! (l + n)!

∣∣∣∣∣t=0

=(l − n)(zu)2(zu)l−n−1

1 (zu)l+n2 + (zu)l−n

1 (l + n)(zu)1(zu)l+n−12

−2i√

(l − n)! (l + n)!

=

√(l − n)(l + n + 1)ql,n+1(zu) +

√(l + n)(l − n + 1)ql,n−1(zu)

−2i.

Thus D1tlmn(u) equals√

(l − n)(l + n + 1)tlm,n+1(u) +√

(l + n)(l − n + 1)tlm,n−1(u)−2i

,


In the sequel, we will work with operators ∂+, ∂−, ∂0 rather than with D1,D2, D3, and the relation between them was given in Definition 11.5.10, which werecall: ⎧⎪⎨⎪⎩

∂+ := iD1 −D2,

∂− := iD1 + D2,

∂0 := iD3,

, i.e.,

⎧⎪⎨⎪⎩D1 = −i

2 (∂− + ∂+) ,

D2 = 12 (∂− − ∂+) ,

D3 = −i∂0.

(11.16)


Theorem 11.9.3 (Derivatives ∂+, ∂−, ∂0 and Laplacian L of tlmn). We have

∂+tlmn = −√

(l − n)(l + n + 1) tlm,n+1,

∂−tlmn = −√

(l + n)(l − n + 1) tlm,n−1,

∂0tlmn = n tlmn,

Ltlmn = −l(l + 1) tlmn.

Proof. Formulae for ∂+, ∂−, ∂0 follow from formulae in Proposition 11.9.2 andformulae (11.16). Since by Remark 11.5.11 we have

L = −∂20 − (∂+∂− + ∂−∂+)/2,

we get

Ltlmn

= −n2 tlmn +12

(√(l + n)(l − n + 1) ∂+tlm,n−1

+√

(l − n)(l + n + 1) ∂−tlm,n+1

)=

−12

(2n2 +

√(l + n)(l − n + 1)

√(l − (n− 1))(l + (n− 1) + 1)

+√

(l − n)(l + n + 1)√

(l + (n + 1))(l − (n + 1) + 1))

tlmn

=−12

(n2 + (l + n)(l − n + 1) + (l − n)(l + n + 1)

)tlmn

=−12

(2n2 + 2(l2 − n2) + (l + n) + (l − n)

)tlmn

= −l(l + 1) tlmn,

completing the proof. �Remark 11.9.4. In Proposition 11.9.2 we saw that

D1tlmn =

√(l − n)(l + n + 1) tlm,n+1 +

√(l + n)(l − n + 1) tlm,n−1

−2i,

D2tlmn =

√(l − n)(l + n + 1) tlm,n+1 −

√(l + n)(l − n + 1) tlm,n−1

2,

D3tlmn = −in tlmn,

so that

D1tlmn =

√(l − n)(l + n + 1) tlm,n+1 +

√(l + n)(l − n + 1) tlm,n−1

+2i,

D2tlmn =

√(l − n)(l + n + 1) tlm,n+1 −

√(l + n)(l − n + 1) tlm,n−1

2,

D3tlmn = +in tlmn,

11.10. Fourier series on SU(2) and on SO(3) 629

implying

∂+tlmn = +√

(l + n)(l − n + 1) tlm,n−1,

∂−tlmn = +√

(l − n)(l + n + 1) tlm,n+1,

∂0tlmn = −n tlmn.

11.10 Fourier series on SU(2) and on SO(3)

By the Peter–Weyl theorem (Theorem 7.5.14), an orthogonal basis for L2(SU(2))consists of functions tlnm for l ∈ N0/2 and −l ≤ m,n ≤ l, where l −m, l − n ∈ Zand

tlnm(ω(φ, θ, ψ)) = e−i(nφ+mψ) P lnm(cos(θ)),

and where

P lnm(z) = 2−l (−1)l−m im−n√

(l −m)!(l + m)!

√(l + n)!(l − n)!

× (1− z)m−n

2

(1 + z)n+m

2

dl−n

dzl−n

[(1− z)l−m(1 + z)l+m

].

Note that here we changed the order of indices (m,n) into (n, m) compared withthe formulation of the Peter–Weyl theorem in order to have (m,n) entries for theFourier coefficients in (11.17) and in the sequel.

Notice that for SO(3), these same formulae are valid with appropriate Eu-ler’s angles, but then l ∈ N0 (not l ∈ N0/2). Nevertheless, tl(u) ∈ U(2l + 1) ⊂C(2l+1)×(2l+1). For instance,

T0(u(φ, θ, ψ)) =(1),

T1/2(u(φ, θ, ψ)) =(

ei(φ+ψ)/2 cos θ2 ei(φ−ψ)/2i sin θ

2


2

)= u(φ, θ, ψ) =

(a bc d

),

T1(u(φ, θ, ψ)) =

⎛⎜⎝ ei(φ+ψ) cos2 θ2 eiφ i sin θ√

2−ei(φ−ψ) sin2 θ

2

eiψ i sin θ√2

cos θ e−iψ i sin θ√2

−e−i(φ−ψ) sin2 θ2 e−iφ i sin θ√

2e−i(φ+ψ) cos2 θ

2

⎞⎟⎠=

⎛⎝ a2√

2ab b2√

2ac ad + bc√

2bd

c2√

2cd d2

⎞⎠ .

Consequently, the collection

{√

2l + 1 tlnm : l ∈ 12

N0, −l ≤ m,n ≤ l, l −m, l − n ∈ Z}


is an orthonormal basis for L2(SU(2)), and thus by the Peter–Weyl theorem (The-orem 7.5.14) any function f ∈ C∞(SU(2)) has a Fourier series representation

f(x) =∑

l∈ 12 N0

(2l + 1)∑m

∑n

f(l)mn tlnm(x), (11.17)

where the Fourier coefficients are computed by

f(l)mn :=∫

SU(2)

f(x) tlnm(x) dx = 〈f, tlnm〉L2(SU(2)),

so that f(l) ∈ C(2l+1)×(2l+1). The series (11.17) converges almost everywhere onSU(2) as well as in L2(SU(2)).

Definition 11.10.1 (Quantum numbers, notation f(l), and summation on SU(2)).In the case of SU(2), we simplify the notation writing f(l) instead of f(tl), etc.In sums (11.17), our convention will be that summations

∑m

∑n are over m,n

such that −l ≤ m,n ≤ l and l −m, l − n ∈ Z. The index l is called the quantumnumber.

On SO(3), the collection

{√

2l + 1 tlnm : l ∈ N0, m, n ∈ Z, −l ≤ m,n ≤ l}

is an orthonormal basis for L2(SO(3)), and thus f ∈ C∞(SO(3)) has a Fourierseries representation

f(x) =∞∑

l=0

(2l + 1)l∑

m=−l

l∑n=−l

f(l)mn tlnm(x),


f(l)mn := 〈f, tlnm〉L2(SO(3)) =∫

SO(3)

f(x) tlnm(x) dμSO(3)(x).

Notice that by Remark 10.2.3 we have

tlnm(x) = (tl(x)∗)mn = tlmn(x−1).

Evidently, the values of f ∈ C∞(S2) ⊂ C∞(SO(3)) do not depend on the Eulerangle ψ, so that in this case f(l)nm = 0 whenever n = 0.

Chapter 12

Pseudo-differential Operatorson SU(2)

In this chapter we carry out the analysis of operators on SU(2) with an applica-tion to the operators on the three-dimensional sphere S3. In particular, we derive amuch simpler symbolic characterisation of pseudo-differential operators on SU(2)than the one given in Definition 10.9.5. In turn, this will also yield a charac-terisation of full symbols of pseudo-differential operators on the 3-sphere S3. Wenote that this approach works globally on the whole sphere, since the version ofthe Fourier analysis that we use is different from the one in, e.g., [110, 122, 111]which covers only a hemisphere, with singularities at the equator. For a generalintroduction and motivation for the analysis on SU(2) we refer the reader to theintroduction in Part IV where the cases of SU(2) and S3 were put in a perspective.

On SU(2), the conventional abbreviations in summation indices are∑l

=∑

l∈ 12 N0

,∑

l

∑m,n

=∑

l∈ 12 N0

∑|m|≤l, l+m∈Z

∑|n|≤l, l+n∈Z

,

where N0 = {0} ∪ N = {0, 1, 2, . . . }. As before, the space of all linear mappingsfrom a finite-dimensional (inner-product) vector space H to itself is denoted byEnd(H), a mapping U ∈ L(H) is called unitary if U∗ = U−1, and the space of allunitary linear mappings on a finite-dimensional inner product space H is denotedby U(H).

12.1 Symbols of operators on SU(2)

First we summarise the approach to symbols from Section 10.4 in the case ofSU(2), also simplifying the notation in this case. We recall that in the case ofSU(2), we simplify the notation writing f(l) instead of f(tl), etc.

632 Chapter 12. Pseudo-differential Operators on SU(2)

By the Peter–Weyl theorem (Theorem 7.5.14) the collection

{√

2l + 1 tlnm : l ∈ 12

N0, −l ≤ m,n ≤ l, l −m, l − n ∈ Z} (12.1)

is an orthonormal basis for L2(SU(2)), and thus f ∈ C∞(SU(2)) has a Fourierseries representation

f(x) =∑

l∈ 12 N0

(2l + 1)∑m

∑n

f(l)mn tlnm(x)

=∑

l∈ 12 N0

(2l + 1) Tr(f(l) tl(x)

),


f(l)mn :=∫

SU(2)

f(x) tlnm(x) dx = 〈f, tlnm〉L2(SU(2)),

so that f(l) ∈ C(2l+1)×(2l+1). We recall that tl ∈ C(2l+1)×(2l+1) by Definition11.10.1 is a matrix with components tlnm, with the convention that indices m andn vary as in (12.1).

Let A : C∞(SU(2)) → C∞(SU(2)) be a continuous linear operator and letRA ∈ D′(SU(2)× SU(2)) be its right-convolution kernel, i.e.,

Af(x) =∫

SU(2)

f(y) RA(x, y−1x) dy = (f ∗RA(x, ·))(x)

in the sense of distributions. According to Definition 10.4.3, by the symbol of Awe mean the sequence of matrix-valued mappings

(x �→ σA(x, l)) : SU(2)→ C(2l+1)×(2l+1),

where 2l ∈ N0, obtained from

σA(x, l)mn =∫

SU(2)

RA(x, y) tlnm(y) dy. (12.2)

That is, σA(x, l) is the lth Fourier coefficient of the function y �→ RA(x, y). Thenby Theorem 10.4.4 we have

Af(x) =∑

l

(2l + 1) Tr(tl(x) σA(x, l) f(l)

)=

∑l

(2l + 1)∑m,n

tl(x)nm

(∑k

σA(x, l)mk f(l)kn

).

12.1. Symbols of operators on SU(2) 633

Alternatively, by Theorem 10.4.6 we have

σA(x, l) = tl(x)∗(Atl

)(x), (12.3)

that isσA(x, l)mn =

∑k

tlkm(x)(Atlkn)(x), (12.4)

by formula (10.21). Formula (10.20) expressing the right-convolution kernel interms of the symbol becomes

RA(x, y) =∑

l

(2l + 1)∑m,n

tlnm(y) σA(x, l)mn, (12.5)

with a similar distributional interpretation for the series. In the case of SU(2), thequantity

⟨tl

⟩in (10.9) for the representation ξ = tl can be calculated as⟨

tl⟩

= (1 + λ[tl])1/2 = (1 + l(l + 1))1/2,

in view of Theorem 11.9.3. Consequently, Definition 10.9.5 of the symbol classΣm(SU(2)) becomes:

Definition 12.1.1. We write that σA ∈ Σm0 (SU(2)) if

sing supp (y �→ RA(x, y)) ⊂ {e} (12.6)

and if ∥∥�αl ∂β

xσA(x, l)∥∥

C2l+1→C2l+1 ≤ CAαβm (1 + l)m−|α| (12.7)

for all x ∈ G, all multi-indices α, β, and l ∈ 12N0. Here

∂βx = ∂β1

0 ∂β2+ ∂β3

− and �αl = �α1

0 �α2+ �α3−

are defined in the general situation in Definition 10.7.1, but in Section 12.3 we dis-cuss the simplification of these difference operators. Moreover, σA ∈ Σm

k+1(SU(2))if and only if

σA ∈ Σmk (SU(2)), (12.8)

[σ∂j, σA] = σ∂j

σA − σAσ∂j∈ Σm

k (SU(2)), (12.9)

(�γl σA) σ∂j

∈ Σm+1−|γ|k (SU(2)), (12.10)

for all |γ| > 0 and j ∈ {0,+,−}. Let

Σm(SU(2)) :=∞⋂

k=0

Σmk (SU(2)),

so that by Theorem 10.9.6 we have OpΣm(SU(2)) = Ψm(SU(2)).


Remark 12.1.2. The ordering of operators ∂0, ∂+, ∂− in ∂βx = ∂β1

0 ∂β2+ ∂β3

− mayseem to be of importance because they do not commute (see Exercise 11.5.12).However, this is not really an issue in view of Remark 10.5.4: indeed applying theircommutator relations from Exercise 11.5.12 iteratively, we see that we can takeany ordering in ∂β

x in (12.7) to obtain the same class of symbols.

Remark 12.1.3. We would like to provide a more direct definition for the sym-bol class Σm(SU(2)), without resorting to classes Σm

k (SU(2)). Condition (12.7) isjust an analogy of the usual symbol inequalities. Conditions (12.6) and (12.8) arestraightforward. We may have difficulties with differences �α

l , but derivatives ∂βx

do not cause problems; if we want, we may assume that the symbols are constant inx. In Section 12.4 we present such a simplification, thus providing a more straight-forward characterisation of operators from Ψm(SU(2)) in terms of quantizationsand full symbols developed here.

Exercise 12.1.4. From the definition of operators �αl and ∂β

x , verify the followingproperties:

�αl ∂β

xσA(x, l) = ∂βx�α

l σA(x, l),∂j (σA(x, l) σB(x, l)) = (∂jσA(x, l))σB(x, l) + σA(x, l) ∂jσA(x, l),

∂βy (σA(x, l) σB(y, l) σC(z, l)) = σA(x, l)

(∂β

y σB(y, l))σC(z, l).

12.2 Symbols of ∂+, ∂−, ∂0 and Laplacian LIn this section we calculate symbols of the creation, annihilation and neutral op-erators ∂+, ∂−, ∂0, and of the Laplacian L. We will use the fact that the symbolof the operator A is obtained by

σA(x, l) = tl(x)∗ (Atl)(x),

that is, σA(x, l)mn =∑

k tlkm(x) (Atlkn)(x), see (12.3) and (12.4).

Theorem 12.2.1. We have

σ∂+(x, l)mn = −√

(l − n)(l + n + 1) δm,n+1

= −√

(l −m + 1)(l + m) δm−1,n,

σ∂−(x, l)mn = −√

(l + n)(l − n + 1) δm,n−1

= −√

(l + m + 1)(l −m) δm+1,n,

σ∂0(x, l)mn = n δmn = m δmn,

σL(x, l)mn = −l(l + 1) δmn,

where δmn is the Kronecker delta: δmn = 1 for m = n and, δmn = 0 otherwise.

12.2. Symbols of ∂+, ∂−, ∂0 and Laplacian L 635

Proof. Let e ∈ SU(2) be the neutral element of SU(2) and let tl be a unitarymatrix representation of SU(2). First we note that

δmn = tl(e)mn = tl(x−1x)mn =∑

k

tl(x−1)mk tl(x)kn =∑

k

tl(x)km tl(x)kn,

see also Remark 10.2.11 for the general form of this identity. Similarly, we notethe identity

δmn =∑

k

tl(x)mk tl(x)nk.

From this, formula (12.4), and Theorem 11.9.3 we get

σ∂+(x, l)mn =∑

k

tlkm(x)(∂+tlkn

)(x)

= −√

(l − n)(l + n + 1)∑

k

tlkm(x) tlk,n+1(x)

= −√

(l − n)(l + n + 1) δm,n+1,

and the case of σ∂−(x, l) is analogous:

σ∂−(x, l)mn =∑

k

tlkm(x)(∂−tlkn

)(x)

= −√

(l + n)(l − n + 1)∑

k

tlkm(x) tlk,n−1(x)

= −√

(l + n)(l − n + 1) δm,n−1,

Finally,

σ∂0(x, l)mn =∑

k

tlkm(x)(∂0t

lkn

)(x) = n

∑k

tlkm(x) tlkn(x) = n δm,n,

and similarly for L. �

Exercise 12.2.2. Complete the proof of Theorem 12.2.1 for the symbol σL of theLaplacian L.

Remark 12.2.3. Notice that σ∂0(x, l) and σL(x, l) are diagonal matrices. The non-zero elements reside just above the diagonal of σ∂+(x, l), and just below the diago-nal of σ∂−(x, l). Because of this the operators ∂0, ∂+ and ∂− may be called neutral,creation and annihilation operators, respectively, and this explains our preferenceto work with them rather than with Dj ’s, which have more non-zero entries.

Finally we note that vector fields D1, D2, D3 related to ∂+, ∂−, ∂0 by Defini-tion 11.5.10 can be conjugated as follows (recall the definition of conjugations andtheir properties, e.g., in Proposition 10.4.18):


Proposition 12.2.4 (Conjugation of D1, D2, D3 and their symbols). We have

(D3)(w1)R= D2, (D1)(w2)R

= D3, (D2)(w3)R= D1. (12.11)

The symbols of the operators D1, D2 can be transformed to that of D3 by takingsuitable conjugations:

σD1(x, l) = tl(w2) σD3(x, l) tl(w2)∗, (12.12)σD2(x, l) = tl(w1)∗ σD3(x, l) tl(w1). (12.13)

Moreover, if D ∈ su(2) there exists u ∈ SU(2) such that

σD(l) = tl(u)∗ σD3(l) tl(u).

Proof. Combining Lemma 10.4.19 with Proposition 11.5.7, we see that formulae(12.11) hold. Since D1, D2, D3 are left-invariant operators, their symbols σDj

(x, l)do not depend on x ∈ G, and by Proposition 10.4.18 we obtain (12.12) and (12.13).The last statement follows from Proposition 10.4.18 since D is a rotation of D3.

�

12.3 Difference operators for symbols

We are now going to introduce “difference operators” �+,�−,�0 acting on sym-bols of operators on SU(2) that resemble first-order forward and backward dif-ferences �α

ξ acting on symbols σA(x, ξ) on a torus, where we had ξ ∈ Zn (seeDefinition 3.3.1). Then we will apply these differences to first-order differentialoperators, as well as to products of special type.

12.3.1 Difference operators on SU(2)

From Theorem 11.7.1 and Theorem 11.8.1 we recall the notation

t1/2 =(

t−− t−+

t+− t++

)=

(t1/2−1/2,−1/2 t

1/2−1/2,+1/2

t1/2+1/2,−1/2 t

1/2+1/2,+1/2

)

=(

cos(θ/2) ei(+φ+ψ)/2 i sin(θ/2) ei(+φ−ψ)/2

i sin(θ/2) ei(−φ+ψ)/2 cos(θ/2) ei(−φ−ψ)/2

).

Definition 12.3.1 (Differences �q for Fourier coefficients). For q ∈ C∞(SU(2))and f ∈ D′(SU(2)), let

�q f(l) := qf(l).

We shall use abbreviations

�+ = �q+ ,�− = �q− and �0 = �q0 ,

12.3. Difference operators for symbols 637

where

q− := t−+ = t1/2−1/2,+1/2,

q+ := t+− = t1/2+1/2,−1/2,

q0 := t−− − t++ = t1/2−1/2,−1/2 − t

1/2+1/2,+1/2.

Thus each trigonometric polynomial q+, q−, q0 ∈ C∞(SU(2)) vanishes at theneutral element e ∈ SU(2). In this sense trigonometric polynomials q− + q+, q− −q+, q0 on SU(2) are analogues of polynomials x1, x2, x3 in the Euclidean space R3.

Now our aim is to let these difference operators act on symbols. For this pur-pose we may only look at symbols independent of x corresponding to left-invariant(right-convolution) operators since the following construction is independent of x:

Definition 12.3.2 (Differences �+,�−,�0 acting on symbols). Let a = a(ξ) bea symbol as in Definition 10.4.3. It follows that a = s for some right-convolutionkernel s ∈ D′(SU(2)) so that the operator Op(a) is given by

Op(a)f = f ∗ s.

We define “difference operators” �+,�−,�0 acting on the symbol a by

�+a := q+ s, (12.14)�−a := q− s, (12.15)�0a := q0 s. (12.16)

Obviously, Definitions 12.3.1 and 12.3.2 are consistent. We note once morethat this construction is analogous to the one producing usual derivatives in Rn

or difference operators on the torus Tn.To analyse the structure of difference operators on SU(2), we first need to

know how to multiply functions tlmn by q+, q−, q0, and the necessary formulae aregiven in Theorem 11.8.1. We recall the notation

t1/2 =(

t−− t−+

t+− t++

)=

(t−− q−q+ t++

), q0 = t−− − t++,

and summarise the multiplication formulae as follows:

Corollary 12.3.3. For x ∈ R, let x± := x± 1/2. Then

(2l + 1)q+tlmn = +√

(l + m + 1)(l − n + 1)tl+

m+n− −√

(l −m)(l + n)tl−

m+n− ,

(2l + 1)q−tlmn = +√

(l −m + 1)(l + n + 1)tl+

m−n+ −√

(l + m)(l − n)tl−

m−n+ ,

(2l + 1)q0tlmn = +

√(l −m + 1)(l − n + 1)tl

+

m−n− +√

(l + m)(l + n)tl−

m−n−

−√

(l + m + 1)(l + n + 1)tl+

m+n+ −√

(l −m)(l − n)tl−

m+n+ .


We also recall the relation between symbols and kernels which follows from(12.2) and (12.5), and where we switch the order of m and n to adopt it to theproof of Theorem 12.3.5:

Corollary 12.3.4 (Kernel and symbol). Symbol a(x, l) = a(l) and kernel s(x) of anoperator are related by

a(x, l)nm = alnm = s(l)nm =

∫SU(2)

s(y) tlmn(y) dy,

and

s(x) =∑

l

(2l + 1) Tr(a(x, l) tl(x)

)=

∑l

(2l + 1)∑m,n

alnm tlmn. (12.17)

Let us now derive explicit expressions for the first-order difference operators�+, �−, �0 defined in (12.14)–(12.16). To abbreviate the notation, we will alsowrite al

nm = a(x, l)nm, even if symbol a(x, l) depends on x, keeping in mind thatthe following theorem holds pointwise in x.

Theorem 12.3.5 (Formulae for difference operators �+,�−,�0). The differenceoperators are given by

(�−a)lnm =

√(l −m)(l + n)

2l + 1al−

n−m+ −√

(l + m + 1)(l − n + 1)2l + 1

al+

n−m+ ,

(�+a)lnm =

√(l + m)(l − n)

2l + 1al−

n+m− −√

(l −m + 1)(l + n + 1)2l + 1

al+

n+m− ,

(�0a)lnm =

√(l −m)(l − n)

2l + 1al−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1al+

n+m+

−√

(l + m)(l + n)2l + 1

al−n−m− −

√(l −m + 1)(l − n + 1)

2l + 1al+

n−m− ,

where k± = k ± 12 , and satisfy commutator relations

[�0,�+] = [�0,�−] = [�−,�+] = 0. (12.18)

Proof. Identities (12.18) follow immediately from (12.14)–(12.16). As discussedbefore, we can abbreviate a(x, l) by a(l) since none of the arguments in the proofwill act on the variable x. In the proof we will heavily rely on the relation betweenkernels and symbols in Proposition 12.3.4. Also, in the calculation below we willnot worry about boundaries of summations, keeping in mind that we can alwaysview finite matrices as infinite ones simply by extending them by zeros. Recallingthat q− = t−+ and using Theorem 11.8.1, we can calculate

q− s(12.17)

=∑

l

(2l + 1)∑m,n

alnm q− tlmn


Cor 12.3.3=∑

l

∑m,n

alnm

[tl

+

m−n+

√(l −m + 1)(l + n + 1)

−tl−

m−n+

√(l + m)(l − n)

]=

∑l

∑m,n

tlmn

[al−

n−m+

√(l −m)(l + n)

−al+

n−m+

√(l + m + 1)(l − n + 1)

].

Since �−a = q−s, we obtain the desired formula for �−:

(�−a)lnm =

√(l −m)(l + n)

2l + 1al−

n−m+ −√

(l + m + 1)(l − n + 1)2l + 1

al+

n−m+ .

Similarly, for �+, we calculate

q+ s =∑

l

(2l + 1)∑m,n

alnm q+ tlmn

=∑

l

∑m,n

alnm

[tl

+

m+n−√

(l + m + 1)(l − n + 1)

−tl−

m+n−√

(l −m)(l + n)]

=∑

l

∑m,n

tlmn

[al−

n+m−√

(l + m)(l − n)

−al+

n+m−√

(l −m + 1)(l + n + 1)].

From this we obtain the desired formula for �+:

(�+a)lnm =

√(l + m)(l − n)

2l + 1al−

n+m− −√

(l −m + 1)(l + n + 1)2l + 1

al+

n+m− .

Finally, for �0, we calculate

q0 s =∑

l

(2l + 1)∑m,n

alnm q0 tlmn

=∑

l

∑m,n

alnm

[tl

+

m−n−√

(l −m + 1)(l − n + 1) + tl−

m−n−√

(l + m)(l + n)

− tl+

m+n+

√(l + m + 1)(l + n + 1)− tl

−m+n+

√(l −m)(l − n)

]=

∑l

∑m,n

tlmn

[al−

n+m+

√(l −m)(l − n) + al+

n+m+

√(l + m + 1)(l + n + 1)

−al−n−m−

√(l + m)(l + n)− al+

n−m−√

(l −m + 1)(l − n + 1)].


From this we obtain the desired formula for �0:

(�0a)lnm =

√(l −m)(l − n)

2l + 1al−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1al+

n+m+

−√

(l + m)(l + n)2l + 1

al−n−m− −

√(l −m + 1)(l − n + 1)

2l + 1al+

n−m− ,

and the proof of Theorem 12.3.5 is complete. �

12.3.2 Differences for symbols of ∂+, ∂−, ∂0 and Laplacian LRemark 10.7.3 and Proposition 10.7.4 said that application of difference operatorsreduces the order of the symbol of differential operators. However, for operators∂+, ∂−, ∂0 and for the Laplacian L we calculate this now more explicitly:

Theorem 12.3.6. We have

σI = �+σ∂+ = �−σ∂− = �0σ∂0 . (12.19)

If μ, ν ∈ {+,−, 0} are such that μ = ν, then

�μσ∂ν= 0, (12.20)

and for every ν ∈ {+,−, 0}, we have

�νσI(x) = 0. (12.21)

Moreover, if L is the bi-invariant Laplacian, then

�+σL = −σ∂− , �−σL = −σ∂+ , �0σL = −2σ∂0 . (12.22)

The proof of this theorem will depend on explicit calculations. In trying tosimplify the presentation, we prove this theorem in the form of several propositionsdealing with different parts of the statement. We recall that the symbols of thefirst-order partial differential operators ∂+, ∂−, ∂0 have many zero elements, andaltogether they are as in Theorem 12.2.1:

σ∂+(l)mn =

{−

√(l − n)(l + n + 1), if m = n + 1,

0, otherwise.(12.23)

σ∂−(l)mn =

{−

√(l + n)(l − n + 1), if m = n− 1,

0, otherwise.

σ∂0(l)mn =

{n, if m = n,

0, otherwise.


Proposition 12.3.7. We have identities (12.19), i.e.,

σI = �+σ∂+ = �−σ∂− = �0σ∂0 .

Proof. From Corollary 12.3.3 we get an expression for q+tlmn, which is used in thefollowing calculation together with (12.17) in Corollary 12.3.4 and Theorem 12.2.1:

q+ s∂+

(12.17)= q+

∑l

(2l + 1)∑m,n

σ∂+(l)mn tlnm

(12.23)=

∑l

∑n

σ∂+(l)n+1,n (2l + 1) q+ tln,n+1

Cor 12.3.3= −∑

l

∑n

(√(l − n)(l + n + 1)

)2 (tl

+

n+n+ − tl−

n+n+

)(12.24)

=∑

l

(2l + 1)∑

k

tlkk

(12.17)= δe

= sI ,

where we made a change in indices and used the identity

(l− − n)(l− + n + 1)− (l+ − n)(l+ + n + 1) = −2l − 1. (12.24)

Hence �+σ∂+ = σI . Similarly, �−σ∂− = σI , because

q− s∂− = q−∑

l

(2l + 1)∑m,n

σ∂−(l)mn tlnm

=∑

l

∑n

σ∂−(l)n−1,n (2l + 1) q− tln,n−1

=∑

l

∑n

−(√

(l − n + 1)(l + n))2 (

tl+

n−n− − tl−

n−n−

)=

∑l

(2l + 1)∑

k

tlkk

= δe

= sI .

Moreover,

q0 s∂0 = q0

∑l

(2l + 1)∑m,n

σ∂0(l)mn tlnm

=∑

l

∑n

σ∂0(l)nn (2l + 1) q0 tlnn


=∑

l

∑n

n(+(l − n + 1) tl

+

n−n− + (l + n) tl−

n−n−

−(l + n + 1) tl+

n+n+ − (l − n) tl−

n+n+

)=

∑l

∑n

(tl

+

n+n+ ((n + 1)(l − n)− n(l + n + 1))

+tl−

n+n+ ((n + 1)(l + n + 1)− n(l − n)))

=∑

l

∑n

(tl

+

n+n+ (l − 2n2 − 2n)

+tl−

n+n+ (l + 1 + 2n2 + 2n))

=∑

l

∑n

tl+

n+n+

((l − 2n2 − 2n) + (l + 2 + 2n2 + 2n)

)=

∑l

(2l + 2)∑

n

tl+

n+n+

=∑

l

(2l + 1)∑

n

tlnn

= δe

= sI . �Proposition 12.3.8. We have identities (12.20), i.e.,

�μσ∂ν= 0,

where μ, ν ∈ {+,−, 0} such that μ = ν.


q+ s∂− = q+

∑l

(2l + 1)∑m,n

σ∂−(l)mn tlnm

=∑

l

∑n

σ∂−(l)n−1,n (2l + 1) q+ tln,n−1

=∑

l

∑n

−√

(l + n)(l − n + 1)

×(√

(l + n + 1)(l − n + 2) tl+1/2n+1/2,n−3/2

−√

(l − n)(l + n− 1) tl−1/2n+1/2,n−3/2

)=

∑l

∑n

tl+1/2n+1/2,n−3/2

×(−

√(l + n + 1)(l − n + 2)

√(l − n + 1)(l + n)

+√

(l + n)(l − n + 1)√

(l + n + 1)(l − n + 2))

= 0.


Analogously,

q− s∂+ = q−∑

l

(2l + 1)∑m,n

σ∂+(l)mn tlnm

=∑

l

∑n

σ∂+(l)n+1,n (2l + 1) q− tln,n+1

=∑

l

∑n

−√

(l − n)(l + n + 1)

×(√

(l − n + 1)(l + n + 2) tl+1/2n−1/2,n+3/2

−√

(l + n)(l − n− 1) tl−1/2n−1/2,n+3/2

)=

∑l

∑n

tl+1/2n−1/2,n+3/2

×(−

√(l − n)(l + n + 1)

√(l − n + 1)(l + n + 2)

+√

(l − n + 1)(l + n + 2)√

(l + n + 1)(l − n))

= 0,

and

q+ s∂0 = q+

∑l

(2l + 1)∑m,n

σ∂0(l)mn tlnm

=∑

l

∑n

σ∂0(l)nn (2l + 1) q+ tlnn

=∑

l

∑n

n(√

(l + n + 1)(l − n + 1) tl+

n−,n+ −√

(l − n)(l + n) tl−

n−,n+

)=

∑l

∑n

n tl+

n−,n+

(√(l + n + 1)(l − n + 1)−

√(l − n + 1)(l + n + 1)

)= 0.

Analogously,

q− s∂0 = q−∑

l

(2l + 1)∑m,n

σ∂0(l)mn tlnm

=∑

l

∑n

σ∂0(l)nn (2l + 1) q− tlnn

=∑

l

∑n

n(√

(l + n + 1)(l − n + 1) tl+

n+,n− −√

(l − n)(l + n) tl−

n+,n−

)=

∑l

∑n

n tl+

n+,n−

(√(l + n + 1)(l − n + 1)−

√(l − n + 1)(l + n + 1)

)= 0.


We also have

q0 s∂− = q0

∑l

(2l + 1)∑m,n

σ∂−(l)mn tlnm

=∑

l

∑n

σ∂−(l)n−1,n (2l + 1) q0 tln,n−1

=∑

l

∑n

−√

(l + n)(l − n + 1)

×(+

√(l − n + 1)(l − n + 2)tl+1/2

n−1/2,n−3/2 +√

(l + n)(l + n− 1)tl−1/2n−1/2,n−3/2

−√

(l + n + 1)(l + n)tl+1/2n+1/2,n−1/2 −

√(l − n)(l − n + 1)tl−1/2

n+1/2,n−1/2

)=

∑l

∑n

tln,n−1

×(+

√(l + n)(l − n + 1) (l − n) + (l + n + 1)

√(l − n + 1)(l + n)

− (l + n− 1)√

(l − n + 1)(l + n)−√

(l + n)(l − n + 1) (l − n + 2))

= 0.

Analogously,

q0 s∂+ = q0

∑l

(2l + 1)∑m,n

σ∂+(l)mn tlnm

=∑

l

∑n

σ∂+(l)n+1,n (2l + 1) q0 tln,n+1

=∑

l

∑n

−√

(l − n)(l + n + 1)

×(+

√(l − n + 1)(l − n)tl+1/2

n−1/2,n+1/2 +√

(l + n)(l + n + 1)tl−1/2n−1/2,n+1/2

−√

(l + n + 1)(l + n + 2)tl+1/2n+1/2,n+3/2 −

√(l − n)(l − n− 1)tl−1/2

n+1/2,n+3/2

)=

∑l

∑n

tln,n+1

×(+(l − n− 1)

√(l + n + 1)(l − n) +

√(l − n)(l + n + 1) (l + n + 2)

−√

(l − n)(l + n + 1) (l + n)− (l − n + 1)√

(l + n + 1)(l − n))

= 0. �


�νσI(x) = 0,

for every ν ∈ {+,−, 0}.


Proof. Similarly to the propositions before, we calculate

q+ sI = q+

∑l

(2l + 1)∑m,n

δmntlnm

=∑

l

∑n

(2l + 1) q+ tlnn

=∑

l

∑n

(√(l + n + 1)(l − n + 1) tl

+

n+,n− −√

(l − n)(l + n) tl−

n+,n−

)=

∑l

∑n

tl−

n+,n−

(√(l + n)(l − n)−

√(l − n)(l + n)

)= 0.

Analogously,

q− sI = q−∑

l

(2l + 1)∑m,n

δmntlnm

=∑

l

∑n

(2l + 1) q− tlnn

=∑

l

∑n

(√(l − n + 1)(l + n + 1) tl

+

n−,n+ −√

(l + n)(l − n) tl−

n−,n+

)=

∑l

∑n

tl−

n−,n+

(√(l − n)(l + n)−

√(l + n)(l − n)

)= 0.

Moreover,

q0 sI = q0

∑l

(2l + 1)∑m,n

δmntlnm

=∑

l

∑n

(2l + 1) q0 tlnn

=∑

l

∑n

(+(l − n + 1) tl

+

n−n− + (l + n) tl−

n−n−

−(l + n + 1) tl+

n+n+ − (l − n) tl−

n+n+

)=

∑l

∑n

tlnn (+(l − n) + (l + n + 1)− (l + n)− (l − n + 1))

= 0. �


�+σL = −σ∂− , �−σL = −σ∂+ , �0σL = −2σ∂0 .


Proof. SinceσL(x, l)mn = −l(l + 1) δmn,

we get

q+ s−L = q+

∑l

(2l + 1)∑m,n

σ−L(x, l)mn tlnm

=∑

l

(2l + 1)∑

n

l(l + 1) q+ tlnn

=∑

l

∑n

l(l + 1)(+

√(l + n + 1)(l − n + 1) tl

+

n−,n+

−√

(l − n)(l + n) tl−

n−,n+

)=

∑l

∑n

tl+

n−,n+

(+l(l + 1)

√(l + n + 1)(l − n + 1)

−(l + 1)(l + 2)√

(l − n + 1)(l + n + 1))

=∑

l

∑n

−2(l + 1)√

(l + n + 1)(l − n + 1) tl+

n−,n+

=∑

l

(2l + 1)∑

n

−√

(l + n)(l − n + 1) tln−1,n

= s∂− .

Analogously,

q− s−L = q−∑

l

(2l + 1)∑m,n

σ−L(x, l)mn tlnm

=∑

l

(2l + 1)∑

n

l(l + 1) q− tlnn

=∑

l

∑n

l(l + 1)(+

√(l − n + 1)(l + n + 1) tl

+

n−,n+

−√

(l + n)(l − n) tl−

n−,n+

)=

∑l

∑n

tl+

n−,n+

(+l(l + 1)

√(l − n + 1)(l + n + 1)

−(l + 1)(l + 2)√

(l + n + 1)(l − n + 1))

=∑

l

∑n

−2(l + 1)√

(l + n + 1)(l − n + 1) tl+

n−,n+

=∑

l

(2l + 1)∑

n

−√

(l + n)(l − n + 1) tln−1,n

= s∂+ .


Moreover,

q0 s−L = q0

∑l

(2l + 1)∑m,n

σ−L(x, l)mntlnm

=∑

l

(2l + 1)∑

n

l(l + 1) q0 tlnn

=∑

l

∑n

l(l + 1)(+(l − n + 1) tl

+

n−n− + (l + n) tl−

n−n−

−(l + n + 1) tl+

n+n+ − (l − n) tl−

n+n+

)=

∑l

l(l + 1)∑

n(tl

+

n+,n+((l − n)− (l + n + 1)) + tl−

n+,n+((l + n + 1)− (l − n)))

=∑

l

l(l + 1)∑

n

−(2n + 1)(tl

+

n+,n+ − tl−

n+,n+

)= −

∑l

∑n

tl+

n+,n+(2n + 1) (l(l + 1)− (l + 1)(l + 2))

=∑

l

∑n

tl+

n+,n+(2n + 1) (2l + 2) =∑

l

(2l + 1)∑

n

2n tlnn = 2 s∂0 ,

completing the proof. �Remark 12.3.11. The proof of Theorem 12.3.6 relied on explicit calculations inrepresentations and we decided to include them in detail for didactic purposes aswell as in preparation for the proof of Theorem 12.3.12. However, we may alsoargue using formula (10.28) in Proposition 10.7.4. There, if q(e) = 0, we get

�qσ∂+ = −(∂+q)(e) σI ,

�qσ∂−(x, ξ) = −(∂−q)(e) σI ,

�qσ∂0(x, ξ) = −(∂0q)(e) σI .

If we write u ∈ SU(2) as u =(

a bc d

)then

q+ = c, q− = b, q0 = a− d,

and by Exercise 11.5.14 we have

∂+u|u=e =(−b 0−d 0

)u=e

=(

0 0−1 0

),

∂−u|u=e =(

0 −a0 −c

)u=e

=(

0 −10 0

),

∂0u|u=e =(−a/2 0

0 d/2

)u=e

=(−1/2 0

0 1/2

).


This implies that

σI = �+σ∂+ = �−σ∂− = �0σ∂0 ,

0 = �+σ∂− = �+σ∂0

= �−σ∂+ = �−σ∂0

= �0σ∂+ = �0σ∂− .

Let us now apply the difference operators to the symbol of the Laplacian L. First,by Remark 11.5.11 we write

L = −∂20 − (∂+∂− + ∂−∂+)/2.

Now

∂20u = ∂0

12

(−a b−c d

)=

14

(a bc d

)u=e=

14

(1 00 1

),

∂+∂−u = ∂+

(0 −a0 −c

)=

(0 b0 d

)u=e=

(0 00 1

),

∂−∂+u = ∂−

(−b 0−d 0

)=

(a 0c 0

)u=e=

(1 00 0

),

so that

�+σ∂20

= q+(e) σ∂20− 2(∂0q+)(e) σ∂0 + (∂2

0q+)(e) σI

= 0,

�+σ∂+∂− = q+(e) σ∂+∂− − [(∂+q+)(e) σ∂−

+(∂−q−)(e) σ∂+ ] + (∂+∂−q+)(e) σI

= σ∂− ,

�+σ∂−∂+ = q+(e) σ∂−∂+ − [(∂+q+)(e) σ∂−

+(∂−q−)(e) σ∂+ ] + (∂−∂+q+)(e) σI

= σ∂− .

Therefore

�+σL = −σ∂− .

Analogously,

�−σL = −σ∂+ .

Finally,

�0σL = −2σ∂0 ,


because

�0σ∂20

= q0(e) σ∂20− 2(∂0q0)(e) σ∂0 + (∂2

0q0)(e) σI

= 2σ∂0 ,

�0σ∂+∂− = q0(e) σ∂+∂− − [(∂+q0)(e) σ∂−

+(∂−q−)(e) σ∂+ ] + (∂+∂−q0)(e) σI

= −σI ,

�0σ∂−∂+ = q0(e) σ∂−∂+ − [(∂+q0)(e) σ∂−

+(∂−q−)(e) σ∂+ ] + (∂−∂+q0)(e) σI

= +σI .

12.3.3 Differences for aσ∂0

Let us now calculate higher-order differences of the symbol aσ∂0 which will beneeded in the sequel.

Theorem 12.3.12. For any α ∈ N30, we have the formula

[�α1

+ �α2− �α30 (aσ∂0)

]l

nm

= (m− α1/2 + α2/2)[�α1

+ �α2− �α30 a

]l

nm+ α3

[�0�α1

+ �α2− �α3−10 a

]l

nm,

where �0 is given by

(�0a)lnm =

12

[√(l −m)(l − n)

2l + 1al−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1al+

n+m+

+

√(l + m)(l + n)

2l + 1al−

n−m− +

√(l −m + 1)(l − n + 1)

2l + 1al+

n−m−

],

and satisfies [�0,�0] = 0.

Proof. First we observe that we have

(a σ∂0)lnm =

∑k

alnk k δkm = m al

nm.

Then using Theorem 12.3.5, we get

�−(aσ∂0)lnm =

√(l −m)(l + n)

2l + 1m+al−

n−m+ −√

(l + m + 1)(l − n + 1)2l + 1

m+al+

n−m+

= (m+�−a)lnm,


and we can abbreviate this by writing �−(aσ∂0) = m+�−a. Further, we have

�−(�−(aσ∂0))lnm =

√(l −m)(l + n)

2l + 1[�−(aσ∂0)]

l−

n−m+

−√

(l + m + 1)(l − n + 1)2l + 1

[�−(aσ∂0)]l+

n−m+

=

√(l −m)(l + n)

2l + 1(m + 1)(�−a)l−

n−m+

−√

(l + m + 1)(l − n + 1)2l + 1

(m + 1)(�−a)l+

n−m+

= (m + 1)(�2−a)l

nm.

Continuing this calculation we can obtain[�k−(aσ∂0)

]l

nm= (m + k/2)(�k

−a)lnm. (12.25)

By Theorem 12.3.5 we also have

[�+(�−(aσ∂0))]lnm =

[�+

(m+�−a

)]l

nm

=

√(l + m)(l − n)

2l + 1(m+�−a

)l−

n+m−

−√

(l −m + 1)(l + n + 1)2l + 1

(m+�−a

)l+

n+m−

= m(�+�−a)lnm.

By induction, and using (12.25), we then get[�k1

+�k2− (aσ∂0)]l

nm= (m− k1/2 + k2/2)(�k1

+�k2− a)lnm. (12.26)

The situation with �0 is more complicated because there are more terms. UsingTheorem 12.3.5 we have

�0(aσ∂0)lnm

=

√(l −m)(l − n)

2l + 1(ma)l−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1(ma)l+

n+m+

−√

(l + m)(l + n)2l + 1

(ma)l−n−m− −

√(l −m + 1)(l − n + 1)

2l + 1(ma)l+

n−m−

=

√(l −m)(l − n)

2l + 1m+al−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1m+al+

n+m+

−√

(l + m)(l + n)2l + 1

m−al−n−m− −

√(l −m + 1)(l − n + 1)

2l + 1m−al+

n−m−


= m(�0a)lnm +

12

[√(l −m)(l − n)

2l + 1al−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1al+

n+m+

+

√(l + m)(l + n)

2l + 1al−

n−m− +

√(l −m + 1)(l − n + 1)

2l + 1al+

n−m−

]= m(�0a)l

nm + (�0a)lnm,

i.e.,�0(aσ∂0)

lnm = m(�0a)l

nm + (�0a)lnm, (12.27)

where �0 is a weighted averaging operator given by

(�0a)lnm =

12

[√(l −m)(l − n)

2l + 1al−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1al+

n+m+

+

√(l + m)(l + n)

2l + 1al−

n−m− +

√(l −m + 1)(l − n + 1)

2l + 1al+

n−m−

].

We want to find a formula for �k0 , and for this we first calculate[

�0(�0a)]l

nm

=

√(l −m)(l − n)

2l + 1(�0a)l−

n+m+ +

√(l + m + 1)(l + n + 1)

2l + 1(�0a)l+

n+m+

−√

(l + m)(l + n)2l + 1

(�0a)l−n−m− −

√(l −m + 1)(l − n + 1)

2l + 1(�0a)l+

n−m−

=

√(l −m)(l − n)

2l + 112

[√(l− −m+)(l− − n+)

2l− + 1al−−

n++m++

+

√(l− + m+ + 1)(l− + n+ + 1)

2l− + 1al−+

n++m++

+

√(l− + m+)(l− + n+)

2l− + 1al−−

n+−m+−

+

√(l− −m+ + 1)(l− − n+ + 1)

2l− + 1al−+

n+−m+−

]

+

√(l + m + 1)(l + n + 1)

2l + 112

12l+ + 1

[√(l+ −m+)(l+ − n+)al+−

n++m++

+√

(l+ + m+ + 1)(l+ + n+ + 1)al++

n++m++

+√

(l+ + m+)(l+ + n+)al+−n+−m+−

+√

(l+ −m+ + 1)(l+ − n+ + 1)al++

n+−m+−

]−

√(l + m)(l + n)

2l + 112

12l− + 1

[√(l− −m−)(l− − n−)al−−

n−+m−+


+√

(l− + m− + 1)(l− + n− + 1)al−+

n−+m−+

+√

(l− + m−)(l− + n−)al−−n−−m−−

+√

(l− −m− + 1)(l− − n− + 1)al−+

n−−m−−

]−

√(l −m + 1)(l − n + 1)

2l + 112

12l+ + 1

[√(l+ −m−)(l+ − n−)al+−

n−+m−+

+√

(l+ + m− + 1)(l+ + n− + 1)al++

n−+m−+

+√

(l+ + m−)(l+ + n−)al+−n−−m−−

+√

(l+ −m− + 1)(l+ − n− + 1)al++

n−−m−−

].

From this we get

[�0(�0a)

]l

nm=

√(l −m)(l − n)

2l + 112

12l

[√(l −m− 1)(l − n− 1)al−−

n++m++

+√

(l + m + 1)(l + n + 1)aln++m++

+√

(l + m)(l + n)al−−nm +

√(l −m)(l − n)al

nm

]+

√(l + m + 1)(l + n + 1)

2l + 112

12l + 2

[√(l −m)(l − n)al

n++m++

+√

(l + m + 2)(l + n + 2)al++

n++m++

+√

(l + m + 1)(l + n + 1)alnm +

√(l −m + 1)(l − n + 1)al++

nm

]−

√(l + m)(l + n)

2l + 112

12l

[√(l −m)(l − n)al−−

nm

+√

(l + m)(l + n)alnm

+√

(l + m− 1)(l + n− 1)al−−n−−m−−

+√

(l −m + 1)(l − n + 1)aln−−m−−

]−

√(l −m + 1)(l − n + 1)

2l + 112

12l + 2

[√(l −m + 1)(l − n + 1)al

nm

+√

(l + m + 1)(l + n + 1)al++

nm

+√

(l + m)(l + n)aln−−m−− +

√(l −m + 2)(l − n + 2)al++

n−−m−−

],

=

√(l −m)(l − n)

2l + 112

12l

[√(l −m− 1)(l − n− 1)al−−

n++m+++

+√

(l + m + 1)(l + n + 1)aln++m++ ]

+

√(l + m + 1)(l + n + 1)

2l + 112

12l + 2

[√(l −m)(l − n)al

n++m++


+√

(l + m + 2)(l + n + 2)al++

n++m++ + ]

−√

(l + m)(l + n)2l + 1

12

12l

[√(l + m− 1)(l + n− 1)al−−

n−−m−−+

+√

(l −m + 1)(l − n + 1)aln−−m−−

]−

√(l −m + 1)(l − n + 1)

2l + 112

12l + 2

×[√

(l + m)(l + n)aln−−m−− +

√(l −m + 2)(l − n + 2)al++

n−−m−−

],

where we used that pairs of terms with al−−nm , al++

nm canceled, and also four termswith al

nm canceled in view of the identity

(l −m)(l − n)(2l + 1)(2l)

+(l + m + 1)(l + n + 1)

(2l + 1)(2l + 2)

− (l + m)(l + n)(2l + 1)(2l)

− (l −m + 1)(l − n + 1)(2l + 1)(2l + 2)

=−2l(m + n)(2l + 1)(2l)

+(2l + 2)(m + n)(2l + 1)(2l + 2)

= 0.

Calculating in the other direction, we get[�0(�0a)

]l

nm

=12

√(l −m)(l − n)

2l + 1(�0a)l−

n+m+ +12

√(l + m + 1)(l + n + 1)

2l + 1(�0a)l+

n+m+

+12

√(l + m)(l + n)

2l + 1(�0a)l−

n−m− +12

√(l −m + 1)(l − n + 1)

2l + 1(�0a)l+

n−m−

=

√(l −m)(l − n)

2l + 112

12l− + 1

[√(l− −m+)(l− − n+)al−−

n++m++

+√

(l− + m+ + 1)(l− + n+ + 1)al−+

n++m++

−√

(l− + m+)(l− + n+)al−−n+−m+−

−√

(l− −m+ + 1)(l− − n+ + 1)al−+

n+−m+−

]+

√(l + m + 1)(l + n + 1)

2l + 112

12l+ + 1

[√(l+ −m+)(l+ − n+)al+−

n++m++

+√

(l+ + m+ + 1)(l+ + n+ + 1)al++

n++m++

−√

(l+ + m+)(l+ + n+)al+−n+−m+−

−√

(l+ −m+ + 1)(l+ − n+ + 1)al++

n+−m+−

]+

√(l + m)(l + n)

2l + 112

12l− + 1

[√(l− −m−)(l− − n−)al−−

n−+m−+


+√

(l− + m− + 1)(l− + n− + 1)al−+

n−+m−+

−√

(l− + m−)(l− + n−)al−−n−−m−−

−√

(l− −m− + 1)(l− − n− + 1)al−+

n−−m−−

]+

√(l −m + 1)(l − n + 1)

2l + 112

12l+ + 1

[√(l+ −m−)(l+ − n−)al+−

n−+m−+

+√

(l+ + m− + 1)(l+ + n− + 1)al++

n−+m−+

−√

(l+ + m−)(l+ + n−)al+−n−−m−−

−√

(l+ −m− + 1)(l+ − n− + 1)al++

n−−m−−

].

From this we get

[�0(�0a)

]l

nm

=

√(l −m)(l − n)

2l + 112

12l

[√(l −m− 1)(l − n− 1)al−−

n++m++

+√

(l + m + 1)(l + n + 1)aln++m++

−√

(l + m)(l + n)al−−nm −

√(l −m)(l − n)al

nm

]+

√(l + m + 1)(l + n + 1)

2l + 112

12l + 2

[√(l −m)(l − n)al

n++m++

+√

(l + m + 2)(l + n + 2)al++

n++m++

−√

(l + m + 1)(l + n + 1)alnm −

√(l −m + 1)(l − n + 1)al++

nm

]+

√(l + m)(l + n)

2l + 112

12l

[√(l −m)(l − n)al−−

nm

+√

(l + m)(l + n)alnm

−√

(l + m− 1)(l + n− 1)al−−n−−m−−

−√

(l −m + 1)(l − n + 1)aln−−m−−

]+

√(l −m + 1)(l − n + 1)

2l + 112

12l + 2

[√(l −m + 1)(l − n + 1)al

nm

+√

(l + m + 1)(l + n + 1)al++

nm

−√

(l + m)(l + n)aln−−m−− −

√(l −m + 2)(l − n + 2)al++

n−−m−−

]=

√(l −m)(l − n)

2l + 112

12l

[√(l −m− 1)(l − n− 1)al−−

n++m++

+√

(l + m + 1)(l + n + 1)aln++m++ ]


+

√(l + m + 1)(l + n + 1)

2l + 112

12l + 2

[√(l −m)(l − n)al

n++m++

+√

(l + m + 2)(l + n + 2)al++

n++m++ ]

+

√(l + m)(l + n)

2l + 112

12l

[√(l + m− 1)(l + n− 1)al−−

n−−m−−

−√

(l −m + 1)(l − n + 1)aln−−m−−

]+

√(l −m + 1)(l − n + 1)

2l + 112

12l + 2

×[−

√(l + m)(l + n)al

n−−m−− −√

(l −m + 2)(l − n + 2)al++

n−−m−−

],

where we used again the fact that terms alnm, al−−

nm and al++

nm canceled. Comparingthese calculations we get the commutativity property

�0�0a = �0�0a.

From this and (12.27) we obtain

�20(ma) = �0(m�0a +�0a) = m�2

0a + 2�0�0a,

and, consequently, by induction we get

�k0(ma) = m�k

0a + k�0�k−10 a.

Let us now apply this to (12.26). Using commutativity of �0,�+ and �− fromTheorem 12.3.5, we get[

�k1+�k2−�k3

0 (aσ∂0)]l

nm

=[�k3

0 �k1+�k2− (aσ∂0)

]l

nm

=[�k3

0

((m− k1/2 + k2/2)�k1

+�k2− a)]l

nm

=[�k3

0

(m�k1

+�k2− a)]l

nm−

[�k3

0

((k1/2− k2/2)�k1

+�k2− a)]l

nm

= m[�k3

0 �k1+�k2− a

]l

nm+ k3

[�0�k3−1

0 �k1+�k2− a

]l

nm

−(k1/2− k2/2)[�k3

0 �k1+�k2− a

]l

nm

= (m− k1/2 + k2/2)[�k1

+�k2−�k30 a

]l

nm+ k3

[�0�k1

+�k2−�k3−10 a

]l

nm,



12.4 Symbol classes on SU(2)

The goal of this section is to simplify the symbol class Σm(SU(2)) from Definition12.1.1, yielding Hormander’s class Ψm(SU(2)) of pseudo-differential operators onSU(2). For this purpose, we introduce and investigate the symbol class Sm(SU(2)).

Definition 12.4.1 (Symbol class Sm(SU(2))). For u ∈ SU(2), we write

Auf := A(f ◦ φ) ◦ φ−1,

where φ(x) = xu; note that by Proposition 10.4.18 we have

RAu(x, y) = RA(xu−1, uyu−1),

σAu(x, l) = tl(u)∗ σA(xu−1, l) tl(u).

The symbol class Sm(SU(2)) consists of the symbols σA of those operators A ∈L(C∞(SU(2))) for which

sing supp (y �→ RA(x, y)) ⊂ {e},

and for which ∣∣�αl ∂β

xσAu(x, l)ij

∣∣ ≤ CAαβmN 〈i− j〉−N (1 + l)m−|α| (12.28)

uniformly in x, u ∈ SU(2), for every N ≥ 0, all l ∈ 12N0, all multi-indices α, β ∈ N3

0,and for all matrix column/row numbers i, j. Thus, the constant in (12.28) maydepend on A,α, β,m and N , but not on x, u, l, i, j.

Remark 12.4.2 (Rapid off-diagonal decay). We note that inequality (12.28) con-tains the rapid off-diagonal decay property since we can take N as large as wewant.

We now formulate the main theorem of this section:

Theorem 12.4.3 (Equality of classes OpSm(SU(2)) = Ψm(SU(2))). We have A ∈Ψm(SU(2)) if and only if σA ∈ Sm(SU(2)). Moreover, we have the equality ofsymbol classes

Sm(SU(2)) = Σm(SU(2)). (12.29)

In fact, we need to prove only the equality of symbol classes (12.29), fromwhich the first part of the theorem would follow by Theorem 10.9.6. In the processof proving equality (12.29), we establish a number of auxiliary results.Remark 12.4.4. By Corollary 10.9.8, if σA ∈ Σm(SU(2)) then

�γl ∂δ

xσA ∈ Σm−|γ|(SU(2)).

We show the analogous result for Sm(SU(2)):

Lemma 12.4.5. If σA ∈ Sm(SU(2)) then σB = �γl ∂δ

xσA ∈ Sm−|γ|(SU(2)).

12.4. Symbol classes on SU(2) 657

Proof. First, let |γ| = 1. Then �γl f(l) = qf(l) for some

q ∈ Pol1(SU(2)) := span{

t1/2ij : i, j ∈ {−1/2,+1/2}

}for which q(e) = 0. Let r(y) := q(uyu−1). Then r ∈ Pol1(SU(2)), because

t1/2ij (uyu−1) =

∑k,m

t1/2ik (u) t

1/2km(y) t

1/2mj (u−1).

Moreover, we have r(e) = 0. Hence f(l) �→ rf(l) is a linear combination of dif-ference operators �0,�+,�− because

{f ∈ Pol1(SU(2)) : f(e) = 0

}is a three-

dimensional vector space spanned by q0, q+, q−. Now let γ ∈ N30 and σB = �γ

l ∂δxσA.

We have

�αl ∂β

xσBu(x, l) = �α

l ∂βx

(tl(u)∗ σB(xu−1, l) tl(u)

)= �α

l ∂βx

(tl(u)∗

(�γ

l ∂δxσA(xu−1, l)

)tl(u)

)=

∑|γ′|=|γ|

λu,γ′ �α+γ′l ∂β+δ

x σAu(x, l),

for some scalars λu,γ′ ∈ C depending only on u ∈ SU(2) and multi-indicesγ′ ∈ N3

0. �

Remark 12.4.6. Let D be a left-invariant vector field on SU(2). From the verydefinition of the symbol classes Σm(SU(2)) =

⋂∞k=0 Σm

k (SU(2)), it is evident that[σD, σA] ∈ Σm(SU(2)) if σA ∈ Σm(SU(2)). We now prove the similar invariancefor Sm(SU(2)).

Lemma 12.4.7. Let D be a left-invariant vector field on SU(2). Let σA∈Sm(SU(2)).Then [σD, σA] ∈ Sm(SU(2)) and σA σD ∈ Sm+1(SU(2)).

Proof. For D ∈ su(2) we write D = iE, so that E ∈ i su(2). By Proposition 12.2.4there is some u ∈ SU(2) such that σE(l) = tl(u)∗ σ∂0(l) tl(u). Now, we have

[σE , σA](l) = tl(u)∗[σ∂0(l), t

l(u) σA(x, l) tl(u)∗]

tl(u)

=[σ∂0 , σAu−1

]u

(l).

Next, notice that Sm(SU(2)) is invariant under the mappings σB �→ σBu andσB �→ [σ∂0 , σB ]; here [σ∂0 , σB ](l)ij = (i− j) σB(l)ij . Finally,

σA(x, l) σE(l) = tl(u)∗ tl(u) σA(x, l) tl(u)∗ σ∂0(l) tl(u)=

(σAu−1 (x, l) σ∂0(l)

)u

.

Just as in the first part of the proof, we see that σA σD belongs to Sm+1(SU(2))since σB σ∂0 ∈ Sm+1(SU(2)) if σB ∈ Sm(SU(2)), by Theorem 12.3.12. �


Proof of Theorem 12.4.3. We have to show that Sm(SU(2)) = Σm(SU(2)), so thatthe theorem would follow from Theorem 10.9.6. Note that both classes Sm(SU(2))and Σm(SU(2)) require the singular support condition (y �→ RA(x, y)) ⊂ {e}, sowe do not have to consider this; moreover, the x-dependence of the symbol is notessential here, and therefore we abbreviate σA(l) := σA(x, l). First, let us show thatΣm(SU(2)) ⊂ Sm(SU(2)). Take σA ∈ Σm(SU(2)). Then also σAu ∈ Σm(SU(2))(either by the well-known properties of pseudodifferential operators and Theorem10.9.6, or by checking directly that the definition of the classes Σm

k (SU(2)) isconjugation-invariant). Let us define cN (B) by

σcN (B)(l)ij := (i− j)N σB(l)ij .

Now σcN (Au) ∈ Σm(SU(2)) for every N ∈ Z+, because σAu ∈ Σm(SU(2)) and

[σ∂0 , σB ](l)ij = (i− j) σB(l)ij .

This implies the “rapid off-diagonal decay” of σAu:

|σAu(x, l)ij | ≤ CAmN 〈i− j〉−N (1 + l)m,

implying the norm comparability

‖· · ·σAu(l)‖op ∼ supi,j|· · ·σAu(l)ij | (12.30)

in view of Lemma 12.6.5 in Section 12.6. Moreover, �αl ∂β

xσAu ∈ Σm−|α|(SU(2)) byCorollary 10.9.8, so that we obtain the symbol inequalities (12.28) from (10.37).Thereby Σm(SU(2)) ⊂ Sm(SU(2)).

Now we have to show that Sm(SU(2)) ⊂ Σm(SU(2)). Again, we may exploitthe norm comparabilities (12.30): thus clearly Sm(SU(2)) ⊂ Σm

0 (SU(2)). Conse-quently, Sm(SU(2)) ⊂ Σm

k (SU(2)) for all k ∈ Z+, due to Lemmas 12.4.5 and12.4.7. �

Remark 12.4.8. Notice that in Definition 12.4.1, we demanded inequalities (12.28)uniformly in u∈SU(2). However, it suffices to assume this for only u∈{e,ω1(π/2)},where

e =(

1 00 1

)and ω1

(π

2

)=√

22

(1 ii 1

).

Exercise 12.4.9. Prove the claim of Remark 12.4.8. Hint: Notice that

u(φ, θ, ψ) = ω3(φ) ω2(θ) ω3(ψ) and that ω2(θ) = w−11 ω3(θ)w1,

where w1 = ω1(π/2) =√

22

(1 ii 1

). Recall also Proposition 10.4.18. Conjugating

a symbol σA(x, l) with tl(ω3(t)) does not affect the “rapid off-diagonal decay”

12.4. Symbol classes on SU(2) 659

property. Let q ∈ Q1 := span{q+, q−, q0} \ {0}, and let �q be the correspondingfirst-order difference operator, i.e.,

�q f(l) = qf(l).

Defining q1 ∈ Q1 by q1(z) := q(u−1zu) and σB := �q1σA, we have �qσAu(l) =

σBu(l). This shows us that difference operators can essentially be moved throughthe tl(u)-conjugations; also, such conjugations do not affect the x-derivatives;moreover, these conjugations behave well with respect to taking commutators ofsymbols.

Remark 12.4.10 (Topology on Sm(SU(2))). It is natural to define the topology onSm(SU(2)) by seminorms

pα,β,m,i,j,N,u(σA) := supx∈SU(2),l∈ 1

2 N0

{〈i− j〉N

∣∣�αl ∂β

xσAu(x, l)ij

∣∣(1 + l)m−|α|

}. (12.31)

Notice that by Exercise 12.4.9, it is sufficient to consider only the cases u ∈{e, ω2(π/2)}.

Compared to Corollary 10.9.9, we can replace the convergence in the Hilbert–Schmidt norm by the pointwise �∞ convergence to relate the convergence of sym-bols to the convergence of operators:

Corollary 12.4.11 (Convergence of symbols and operators). Let σ ∈ S0(SU(2)),and assume that a sequence σk ∈ S0(SU(2)) satisfies inequalities (12.28) uniformlyin k (i.e., with constants independent of k). Assume that for all |β| ≤ 2 we havethe convergence

∂βxσk(x, l)→ ∂β

xσ(x, l) as k →∞ (12.32)

in the �∞ norm, uniformly over all x ∈ G and all l ∈ 12N0. Then Opσk → Opσ

strongly on L2(SU(2)).Moreover, if the convergence (12.32) holds for all β, then Op σk → Opσ

strongly on Hs(SU(2)) for any s ∈ R.

Proof. We observe that Theorem 10.5.5 implies that

‖Opσk −Opσ‖L(L2(SU(2)))

≤ C1 supx∈SU(2),l∈ 1

2 N0,|β|≤2

‖∂βxσk(x, l)− ∂β

xσ(x, l)‖op

≤ C2 supx∈SU(2),l∈ 1

2 N0,|β|≤2

‖∂βxσk(x, l)− ∂β

xσ(x, l)‖�∞ ,

where the last estimate follows from Lemma 12.6.5 in Section 12.6, with constantC2 independent of k. The strong convergence on L2(SU(2)) now follows directlyfrom (12.32). The strong convergence on Hs(SU(2)) follows from Theorem 10.8.1by the same argument. �


12.5 Pseudo-differential operators on S3

In this section we discuss how the construction of Section 10.10 yields full symbolsof pseudo-differential operators on the 3-sphere S3. For this, we will use a globalisomorphism S3 ∼= SU(2) from Proposition 11.4.2. First we recall the quaternionspace H from Section 11.4 which is the associative R-algebra with a vector spacebasis {1, i, j,k}, where 1 ∈ H is the unit and

i2 = j2 = k2 = −1 = ijk.

The mapping x = (xm)3m=0 �→ x01 + x1i + x2j + x3k identifies R4 with H. Inparticular, the unit sphere S3 ⊂ R4 ∼= H is a multiplicative group. A bijectivehomomorphism Φ−1 : S3 → SU(2) in (11.6) is defined by

x �→ Φ−1(x) =(

x0 + ix3 x1 + ix2

−x1 + ix2 x0 − ix3

),

and its inverse Φ : SU(2) → S3 gives rise to the global quantisation of pseudo-differential operators on S3 induced by that on SU(2), as shown in Section 10.10.

The diffeomorphism Φ induces the Fourier analysis on S3 in terms of therepresentations of SU(2). To fix the notation for this in terms of S3, let

tl : S3 → U(2l + 1) ⊂ C(2l+1)×(2l+1),

l ∈ 12N0, be a family of group homomorphisms, which are the irreducible contin-

uous (and hence smooth) unitary representations of S3 when it is endowed withthe SU(2) structure via the quaternionic product, see Section 12.5 for details. TheFourier coefficient f(l) of f ∈ C∞(S3) is defined by

f(l) =∫

S3f(x) tl(x)∗ dx,

where the integration is performed with respect to the Haar measure, and f(l) ∈C(2l+1)×(2l+1). The corresponding Fourier series is given by

f(x) =∑

l∈ 12 N0

(2l + 1) Tr(f(l) tl(x)

).

Now, if A : C∞(S3) → C∞(S3) is a continuous linear operator, we define its fullsymbol as a mapping

(x, l) �→ σA(x, l), σA(x, l) = tl(x)∗(Atl)(x) ∈ C(2l+1)×(2l+1).

Then we have the representation of operator A in the form

Af(x) =∑

l∈ 12 N0

(2l + 1) Tr(tl(x) σA(x, l) f(l)

),

12.5. Pseudo-differential operators on S3 661

see Theorem 10.4.4. We also note that if

Af(x) =∫

S3KA(x, y) f(y) dy =

∫S3

f(y) RA(x, y−1x) dy,

where RA is the right-convolution kernel of A, then

σA(x, l) =∫

S3RA(x, y) tl(y)∗ dy

by Theorem 10.4.6, where, as usual, the integration is performed with respect tothe Haar measure with a standard distributional interpretation.

We now introduce symbol classes Sm(S3) which allow us to characterise op-erators from Hormander’s class Ψm(S3).

Definition 12.5.1 (Symbol class Sm(S3)). We write σA ∈ Sm(S3) if the corre-sponding kernel KA(x, y) is smooth outside the diagonal x = y and if we have theestimate∣∣�α

l ∂βxσAu

(x, l)ij

∣∣ ≤ CAαβmN (1 + |i− j|)−N (1 + l)m−|α|, (12.33)

for every N ≥ 0, every u ∈ S3, and all multi-indices α, β, where the symbol σAu

is the symbol of the operator

Auf = A(f ◦ ϕu) ◦ ϕ−1u ,

with ϕu(x) = xu the quaternionic product. We write �αl = �α1

+ �α2− �α30 , where

the operators �+,�−,�0 are discrete difference operators acting on matricesσA(x, l) in the variable l, and explicit formulae for them and their propertiesare given in Definition 12.3.2 and Theorem 12.3.5, with polynomials q+, q− andq0 defined as in Remark 10.6.2. Constants CAαβmN in (12.33) may depend onA,α, β,m,N but not on i, j, l.

Remark 12.5.2. As in the case of SU(2), the symbols of Au and A are related by

σAu(x, l) = tl(u)∗σA(xu−1, l) tl(u),

see Proposition 10.4.18. We notice that imposing the same conditions on all sym-bols σAu in (12.33) simply refers to the well-known fact that the class Ψm(S3)should be in particular “translation”-invariant (i.e., invariant under the changesof variables induced by quaternionic products ϕu), namely that A ∈ Ψm(S3) ifand only if Au ∈ Ψm(S3), for all u ∈ S3. Condition (12.33) is the growth conditionwith respect to the quantum number l combined with the condition that matricesσA(x, l) must have a rapid off-diagonal decay.

With Definition 12.5.1, we have the following characterisation, which followsimmediately from Theorem 12.4.3:

Theorem 12.5.3 (Equality OpSm(S3) = Ψm(S3)). We have A ∈ Ψm(S3) if andonly if σA ∈ Sm(S3).


12.6 Appendix: infinite matrices

In this section we discuss infinite matrices. The main conclusion that we need isthat the operator-norm and the l∞-norm are equivalent for matrices arising asfull symbols of pseudo-differential operators in Ψm(SU(2)), used in (12.30) in theproof of Theorem 12.4.3.

The reader should already know basic linear algebra, but for the sake ofcompleteness, we review necessary matrix operations. Let Cm×n denote the com-plex vector space of matrices with m rows and n columns; the rows are numbered1, . . . ,m downwards, the columns 1, . . . , n from left to right. Let Aij ∈ C de-note the element of matrix A ∈ Cm×n on row i and column k. Let λ ∈ C andA,B ∈ Cm×n; let matrices λA, A + B ∈ Cm×n and the adjoint A∗ ∈ Cn×m bedefined by

(λA)ij := λAij ,

(A + B)ij := Aij + Bij ,

(A∗)ij := Aji.

The product of A ∈ Cm×p and B ∈ Cp×n is AB ∈ Cm×n defined by

(AB)ij :=p∑

k=1

AikBkj .

The trace of A ∈ Cn×n is

Tr(A) :=n∑

j=1

Ajj .

We may naturally identify vector space Cn with Cn×1, and a mapping (x �→ Ax) :Cm×1 → Cn×1 can be seen as a linear mapping Cm → Cn.

The Euclidean inner product (or Hilbert–Schmidt inner product) of A,B ∈Cm×n is a special case of that in Subsection B.5.1, and is given by

〈A,B〉HS := Tr(B∗A)1/2 =m∑

i=1

n∑j=1

BijAij

and the corresponding norm of A is ‖A‖HS := 〈A,A〉1/2HS . The operator norm of

A ∈ Cm×n is

‖A‖ = ‖A‖op := sup{‖Ax‖�2 : x ∈ Cn×1, ‖x‖�2 ≤ 1

},

where ‖x‖�2 = (∑n

j=1 |xj |2)1/2 is the usual Euclidean norm. Of course, due to thefinite dimensionality here, supremum could be replaced by maximum.

12.6. Appendix: infinite matrices 663

Theorem 12.6.1. Let A,B ∈ Cn×n. Then

‖AB‖HS ≤ ‖A‖ ‖B‖HS .

Moreover, ‖A‖ = sup {‖AX‖HS : X ∈ Cn×n, ‖X‖HS ≤ 1} .

Proof. Let bj ∈ Cn×1 denote the jth column vector of the matrix B. Then Abj ∈Cn×1 is the jth column vector of the matrix AB, so that

‖AB‖2HS =n∑

j=1

‖Abj‖2HS ≤ ‖A‖2n∑

j=1

‖bj‖2HS = ‖A‖2 ‖B‖2HS .

Let x ∈ Cn×1 be such that ‖x‖HS = 1 and ‖Ax‖HS = ‖A‖. Let each columnvector of X ∈ Cn×n be x/

√n. Then ‖X‖HS = 1 and

‖AX‖2HS =n∑

j=1

‖Ax‖2HS/n = ‖A‖2.

Taking square roots completes the proof. �

Definition 12.6.2 (Operator as an infinite matrix). Let CZ denote the space ofcomplex sequences x : Z→ C. We will write x = (xj)j∈Z, where xj = x(j). Let

V ={x ∈ CZ : {j ∈ Z : xj = 0} finite

}.

A matrix A ∈ CZ×Z is a function A : Z × Z → C, sometimes presented as aninfinite table

A =(Aij

)i,j∈Z

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

. . ....

......

......

· · · A−1,−1 A−1,0 A−1,1 A−1,2 A−1,3 · · ·· · · A0,−1 A00 A01 A02 A03 · · ·· · · A1,−1 A10 A11 A12 A13 · · ·· · · A2,−1 A20 A21 A22 A23 · · ·· · · A3,−1 A30 A31 A32 A33 · · ·

......

......

.... . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where Aij := A(i, j). The (standard) matrix of a linear operator A : V → CZ isthe matrix

(Aij

)i,j∈Z

where the numbers Aij ∈ C are obtained from

(Ax)i =∑j∈Z

Aijxj ;

there should be no confusion in denoting the linear operator and the correspondingstandard matrix with the same letter.


Remark 12.6.3. Let δi = (δij)j∈Z ∈ �2 = �2(Z), where δii = 1 and δij = 0 if i = j.Then for a linear operator A : �2 → �2, the standard matrix is

Aij = 〈Aδj , δi〉�2 , (12.34)

i.e., the standard matrix is the matrix of the operator A with respect to the basis{δi : i ∈ Z}.Definition 12.6.4. Let A : V → CZ be a linear operator. For each k ∈ Z, let usdefine a linear operator A(k) : V → CZ by

A(k)ij =

{Aij , if i− j = k,

0, if i− j = k.

Notice that now A =∑

k∈Z A(k) and

‖A(k)‖�2→�2 = supj|A(k)j+k,j |. (12.35)

Let‖A‖�∞ := sup

i,j∈Z

|Aij | ,

and recall the notation 〈i− j〉 = (1 + |i− j|2)1/2.

Lemma 12.6.5. Let A : V → CZ be a linear operator. Then

‖A‖�∞ ≤ ‖A‖�2→�2 .

Moreover, if |Aij | ≤ cr〈i− j〉−r||A||�∞ for a constant cr <∞ where r > 1, then

‖A‖�2→�2 ≤ c′r ‖A‖�∞

for the constant c′r = cr

∑k∈Z〈k〉−r < ∞; hence in this case the norms ‖ · ‖�2→�2

and ‖ · ‖�∞ are equivalent.

Proof. The first claim follows from the Cauchy–Schwarz inequality:

|Aij |(12.34)

=∣∣(Aδj , δi)�2

∣∣ ≤ ‖A‖�2→�2 .

Next, we can assume that ||A||�∞ < ∞ since otherwise there is nothing to prove.Since A =

∑k∈Z A(k), we get

‖A‖�2→�2 ≤∑k∈Z

‖A(k)‖�2→�2

(12.35)=

∑k∈Z

supj|A(k)j+k,j |

≤∑k∈Z

cr〈k〉−r||A||�∞ =: c′r||A||�∞ ,

this last sum converging since r > 1. This concludes the proof. �

12.6. Appendix: infinite matrices 665

Definition 12.6.6 (Matrices with rapid off-diagonal decay). A matrix A ∈ CZ×Z issaid to decay (rapidly) off-diagonal if

|Aij | ≤ cAr 〈i− j〉−r

for every i, j ∈ Z and r ∈ N, where constants cAr <∞ depend on r, A, but not oni, j. The set of off-diagonally decaying matrices is denoted by D.

Proposition 12.6.7. Let A,B ∈ D. Then AB ∈ D.

Proof. In principle, we must be cautious here, since linear operators A,B : V → CZ

in general cannot be composed to get AB : V → CZ. Here, however, there is noproblem as A,B ∈ D, so that(∑

k

(AB)ikxk

)i∈Z

= (AB)x = A(Bx) =

⎛⎝∑j

Aij

∑k

Bjkxk

⎞⎠i∈Z

,

where

|(AB)ik| =

∣∣∣∣∣∣∑

j

Aij Bjk

∣∣∣∣∣∣ ≤∑

j

|Aij | |Bjk|

≤ cAr cBs

∑j

〈i− j〉−r 〈j − k〉s

Peetre 3.3.31≤ 2|r|cAr cBs

∑j

〈i− k〉−r 〈k − j〉|r| 〈j − k〉s

= 2|r|cAr cBs 〈i− k〉−r∑

j

〈j〉|r|+s,

which converges if |r|+ s < −1. This shows that AB ∈ D. �Remark 12.6.8. Proposition 12.6.7 dealt with matrix multiplication in D. Formatrices A,B ∈ CZ×Z in general, notice that{

(A + B)ij = Aij + Bij ,

(λA)ij = λAij .

Moreover, we may define involution A �→ A∗ by

(A∗)ij := Aji.

Of course, on the algebra L(H) this corresponds to the usual adjoint operationA �→ A∗, where 〈A∗x, y〉H = 〈x, Ay〉H for every x, y ∈ H. We may collect theseobservations:

Theorem 12.6.9. D ⊂ L(�2) is a unital involutive algebra. Moreover, for A ∈ D,the norms ‖A‖op and ‖A‖�∞ are equivalent.

Chapter 13

Pseudo-differential Operators onHomogeneous Spaces

In this chapter we discuss pseudo-differential operators on homogeneous spaces.The main question addressed here is how operators on such a space are related topseudo-differential operators on the group that acts on the space. Once such a cor-respondence is established, one can use it to map the whole construction developedearlier from the group to the homogeneous space. We also note that among otherthings, this chapter provides an application to the characterisation of pseudo-differential operators in terms of Σm-classes in Theorem 10.9.6. An importantclass of examples to keep in mind here are the spheres Sn ∼= SO(n)\SO(n + 1) ∼=SO(n + 1)/SO(n).

13.1 Analysis on closed manifolds

We start with closed manifolds. Let M be a C∞-smooth, closed (i.e., compact,without a boundary) orientable manifold. We refer to Section 5.2 for the basicconstructions on M , so we now review only a few of them. The test function spaceD(M) is the space of C∞(M) endowed with the usual Frechet space topology. Itsdual D′(M) = L(D(M), C) is the space of distributions, endowed with the weak-∗-topology, see Remark 5.2.15. The duality is expressed by the bracket 〈f, ϕ〉 = f(ϕ)(ϕ ∈ D(M), f ∈ D′(M)). The embedding D(M) ↪→ D′(M) is interpreted by

〈ψ, ϕ〉 :=∫

M

ψ(x) ϕ(x) dx.

The Schwartz kernel theorem states that L(D(M)) is isomorphic to D(M) ⊗D′(M); the isomorphism is given by

〈Aϕ, f〉 = 〈KA, ϕ⊗ f〉,

668 Chapter 13. Pseudo-differential Operators on Homogeneous Spaces

where A ∈ L(D(M)), ϕ ∈ D(M), f ∈ D′(M), and distribution KA ∈ D(M) ⊗D′(M) is called the Schwartz kernel of A. Then A can be uniquely extended (byduality) to A ∈ L(D′(M)), and it is customary to write informally

(Af)(x) =∫

M

KA(x, y) f(y) dy

instead of ϕ �→ 〈Af, ϕ〉 (ϕ ∈ D(M)). Recall that L2(M) = H0(M), D′(M) =∪s∈RHs(M) and D(M) = ∩s∈RHs(M), where Hs(M) is the (L2-type) Sobolevspace of order s ∈ R, see Definition 5.2.16. There are different spaces of distribu-tions available more specifically on homogeneous spaces, see, e.g., [93] for spacesD′L1(M) of summable distributions.

An operator A ∈ L(D(M)) is a pseudo-differential operator of order m ∈ Ron M , A ∈ Ψm(M), if (MφAMψ)κ ∈ Ψm(Rdim(M)) for every chart (U, κ) of Mand for every φ, ψ ∈ C∞0 (U), where Mφ is the multiplication operator f �→ φf ,and

(MφAMψ)κf := (MφAMψ(f ◦ κ)) ◦ κ−1 (f ∈ C∞(κU)).

We sometimes write MφAMψ ∈ Ψm(Rdim(M)), thus omitting the subscript κ andleaving the chart mapping implicit. Equivalently, pseudo-differential operators canbe characterised by commutators (see Theorem 5.3.1): A ∈ L(D(M)) belongs toΨm(M) if and only if (Ak)∞k=0 ⊂ L(Hm(M),H0(M)) for every sequence of smoothvector fields (Dk)∞k=1 on M , where A0 := A and Ak+1 := [Dk+1, Ak].

Definition 13.1.1 (Right transformation group). A smooth right transformationgroup is

(G, M, m),

where G is a Lie group, M is a C∞-manifold and m : M×G→M is a C∞-mappingcalled a right action, satisfying m(p, e) = p and m(m(p, y), x) = m(p, yx) for allx, y ∈ G and p ∈M , where e ∈ G is the neutral element of the group. The action isfree, if m(p, x) = p implies x = e. It is evident how one defines a left transformationgroup (G, M, m) with a left action m : G×M →M .

Definition 13.1.2 (Fiber bundles). A smooth fiber bundle is

(E,B, F, pE→B),

where E,B, F are C∞-manifolds and pE→B ∈ C∞(E,B) is a surjective mappingsuch that there exists an open cover U = {Uj | j ∈ J} of B and diffeomor-phisms φj : p−1(Uj) → Uj × F satisfying φj(x) = (pE→B(x), ψj(x)) for everyx ∈ p−1

E→B(Uj). The spaces E,B, F are called the total space, the base space, andthe fiber of the bundle, respectively. The cover U is called a locally trivialisingcover of the bundle. Sometimes only the mapping pE→B is called the fiber bundle.

Definition 13.1.3 (Principal fiber bundles). A principal fiber bundle is

(E,B, F, pE→B ,m),

13.2. Analysis on compact homogeneous spaces 669

where (E,B, F, pE→B) is a smooth fiber bundle with cover U and mappings φj , ψj

as above and (F,E,m) is a smooth right transformation group with a free actionsatisfying pE→B(m(x, y)) = pE→B(x) for every (x, y) ∈ E × F and ψj(m(x, y)) =ψj(x)y for every (x, y) ∈ p−1

E→B(Uj)× F .

13.2 Analysis on compact homogeneous spaces

Here we review some elements of the analysis on homogeneous spaces. The groupwill be acting on the right to adopt the construction to the previously constructedsymbolic calculus on groups.

Definition 13.2.1 (Homogeneous spaces I). Let (G, M, m) be a smooth right trans-formation group. The manifold M is called a homogeneous space if the actionm : M × G → M is transitive, i.e., if for every p, q ∈ M there exists x ∈ G suchthat m(p, x) = q.

For this line of thought we can refer to, e.g., [149]. However, let us also giveanother, equivalent definition for a homogeneous space:

Definition 13.2.2 (Homogeneous spaces II). Let G be a Lie group with a closedsubgroup K. The homogeneous space K\G is the set of classes Kx = {kx | k ∈ K}(x ∈ G) endowed with the topology co-induced by x �→ Kx, and equipped withthe unique C∞-manifold structure such that the mapping (x, Ky) �→ Kyx belongsto C∞(G × (K\G),K\G), and such that there is a neighbourhood U ⊂ K\G ofKe ∈ K\G and a mapping ψ ∈ C∞(U,G) satisfying Kψ(Kx) = Kx. The groupG acts smoothly from the right on the manifold K\G by (Ky, x) �→ Kyx.

Exercise 13.2.3. Actually a smooth homogeneous space M is diffeomorphic toGp\G, where Gp = {x ∈ G | m(p, x) = p} is the isotropy subgroup (see Definition6.3.3 and Theorem 6.3.4). Thus, show that two definitions are equivalent.

Exercise 13.2.4. Show that (G, K\G, K, x �→ Kx, (x, k) �→ kx) has a structureof a principal fiber bundle. For a further development of this point of view see,e.g., [20].

Remark 13.2.5 (Homogeneous spaces K\G vs G/K). Clearly one can considerhomogeneous spaces G/K with the action of the left transformation group. As itturns out, once we have chosen to identify the Lie algebra of a Lie group withthe left invariant vector fields, the further analysis is fixed from the point ofview of “right”/“left”, see Remark 10.4.2 for the starting point of this choice.Moreover, since on the group we wanted to have the composition formulae forpseudo-differential operators in the usual form σA◦B = σAσB + · · · and not in theform σA◦B = σBσA + · · · , also the choice of the definition of the Fourier trans-form was fixed, see Remark 10.4.13. However, the right/left constructions are verysymmetric, and since the notation G/K recalling the division of numbers may bemore familiar, we chose to introduce homogeneous spaces in this setting in the


definition of the right quotient in Definition 6.2.12. Consequently, this led to thecorresponding definition of the quotient topology in Definition 7.1.8, as well as thecorresponding discussion of the group actions in Section 6.3 and the invariant in-tegration in Section 7.4.1. However, as we pointed out in Remark 6.2.14 the choicebetween “right” and “left” is completely symmetric, so the reader should have nodifficulty in translating those results to the setting of the right actions consideredhere.

From now on we assume the Lie group G to be compact, and we observethat by Remark 7.3.2, (8), and by Proposition 7.1.10 the space K\G is a compactHausdorff space.

We can regard functions (or distributions) constant on the cosets Kx (x ∈ G)as functions (or distributions) on K\G; it is obvious how one embeds the spacesC∞(K\G) and D′(K\G) into the spaces C∞(G) and D′(G), respectively. Let usdefine PK\G ∈ L(C∞(G)) by

(PK\Gf)(x) :=∫

K

f(kx) dμK(k), (13.1)

where dμK is the Haar measure on the compact Lie group K. Hence PK\Gf ∈C∞(K\G), and PK\G extends uniquely to the orthogonal projection of L2(G)onto the subspace L2(K\G). Let us consider an operator A ∈ L(C∞(G)) with thesymbol satisfying

σA(kx, ξ) = σA(x, ξ) (x ∈ G, k ∈ K, ξ ∈ Rep(G)); (13.2)

this condition is equivalent to

RA(kx, y) = RA(x, y) (13.3)

in the sense of distributions for the right-convolution kernels, in view of (10.18) inSection 10.4.1. Consequently, the Schwartz integral kernel KA of A satisfies

KA(kx, kxy−1) = KA(x, xy−1)

in view of Proposition 10.4.1. Replacing xy−1 by y we have

KA(kx, ky) = KA(x, y).

This means that A maps the space C∞(K\G) into itself. Of course, for a generalA ∈ L(C∞(G)) the equality (13.2) does not have to be true, but then we candefine an operator AK\G ∈ L(D(G)) by the right convolution kernel

RAK\G:= (PK\G ⊗ id)RA,

with PK\G as in (13.1). We note that for A ∈ Ψm(G) its right operator-valuedsymbol rA in Definition 10.11.9 satisfies the property that

rA ∈ C∞(G,L(Hm(G),H0(G))),

13.2. Analysis on compact homogeneous spaces 671

so that the right operator-valued symbol

rAK\G(x) =

∫K

rA(kx) dμK(k)

of AK\G exists as a weak integral (Pettis integral), with the interpretation asin Remark 7.9.3. Consequently, by (10.60) in Theorem 10.11.16, or directly byDefinition 10.4.3, the symbol of AK\G satisfies

σAK\G(x, ξ) =

∫K

σA(kx, ξ) dμK(k)

for all x ∈ G and ξ ∈ Rep(G).

Remark 13.2.6 (Calculus of K-invariant operators). Suppose we are given symbolsof pseudodifferential operators A1, A2 on G satisfying the K-invariance (13.2). Ifwe look at the asymptotic expansion formulae for σA1A2 , σA∗

1and σAt

1in Section

10.7.3, we see that all the terms there are K-invariant in the same sense. Moreover,for an elliptic K-invariant symbol the terms in the asymptotic expansion for aparametrix in Theorem 10.9.10 are also K-invariant. In this way the calculus ofthe K-invariant operators is immediately obtained from the corresponding calculusof operators on the group G.

Theorem 13.2.7 and Corollary 13.2.8 below show how to “project” pseudo-differential operators on G to pseudo-differential operators on K\G. The historyof such averaging processes for pseudo-differential operators can be traced at leastback to the work of M.F. Atiyah and I.M. Singer in the 1960s, and H. Stetkærstudied related topics for classical pseudo-differential operators in [119].

Theorem 13.2.7 (Averaging of operators). Let G be a compact Lie group with aclosed Lie subgroup K. If A ∈ Ψm(G), then AK\G ∈ Ψm(G).

Proof. We will use Theorem 10.9.6 characterising symbols of operators fromΨm(G). First, notice that PK\G is right-invariant, and hence

(∂βx ⊗�α

ξ )(PK\G ⊗ id)σA = (PK\G ⊗ id)(∂βx ⊗�α

ξ )σA

for a left-invariant partial differential operator ∂βx and a difference operator �α

ξ ,

for all α, β ∈ Ndim(G)0 . Therefore

Op(�αξ ∂β

xσAK\G) =

(Op(�α

ξ ∂βxσA)

)K\G .

Since A ∈ Ψm(G), by Theorem 10.9.6 we have

‖�αξ ∂β

xσA(x, ξ)‖op ≤ CAαβm〈ξ〉m−|α|,


and hence we can estimate

‖�αξ ∂β

xσAK\G(x, ξ)‖op =

∥∥∥∥∫K

�αξ ∂β

xσA(kx, ξ) dμK(k)∥∥∥∥

op

≤∫

K

‖�αξ ∂β

xσA(kx, ξ)‖op dμK(k)

≤ CAαβm〈ξ〉m−|α|.

At the same time, formula (13.3) implies that the right-convolution kernel ofOp(σAK\G

) has singularities only at y = e. This proves that σAK\G∈ Σm

0 (G).Let now B ∈ L(C∞(G)) be a left-invariant (right-convolution) pseudo-differ-

ential operator. Then σB(x, ξ) = σB(ξ) is independent of x ∈ G in view of Remark10.4.10, and hence B = BK\G. Consequently, we have

(Op(σAσB))K\G = Op(σAK\GσB)

and(Op(σBσA))K\G = Op(σBσAK\G

).

To argue by induction, assume now that for some k ∈ N0 we have proven thatσCK\G

∈ Σrk(G) for every C ∈ Ψr(G), for every r ∈ R. By Remark 13.2.6 we hence

get[σ∂j

, σAK\G] = [σ∂j

, σA]K\G ∈ Σmk (G),

(�γξ σ∂j

)σAK\G= ((�γ

ξ σ∂j)σA)K\G ∈ Σm+1−|γ|

k (G)

and(�γ

ξ σAK\G)σ∂j

= ((�γξ σA)σ∂j

)K\G ∈ Σm+1−|γ|k (G);

this means that σAK\G∈ Σm

k+1(G), so that by induction we get σAK\G∈ Σm(G) =⋂∞

k=0 Σmk (G). �

Once we get a pseudo-differential operator of the form AK\G, it can be pro-jected to the homogeneous space K\G:

Corollary 13.2.8 (Projection of operators). Let K\G be orientable. Then

AK\G|C∞(K\G) ∈ Ψm(K\G)

for every A ∈ Ψm(G).

Proof. Let us write

Ψm(G)K\G := {AK\G | A ∈ Ψm(G)}

andΨm(G)K\G|C∞(K\G) = {AK\G|C∞(K\G) : A ∈ Ψm(G)}.

13.3. Analysis on K\G, K a torus 673

By Theorem 13.2.7 we know that Ψm(G)K\G ⊂ Ψm(G). Let D be a smooth vectorfield on K\G. Since by Exercise 13.2.4, (G, K\G, K, x �→ Kx, (x, k) �→ kx) is aprincipal fiber bundle, there exists a smooth vector field X = XK\G on G suchthat X|C∞(K\G) = D (see [115]). Then we have

[D,Ψm(G)K\G|C∞(K\G)] = [X, Ψm(G)K\G]|C∞(K\G) ⊂ Ψm(G)K\G|C∞(K\G),

and this combined with Ψm(G)K\G|C∞(K\G) ⊂ L(Hm(K\G),H0(K\G)) yieldsthe conclusion due to the commutator characterisation of pseudo-differential op-erators on closed manifolds in Theorem 5.3.1. �

Definition 13.2.9 (Lifting of operators). We will say that the operator A ∈ Ψm(G)is a lifting of the operator B ∈ Ψm(K\G) if A = AK\G and if A|C∞(K\G) = B.

Remark 13.2.10 (Calculus of liftings). It already follows from Corollary 13.2.8 thatat least sometimes a pseudo-differential operator on K\G can be (possibly non-uniquely) lifted to a pseudo-differential operator on G. If Bj ∈ Ψmj (K\G) can belifted to Cj = (Cj)K\G ∈ Ψmj (G) (i.e., Cj |C∞(K\G) = Bj), then C∗j ∈ Ψmj (G)is a lifting of the adjoint operator B∗j ∈ Ψmj (K\G), and B1B2 ∈ Ψm1+m2(K\G)is lifted to C1C2 ∈ Ψm1+m2(G). Moreover, if C1 is elliptic with a parametrixD ∈ Ψ−m1(G) as in Theorem 10.9.10, then D = DK\G and B1 ∈ Ψm1(K\G) iselliptic with a parametrix D|C∞(K\G) ∈ Ψ−m1(K\G).

13.3 Analysis on K\G, K a torus

In this section we assume that the subgroup K of G is a torus, K ∼= Tq. Forexample, K may be the maximal torus which has an additional importance in therepresentation theory of G in view of Cartan’s maximal torus theorem (Theorem7.8.8). However, it may be a lower-dimensional torus as well.

Remark 13.3.1 (Sphere S2). Let Bn be the unit ball of the Euclidean space Rn,and Sn−1 its boundary, the (n−1)-sphere. The two-sphere S2 can be considered asthe base space of the Hopf fibration S3 → S2, where the fibers are diffeomorphic tothe unit circle S1 ⊂ R2. In the context of harmonic analysis, S3 is diffeomorphic1

to the compact non-commutative Lie group G = SU(2), having a maximal torusK ∼= S1 ∼= T1. Then the homogeneous space K\G is diffeomorphic to S2, so thatthe canonical projection pG→K\G : x �→ Kx is interpreted as the Hopf fiber bundleG → K\G; in the sequel we treat the two-sphere S2 always as the homogeneousspace K\G. Notice that also S2 ∼= T1\SO(3). For a sketch of operators there see[140].

Remark 13.3.2 (Spherical symbols). In [125] a subalgebra of Ψm(S2) was describedin terms of the so-called spherical symbols. Functions f ∈ C∞(S2) can be expanded

1See Proposition 11.4.2.


in series

f(φ, θ) =∞∑

l=0

l∑m=−l

f(l)m Y ml (φ, θ),

where (φ, θ) ∈ [0, 2π]× [0, π] are the spherical coordinates, and the functions Y ml

are the spherical harmonics with “spherical” Fourier coefficients

f(l)m :=∫ π

0

∫ 2π

0

f(φ, θ) Y ml (φ, θ) sin(θ) dφ dθ.

Let us define

(Af)(φ, θ) :=∞∑

l=0

l∑m=−l

a(l) f(l)m Y ml (φ, θ),

where a : N0 → C is a rational function; in [125], Svensson states that A ∈ Ψm(S2)if and only if

|a(l)| ≤ CA,m(l + 1)m.

Let us now present another proof for a special case of Theorem 13.2.7 andCorollary 13.2.8 for the torus subgroup K; this method of proof turns out tobe useful when we develop an analogous method for showing that the mapping(A �→ AK\G|C∞(K\G)) : Ψm(G)→ Ψm(K\G) is surjective if K is a torus subgroup(see the proof of Theorem 13.3.5).

Theorem 13.3.3. Let G be a compact Lie group with a torus subgroup K. If A ∈Ψm(G), then AK\G ∈ Ψm(G) and the restriction AK\G|C∞(K\G) ∈ Ψm(K\G).

Proof. Let dim(G) = p + q, where K ∼= Tq. Let V = {Vi | i ∈ I} be a lo-cally trivialising open cover of the base space K\G for the principal fiber bundle(G, K\G, K, x �→ Kx, (x, k) �→ kx). Let U = {Uj | 1 ≤ j ≤ N} be an open coverof K\G such that for every j1, j2 ∈ {1, . . . , N} there exists Vi ∈ V containingUj1 ∪Uj2 whenever Uj1 ∩Uj2 = ∅; notice that we can always refine any open coveron a finite-dimensional manifold to get a new cover satisfying this additional re-quirement. Then each Ui∪Uj (1 ≤ i, j ≤ N) is a chart neighbourhood on K\G, andfurthermore there exist diffeomorphisms φij : (Ui ∪ Uj)×K → p−1

G→K\G(Ui ∪ Uj)such that pG→K\G(φij(x, k)) = x for every x ∈ Ui ∪ Uj and k ∈ K. To simplifythe notation, we treat the neighbourhood Ui ∪ Uj ⊂ K\G as a set Ui ∪ Uj ⊂ Rp,and p−1

G→G/K(Ui ∪ Uj) ⊂ G as a set (Ui ∪ Uj)× Tq ⊂ Rp × Tq.Let {(Uj , ψj) | 1 ≤ j ≤ N} be a partition of unity subordinate to U , and

let Aij = MψiAMψj

∈ Ψm(G). With the localised notation we consider Aij ∈Ψm(Rp×Tq; Rp×Zq), so that it has the symbol σAij

∈ Sm(Rp×Tq; Rp×Zq). Wenote that the notation we use for symbols here is slightly different from before:Rp × Tq stands for the space variables, and Rp × Zq is for dual frequencies. Then

σ(AK\G)ij(x, ξ) = σ(Aij)K\G

(x, ξ)

=∫

Tq

σAij(x1, . . . , xp, xp+1 + z1, . . . , xp+q + zq; ξ) dz1 · · · dzq,


and it is now easy to check that σ(AK\G)ij∈ Sm(Rp × Tq; Rp × Zq). This yields

(AK\G)ij ∈ Ψm(G), and hence

AK\G =∑i,j

(AK\G)ij ∈ Ψm(G),


Theorem 13.3.3 has the inverse which will be given in Theorem 13.3.5. Butfirst, we prepare a lemma on the extension of symbols in the Euclidean space.Because of the commutator characterisations in Chapter 5 (especially the equality(5.4) in Theorem 5.4.1), and in view of Corollary 4.6.13, all of the symbol classeson Tn in both the Euclidean and toroidal quantizations coincide. That is why, tosimplify the notation, we will skip writing the space for the frequency variableand will only write the space which will usually be Rp × Tq. Thus, the classΨm(Rp×Tq) will stand for either Ψm(Rp×Tq; Rp×Zq) or for Ψm(Rp×Tq; Rp×Rq),which we know to be equal, with the correspondence between the Euclidean andtoroidal symbols given in Theorem 4.5.3. The same will apply for symbols, withthe quantization clear from the context.

Lemma 13.3.4 (Extension of symbols). Let χ ∈ C∞(Rp+q) be homogeneous oforder 0 in Rp+q \ B(0, 1), i.e.,2 χ(ξ) = χ(ξ/‖ξ‖) when ‖ξ‖ ≥ 1. Furthermore,assume that χ satisfies χ|(U×Rq)\B(0,1) ≡ 0 and χ|Rp×V ≡ 1, where U ⊂ Rp andV ⊂ Rq are neighbourhoods of zeros. Let σB ∈ Sm(Rp) and write

σA(x, ξ) := χ(ξ) σB(Px, Pξ),

where P : Rp+q → Rp is defined by P (x1, . . . , xp+q) = (x1, . . . , xp). Then σA ∈Sm(Rp+q) and σA|(Rp×Rq)×(Rp×Zq) ∈ Sm(Rp × Tq).

Proof. We shall first prove that

|(∂γξ χ)(ξ)| ≤ Cγr 〈Pξ〉−r 〈ξ〉r−|γ| (13.4)

for every r ∈ R and for every γ ∈ Np+q0 . It is trivial that (x, ξ) �→ χ(ξ) belongs to

S0(Rp+q). If r ≥ 0 then obviously (13.4) is true. Since we are not interested in thebehaviour of the symbols when ‖ξ‖ is small, we assume that ‖ξ‖ > 1 from hereon. There exists r0 ∈ (0, 1) such that χ(ξ) = 0 when ‖Pξ‖ < r0. Let r < 0 andξ ∈ supp(χ). Then ‖Pξ‖ ≥ r0‖ξ‖, and thus

|(∂γξ χ)(ξ)| ≤ Cγ 〈ξ〉−|γ|

= Cγ 〈Pξ〉−r 〈Pξ〉r 〈ξ〉−|γ|

≤ Cγ 〈Pξ〉−r 〈r0ξ〉r 〈ξ〉−|γ|

≤ Cγ rr0 〈Pξ〉−r 〈ξ〉r−|γ|.

2Here B(0, 1) stands for the unit ball in Rp+q centred at the origin and of radius 1.


Hence the inequality (13.4) is proven. Now

|∂αξ ∂β

xσA(x, ξ)| ≤∑γ≤α

(α

γ

)|(∂γ

ξ χ)(ξ)| |(∂α−γξ ∂β

xσB)(Px, Pξ)|

≤∑γ≤α

(α

γ

)Cγrγ 〈Pξ〉−rγ 〈ξ〉rγ−|γ| CB(α−γ)βm 〈Pξ〉m−|α−γ|

≤ CBαβmχ 〈ξ〉m−|α|,if we choose rγ = m − |α − γ|. Thereby σA ∈ Sm(Rp+q). Clearly we can regardthis symbol as a function σA : (Rp×Tq)× (Rp×Rq)→ C and study its restrictionσA|(Rp×Tq)×(Rp×Zq). We claim that this restriction belongs to Sm(Rp×Tq). Indeed,the Taylor expansion of a function σ ∈ C∞(Rq) yields

�γξ σ(ξ) =

∑δ≤γ

(γ

δ

)(−1)|γ−δ| σ(ξ + δ)

=∑δ≤γ

(γ

δ

)(−1)|γ−δ|

⎛⎝ ∑|ρ|<|γ|

1ρ!

δρ (∂ρξ σ)(ξ) +

∑|ρ|=|γ|

1ρ!

δρ (∂ρξ σ)(ξ + θδδ)

⎞⎠=

∑|ρ|<|γ|

1ρ!

(∂ρξ σ)(ξ)

∑δ≤γ

(γ

δ

)(−1)|γ−δ|δρ +

∑δ≤γ

∑|ρ|=|γ|

1ρ!

δρ (∂ρξ σ)(ξ + θδδ)

=∑δ≤γ

∑|ρ|=|γ|

1ρ!

δρ (∂ρξ σ)(ξ + θδδ),

because ∑δ≤γ

(γ

δ

)(−1)|γ−δ|δρ = �γ

ξ ξρ|ξ=0 = 0

whenever |ρ| < |γ|. Therefore

|�γξ σ(ξ)| ≤

∑δ≤γ

∑|ρ|=|γ|

1ρ!

δρ |(∂ρξ σ)(ξ + θδδ)| ≤ cγ sup

η∈Sγ ,|ρ|=|γ||(∂ρ

ξ σ)(ξ + η)|,

where Sγ is the rectangle∏q

j=1[0, γj ]. Let α′ = (Pα, 0, . . . , 0) and let α′′ = α−α′;then

|∂α′ξ �α′′

ξ ∂βxσA(x, ξ)| ≤ Cα sup

η∈Sα′′ ,|ρ|=|α′′||∂α′+ρ

ξ ∂βxσA(x, ξ + η)|

≤ Cα CAαβm supη∈Sα

〈ξ + η〉m−|α|

≤ Cα CAαβm 2|m−|α|| supη∈Sα

〈η〉|m−|α|| 〈ξ〉m−|α|

≤ Cα CAαβm 2|m−|α|| 〈α〉|m−|α|| 〈ξ〉m−|α|

= C ′Aαβm 〈ξ〉m−|α|;


notice the application of the Peetre inequality (Proposition 3.3.31):

〈ξ + η〉s ≤ 2|s| 〈ξ〉s 〈η〉|s|.

Hence σA|(Rp×Tq)×(Rp×Zq) ∈ Sm(Rp × Tq). �

Now we are ready to give the converse to Theorem 13.3.3:

Theorem 13.3.5 (Lifting of operators). Let G be a compact Lie group with a torussubgroup K. Let B ∈ Ψm(K\G). Then there exists an operator A = AK\G ∈Ψm(G) such that A|C∞(K\G) = B.

Proof. Let K ∼= Tq, dim(G) = p + q, and let {(Uj , ψj) | 1 ≤ j ≤ N} be thesame partition of unity as in the proof of Theorem 13.3.3. Let Bij = Mψi

BMψj∈

Ψm(K\G). With the localised notation we consider Bij ∈ Ψm(Rp; Rp), so thatit has the symbol σBij : Rp × Rp → C, and the mapping (x, ξ) �→ σBij (x, ξ) iszero when x ∈ Rp \ (Ui ∪ Uj). By Lemma 13.3.4 there exists a pseudo-differentialoperator Aij ∈ Ψm(Rp×Tq; Rp×Zq) such that σAij

: (Rp×Tq)× (Rp×Zq)→ Csatisfies

σAij(x;Pξ, 0, . . . , 0) = σBij

(Px;Pξ),

where Py = (y1, . . . , yp) (y ∈ Rp+q). Because Aij are independent of the K-variables, we have A = AK\G =

∑i,j Aij ∈ Ψm(G) and A|C∞(K\G) ∈ Ψm(K\G).

Let f =∑

k fk ∈ C∞(K\G) ⊂ C∞(G), fk = fψk. Then we have

(Af)(x) =∑i,j,k

(Aijfk)(x)

=∑i,j,k

∫Rp

∑ξp+1,...,ξp+q∈Z

σAij(x, ξ) fk(ξ) ei2πx·ξ dξ1 · · ·dξp

=∑i,j,k

∫Rp

σAij (x;Pξ, 0, . . . , 0) fk(Pξ, 0, . . . , 0) ei2π(Px)·(Pξ) dξ1 · · ·dξp

=∑i,j,k

∫Rp

σBij(Px;Pξ) fk(Pξ, 0, . . . , 0) ei2π(Px)·(Pξ) dξ1 · · ·dξp

=∑i,j,k

(Bijfk)(Px)

= (Bf)(Kx),


Remark 13.3.6. Theorem 13.3.5 provides just one way of lifting operators inΨm(K\G) to operators in Ψm(G), unfortunately destroying ellipticity: this is dueto the apparent non-ellipticity of the symbol χ in Lemma 13.3.4. We now discussthis problem and a possibility of other liftings.


Remark 13.3.7 (Lifting the identity). Let us lift the identity operator I ∈ Ψ0(Rp)using the process suggested by Lemma 13.3.4. Of course, it would be desirable ifI ∈ Ψ0(Rp) could be extended to the identity in Ψ0(Rp+q), but now σI(x, ξ) ≡ 1,and thereby its lifting A ∈ Ψ0(Rp+q) has the non-elliptic homogeneous symbolσA = χ ∈ S0(Rp+q).

Given an elliptic symbol σB ∈ Sm(Rp) one can occasionally modify theconstruction in Lemma 13.3.4 to get an extended elliptic symbol in Sm(Rp+q).Sometimes the following trick helps: Let σA1 ∈ Sm(Rp+q) be an extension of σB1

as in Lemma 13.3.4,

σA1(x, ξ) = χ1(ξ) σB1(x1, . . . , xp; ξ1, . . . , ξp),

where χ1 ∈ S0(Rp+q) is a homogeneous symbol such that χ1|(U×Rq)\B(0,1) ≡ 0,χ1|Rp×V ≡ 1, where U ⊂ Rp and V ⊂ Rq are neighbourhoods of zeros. Take anyelliptic symbol σB2 ∈ Sm(Rq), and modify Lemma 13.3.4 to construct an extensionσA2 ∈ Sm(Rp+q) such that

σA2(x, ξ) = χ2(ξ) σB2(xp+1, . . . , xp+q; ξp+1, . . . , ξp+q)

for a homogeneous symbol χ2 ∈ S0(Rp+q) satisfying χ2|(U×Rq)\B(0,1) ≡ 1 andχ2|(Rp×V )\B(0,1) ≡ 0. Then σA1 +σA2 ∈ Sm(Rp+q) is an extension for σB1 (moduloinfinitely smoothing operators). For instance, if B1 = I ∈ Ψ0(Rp), let B2 = I ∈Ψ0(Rq) and χ2(ξ) = 1−χ1(ξ) (for |ξ| > 1), then A1+A2 = I ∈ Ψ0(Rp+q) (moduloinfinitely smoothing operators).Remark 13.3.8 (No elliptic liftings). It may happen that any lifting process for anelliptic symbol σB ∈ Sm(Rp) yields a non-elliptic symbol in Sm(Rp+q). Consider,for instance, a case where B ∈ Ψm(R2) is an elliptic convolution operator andξ �→ f(ξ) ≡ σB(x, ξ) is homogeneous outside the unit ball B(0, 1) ⊂ R2. If therestricted mapping f |S1 : S1 → C \ {0} is not homotopic to a constant mapping(i.e., f |S1 has a non-zero winding number) then no lifting σA ∈ Sm(R3) of σB canbe elliptic.

Multiplications on K\G have already been lifted to multiplications on Gvia x �→ Kx, and A = AK\G for any right-convolution operator (multiplier)A ∈ L(C∞(G)) (in fact, then σA(x, ξ) = σA(ξ) for every x ∈ G). Sometimeson K\G we have operators that resemble convolution operators. Suppose we aregiven a right-convolution operator A ∈ Ψm(SU(2)). Then the restriction B =A|C∞(S2) ∈ Ψm(S2) is of the form

(Bf)(φ, θ) =∞∑

l=0

l∑m=−l

(l∑

n=−l

a(l)mn f(l)n

)Y m

l (φ, θ), (13.5)

where the coefficients a(l)mn ∈ C can be calculated from the data

{BY ml | l ∈ N0, m ∈ {−l,−l + 1, . . . , l − 1, l}}.

13.4. Lifting of operators 679

It is even true that the original operator A can be retrieved from the coefficientsa(l)mn. In fact, any operator B ∈ L(C∞(S2)) of the form (13.5) can be lifted toa unique right-convolution operator belonging to L(C∞(SU(2))). An interestingspecial case is

(Bf)(x) =∫

S2κ(x · y) f(y) dy,

where κ ∈ D′(S2), (x, y) �→ x · y is the scalar product of R3, and the integration iswith respect to the angular part of the Lebesgue measure of R3. Then

(Bf)(φ, θ) =∞∑

l=0

l∑m=−l

cl κ(l)0 f(l)m Y ml (φ, θ)

for some normalising constants cl depending only on l ∈ N0.

13.4 Lifting of operators

We now describe the lifting of operators from K\G to G for general closed sub-group K, which can be done similar to the proof of Theorem 13.3.5:

Theorem 13.4.1 (Lifting of operators). Let G be a compact Lie group with a closedsubgroup K. Let B ∈ Ψm(K\G). Then there exists an operator A = AK\G ∈Ψm(G) such that A|C∞(K\G) = B.

The rest of this section is devoted to the proof of this theorem. Since weproved Theorem 13.3.5 and Lemma 13.3.4 in detail, we sketch the constructionhere. We start with the following

Lemma 13.4.2 (Some properties of representations). Let φ0 ∈ Rep(G) be the trivialone-dimensional representation given by φ0(x) = 1 for all x ∈ G. Then for everyφ ∈ Rep(G) we have

1(φ) = δφ,φ0 Idim φ ={

1, if φ = φ0,0, if φ = φ0,

where 0 on the right-hand side is the zero operator 0 ∈ L(Hφ). Moreover, for everynon-trivial φ ∈ Rep(G), i.e., φ ∈ [φ0], we have

∫G

φ(x) dx = 0.

Proof. If φ ∈ Rep(G) is such that φ ∈ [φ0], then∫

Gφ(x) dx = 0 follows from the or-

thogonality of φij , 1 ≤ i, j ≤ dim φ, and φ0, given in Lemma 7.5.12. Consequently,1(φ) =

∫G

φ(x)∗ dx = 0 ∈ L(Hφ) if φ ∈ [φ0], and 1(φ0) =∫

Gφ0(x)∗ dx = 1. �

Exercise 13.4.3. Show that for every φ ∈ Rep(G) we have φ(φ) = Idim φ, theidentity operator on L(Hφ).


By the argument similar to that in the proof of Theorem 13.3.5 we can usethe partition of unity to reduce the question to the extension of symbols from Rp

to Rp × K. Let x = (x′, x′′) ∈ Rp × K, let ξ ∈ Rp and φ ∈ Rep(K). Assumethat σB = σ(x′, ξ) ∈ Sm(Rp) has an extension to Rp ×K, i.e., that there exists asymbol σA = σA(x′, x′′, ξ, φ) ∈ Sm(Rp)⊗ Σm(K) such that

σA(x′, x′′, ξ, φ0) = σB(x′, ξ), (13.6)

where φ0 ∈ Rep(K) is the trivial representation. Then by the argument of Theorem13.3.5, and using Lemma 13.4.2, we have

(Af)(x) =∑i,j,k

(Aijfk)(x)

=∑i,j,k

∫Rp

∑[φ]∈K

σAij (x′, x′′, ξ, φ) fk ⊗ 1(ξ, φ) ei2πx′·ξφ(x′′) dξ

=∑i,j,k

∫Rp

σAij(x′, x′′, ξ, φ0) fk(ξ) ei2πx′·ξ dξ

=∑i,j,k

(Bijfk)(x′)

= (Bf)(Kx).

Thus, we need to construct an extension σA of σB from Rp to Rp ×K satisfyingproperty (13.6). We note that it is enough to do it for symbols σB with compactsupport in x′. First, let us define operator C ∈ Ψ0(Rp ×K) by

C := (I − 2LK)(I − LRp×K)−1,

where LK is the Laplace operator on K and LRp×K = LRp + LK is the Laplaceoperator on Rp × K. For each φ ∈ Rep(K), φ : K → U(Hφ), let λφ ≥ 0 be theeigenvalue of −LK corresponding to the eigenspace Hφ ⊂ L2(K). For the detailsof this construction on compact groups we refer to Theorem 8.3.47. Consequently,we have σI−2LK

(ξ, φ) = (1+2λφ)Idim φ and σI−LRp×K(ξ, φ) = (1+ |ξ|2 +λφ)Idim φ,

so that

σC(x′, x′′, ξ, φ) = σC(ξ, φ) = (1 + 2λφ)(1 + |ξ|2 + λφ)−1Idim φ.

Now, let χ ∈ C∞0 (R) be such that χ(t) = 1 for |t| ≤ 1/2 and χ(t) = 0 for |t| ≥ 1.Let us denote

A0 := χ(C) = χ((I − 2LK)(I − LRp×K)−1

).

By writing χ(t) =∫

Re2πiτt χ(τ) dτ , we have A0 =

∫R

e2πiτC χ(τ) dτ ∈ Ψ0(Rp ×K). This follows from the fact that the operator u(τ) := e2πiτC is the solution ofthe Cauchy problem

i∂τu(τ) + 2πCu(τ) = 0, u(0) = I,

13.4. Lifting of operators 681

and hence can be constructed by solving the transport equations. We note thatsuch an argument is a special case of solving the hyperbolic Cauchy problems interms of Fourier integral operators but the situation at hand is simpler becauseoperator C is of order zero. Thus, A0 ∈ Ψ0(Rp×K) follows, e.g., from [130, SectionXII.1]. For the representation of χ(C) on the Fourier transform side one can alsosee that

σA0(x′, x′′, ξ, φ) = σA0(ξ, φ) = χ

((1 + 2λφ)(1 + |ξ|2 + λφ)−1

)Idim φ

is a diagonal symbol. We claim that the operator Op(σA) with σA := σA0σB

satisfies the required properties. Let us first show that A ∈ Ψm(G). For this wecan apply Theorem 10.9.6 and use the characterisation given in Definition 10.9.5.On one hand we have

||�αφσA||op = ||(�α

φσA0)σB ||op ≤ Cα(〈φ〉+ 〈ξ〉)−|α|〈ξ〉m

which implies ||�αφσA||op ≤ C ′α(〈φ〉 + 〈ξ〉)m−|α|, if we use Theorem 10.9.6 for

the operator A0 ∈ Ψ0(G) as well as the fact that λφ ≤ |ξ|2, and hence 〈φ〉 =(1 + λφ)1/2 ≤ 〈ξ〉 on the support of σA0(ξ, φ). On the other hand we have

||∂ξjσA||op ≤ ||∂ξj

σA0 ||op |σB |+ |χ((1 + 2λφ)(1 + |ξ|2 + λφ)−1)| |∂ξj

σB |≤ C1(〈φ〉+ 〈ξ〉)−1〈ξ〉m + C2〈ξ〉m−1

,

and again this implies ||∂ξjσA||op ≤ C3(〈φ〉 + 〈ξ〉)m−1 if we use that 〈φ〉 ≤ 〈ξ〉

on the support of σA0(ξ, φ). Similarly, one can extend this to the higher-orderderivatives ∂α

ξ . Let us show (10.36). Using the usual Euclidean formula togetherwith (10.18), the right-convolution kernel of A can be written as

RA(x′, x′′, y′, y′′) =∫

Rp

e2πiy′·ξ ∑[φ]∈K

dim(φ) Tr (φ(y′) σA0(ξ, φ)) σB(x′, ξ) dξ.

Now, if y′ = 0 we can integrate by parts in ξ any number of times. At the sametime, viewing σA0 as a ξ-dependent symbol on K implies that RA is smooth fory′′ = e because the same property holds for σA0 . Hence the singular supportof y �→ RA(x, y) is contained in (0, e). The other properties in Definition 10.9.5follow from the diagonality of the symbol σA0 . It remains to show (13.6). For thiswe note that φ0 ≡ 1 is the eigenfunction of −LK with the eigenvalue λφ0 = 0.Consequently, we have

σA0(x′, x′′, ξ, φ0) = χ

((1 + |ξ|2)−1

)= 1

for |ξ| ≥ 1. Finally we note that we can modify σA(x′, x′′, ξ, φ0) arbitrarily forsmall ξ, in particular setting it to be zero for |ξ| ≤ 1, similarly to Lemma 13.3.4,thus completing the proof. �

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Notation

Part I

∅, x ∈ A, x ∈ A, A ⊂ B, 10N, N0, Z, Q, R, C, Z+, R+, 10A ∪B, A ∩B, A = B, 10P(X), 10Ac, 11f |A, 13∼, 14min, max, inf, sup, 15lim inf, lim sup, 16∏

j∈J Xj , 18card(A), 20|A|, 20d(x, y), 26Br(x), Bd(x, r), 26dp, d2, d∞, 27C([a, b]), B([a, b]), 27d(A,B), 28(xk)∞k=1, 29

limk→∞

xk = p, xk → p, xkd−−−−→

k→∞p, 29

τ , τ∗, 31τd, τ(d), 32τA, 33τ1 ⊗ τ2, 34intd(A), extd(A), ∂d(A), 35Vτ (x), V(x), 36NEFIS(X), 72K, 79Kn, 80V X , 80span(S), 80

L(V,W ), L(V ), 82Ker(A), Im(A), 82σ(A), 82x �→ ‖x‖X , ‖x‖, 92C(K), 93BX(x, r), B(x, r), 94‖A‖op, 94L(X, Y ), L(X), 94V ′, L(V, K), 94LC(X, Y ), LC(X), 95Γ(f), 99〈x, y〉, 103x⊥y, M⊥N , 103PM , PM (x), 105M⊥, M ⊕M⊥, 106M1 ⊕M2,

⊕j∈J Hj , 106

S1, Tr(A), 111S2, 〈A,B〉S2 , 112‖A‖HS , 112a⊗ b, X1 ⊗ · · · ⊗Xr, 83A⊗B, 84X ⊗π Y , X⊗πY , 90X ⊗ε Y , X⊗εY , 91∑

j∈J aj , 115m∗, 116M(ψ), 118Σ(τ), 119μ-a.e., 137f ∼μ g, 137f+, f−, 141∫

f dμ, 143Lp(μ), L∞(μ), ‖f‖Lp(μ), 152

694 Notation

ν+, ν−, |ν|, 159μ⊥λ, 1611A, 191[A,B], 192σA(x), 192A ∼= B, Hom(A,B), 195Spec(A), 208ρ(x), 205R(λ), 204Gel, Gel−1, 216H(Ω), 217

Part II

Spaces, sets

Z+, N0, Z, R, C, 298U|W , 330τX , (X, τX), 330(uj)j∈Z+ , 330L(X, Y ), L(X, Y ), 330supp, 239Ker, Im, 381Rn/Zn, 300Cm

1 , C∞(Tn), 300S1, 299Tn = Rn/Zn, 299L2, L2(Tn), 302TrigPol(Tn), 305Hs(Tn), 307Hs,t(Tn × Tn), 309C∞(Tn × Tn), 309C∞(Tn × Zn), 338Sm(Tn × Zn), Sm

ρ,δ(Tn × Zn),

S∞ρ,δ(Tn × Zn), S−∞(Tn), 338

Sm(Rn × Rn), 260Sm

ρ,δ(Rn × Rn), 261

Op(Sm(Tn × Zn)),Op(Sm

ρ,δ(Tn × Zn)), 338

Op(S∞(Tn)), Op(S−∞(Tn)), 338Am(Rn), Am

ρ,δ(Rn), 275

Am(Tn), Amρ,δ(T

n), A−∞(Tn), 341

Op(Am(Tn)), Op(Amρ,δ(T

n)), 341Op(A∞(Tn)), Op(A−∞(Tn)), 341S(Rn), 224Kers, 382sing supp, 383Hs(x), Hs(U), 383sing suppt, 383

Operators, etc.

clX(·), cl(·), U �→ U , 330‖ · ‖X , ‖ · ‖L(X,Y ), 330u �→ u, 302τx, R, 244(·, ·)L2(Tn) = (·, ·)H0(Tn), 302, 309‖ · ‖Hs(Tn), 307ϕs, 308(·, ·)Hs(Tn), 308〈·, ·〉, 308A∗, A(∗B), A(∗H), 309‖ · ‖s,t, 309�, �ξ, 310�, �ξ, 310‖ · ‖�p , 319δj,k, 322

∂kx , ∂

(k)x , ∂

(−k)x , 327

σ, A �→ σA, 335Op, σ �→ Op(σ), 335a �→ a, 342, 346a �→ a1, a �→ a2, 346f �→ f , FRn , 222F−1

Rn , 225FTn , F−1

Tn , 301exp, 368a∗, a(∗B), a(∗H), 370[·, ·], 371Lj , Rk, 423[·, ·]θ, 385σA, 262Dα

x , 327(−Dy)(α), 394

Notation 695

Other notation

p.v. 1x , 1

x±i0 , 238A|W , 330A(x)|x=x0 , 330uj

τ−→ u, 330Ind, 381dim, codim, 381z, 302〈ξ〉, 221, 300ξ(j), 313S

(j)k ,

{kj

}, 321

K, 340, 350

∼, m∼,m,ρ,δ∼ , 342

[·], 342σ ∼∑∞

j=0 σj , 352Op(σ) ∼∑∞

j=0 Op(σj), 352pj →∞, 386α!, 223α ≤ β, 223|α|, 223Dα, 224L, 225

Part III

K, 430Aut(V ), 431Aff(V ), 431xA, Ax, AB, A0, A−1, An, A−n, 432H < G, H � G, 432Z(G), 432GL(n, R), O(n), SO(n), 433GL(n, C), U(n), SU(n), 433G/H, 433H\G, 434Gq, 437U(H), 438πL, πR, 439φ ∼ ψ, 441τG/H , 447

HOM(G1, G2), 448⊕j∈J φ|Hj , 450

Haar(f), 459PG/H , 462HaarG/H , 463

G, 468TrigPol(G), 474L2(G), 477ResG

Hψ, 482Cφ(G,H), 484IndG

φH, 484exp(X), 492LieK(A), 499Lie(G), g, 499gl(n, K), o(n), so(n), u(n), su(n), 500SL(n, K), sl(n, K), 501Ad(A)X, 505ad(X)Y , 506, 508U(g), 507

Part IV

Re G, 530Hξ, ξ : G→ U(Hξ), 530f(ξ), 531φu, 531DY f , 532f(ξ)mn, 533δmn, 533LG, L, 534D′(G), 534〈·, ·〉G, 535Hs(G), 535Hξ, 536dim(ξ), 536λξ, 537M(G), 537L2(G), 537(·, ·)L2(G), 537〈ξ〉, λ[ξ], 538

696 Notation

S(G), pk, 539S ′(G), 543〈·, ·〉G, 543

Lp(G), �p(G, dimp( 2

p− 12 )

), 546

Lpk(G), 550

KA(x, y), LA(x, y), RA(x, y), 550l(f), r(f), 551, 579∂α, 534, 560σA(x, ξ), 552fφ, Aφ, 556uL, uR, 556〈A,B〉HS , ||A||HS , ||A||op, 559�α

ξ , 564�q, 564Am

k (M), 566Σm(G), Σm

k (G), 575πL, πR, 580RA(x), LA(x), 582la, la(x), rA, rA(x), 583D(G), 591SU(2), 599H, 603Sp(n), 606Sp(n, C), 606w1, w2, w3, 607Y1, Y2, Y3, 607D1, D2, D3, 609∂+, ∂−, ∂0, 611Vl, Tl, 612tl, tlmn, P l

mn, 617t−−, t−+, t+−, t++, 621f(l)mn, 632σA(x, l), σA(x, l)mn, 632σ∂+ , σ∂− , σ∂0 , 634Sm(SU(2)), 656Sm(S3), 661D′L1(M), 668pE→B , 668K\G, 669

Index

∗-algebra, 213Ψ(M), 423Diff(M), 419of pseudo-differential operators

on Tn, 380∗-homomorphism, 213〈ξ〉 on a group, 538⟨tl

⟩on SU(2), 633

μ-almost everywhere, 137μ-integrable, 143∂α on groups, 560σ-algebra, 119

Abel–Dini theorem, 386action

free, 668left, right, 437linear, 437of a group, 436transitive, 437, 669

adjoint operator, 107Banach, 101on a group, 569, 591

adjoints (Banach, Hilbert), 309Ado–Iwasawa theorem, 508affine group, 431algebra, 191∗-algebra, involutive, 213Spec(A), spectrum of, 208Banach, 200C∗-algebra, 213character of, 208commutative, 191derivations of, 499homomorphism, 195Hopf, 520

involution, 213isomorphism, 195Lie, 498quotient, 194, 199radical of, 193, 211semisimple, 193tensor product, 196topological, 196unit, inverse, 191unital, 191universal enveloping, 507

algebra of periodic ΨDOs, 367, 380algebra reformulation, 518

associativity diagram of, 518co-algebra, 519co-associativity diagram, 519multiplication mapping, 518tensor product, 518unit mapping, 518

algebraicbasis, 80dimension, 81number, 24tensor product, 84

almost orthogonality lemma, 95amplitude

of adjoints, 370of periodic integral operator, 388operator, 275, 341toroidal, 340

amplitudes Am(Rn), Amρ,δ(R

n), 275amplitudes Am(Tn), Am

ρ,δ(Tn), 340

Arzela–Ascoli theorem, 57asymptotic equivalence, 342

698 Index

asymptotic expansion, 352, 353of adjoint, 279, 370of parametrix, on a group, 577of parametrix, toroidal, 380of product, 371of transpose, 280, 369

asymptotic sums, 351atlas, 416automorphism, 430

inner, 531space Aut(V ), 431

automorphism group, 431Axiom of Choice, 18, 25, 73

for Cartesian products, 18

Baire’s theorem, 96balls Br(x), Bd(x, r), 26Banach

adjoint, 101algebra, 200duality, 308fixed point theorem, 43injective tensor product, 91projective tensor product, 91

Banach space, 94dual of, 101reflexive, 102

Banach–Alaoglu theorem, 99in Hilbert spaces, 109in topological vector spaces, 89

Banach–Steinhaus theorem, 97barrel, 90basis

algebraic, 80orthonormal, 110

Bernstein’s theorem, 306bijection, 13bilinear mapping, 83Borel

σ-algebra, 119measurable function, 135sets, 119

Borel–Cantelli lemma, 122boundary, 36

bounded inverse theorem, 99

C∗-algebra, 213Calderon–Zygmund covering lemma,

257canonical mapping of a Lie algebra,

507Caratheodory condition, 125Caratheodory–Hahn extension, 123cardinality, 20Cartan’s maximal torus theorem,

481Cartesian product, 12, 18, 71Casimir element, 510Cauchy’s inequality, 229Cauchy–Schwarz inequality, 103,

229, 230chain, 15character

of a representation, 479characterisation of S−∞(Tn), 348characteristic function, 13, 135characters, 582Chebyshev’s inequality, 148choice function, 17closed graph theorem, 99closure, 36closure operator, interior operator,

37co-algebra, 519

monoid, 521co-induced

family, 14topology, 69

commutant, 198commutator, 192, 371commutator characterisation

Euclidean, 414on a group, 566on closed manifolds, 421toroidal, 424

complemented subspace, 101complete topological vector space,

86

Index 699

completion, 44of a topological vector space, 86

component, 448composition, 13composition formula

Euclidean, 271for Fourier series operators, 394,

397on a group, 567, 568toroidal, 371

continuitymetric, 30topological, 46uniform on a group, 452

continuum hypothesis, 26generalised, 26

contraction, 43convergence

almost everywhere, 138almost uniform, 138in Lp(μ), 155in measure, 138in metric spaces, 29in topological spaces, 32metric uniform, 42of a net, 77pointwise, 41, 138uniform, 42, 138

convex hull, 89convolution, 228

left, right, l(f), r(f), 579associativity of, 228non-associativity of, 245of distributions, 244of linear operators, 520of sampling measures, 456on a group, 478, 532translations of, 246

convolution kernel, left, right, 582convolution operators, 551, 579Cotlar’s lemma, 406cover, 49

locally trivialising, 668cyclic vector, representation, 450

de Morgan’s rules, 12density, 36derivations of operator-valued

symbols, 587derivatives and differences, 325diameter, 28difference operators

forward, backward, 310on SU(2), �q, �+, �−, �0, 636on SU(2), formulae for, 638on a group, 564

Diracdelta, 239, 243delta comb, 306, 364

direct sum, 101, 106algebraic, 440of representations, 450

discretecone, 389fundamental theorem of calculus,

314integration, 314partial derivatives D

(α)x , 327

polynomials, 313Taylor expansion, 315

disjoint family, 119distance, 26

between sets, 28distribution function, 255distributionsD′(Ω), 242E ′(Ω), 242D′(Tn), periodic, 304on manifolds, 419periodic, 307, 308summable, D′L1(M), 668

dualalgebraic, 84Banach, Hilbert, 308of Lp(μ), 170of a Banach space, 101second, 102space, 85

700 Index

unitary, 468duality〈·, ·〉G, 535〈·, ·〉G, 543

Egorov’s theorem, 139ellipticity, 376

on a group, 577embedding, 309embedding theorem, 294endomorphism, 430equicontinuous family, 57, 85equivalence relation, 14Euler’s angles

on S3, 604on SO(3), 597on SU(2), 601

Euler’s identity, 284exponential coordinates, 500exponential of a matrix, 492extreme set, 89

family, 9family induced, co-induced, 14, 134Fatou’s lemma, 146

reverse, 147Fatou–Lebesgue theorem, 151fiber, 668fiber bundle, 668

principal, 668finite intersection property, 50, 72Fourier coefficients

on a group, 475Fourier coefficients, series, 302Fourier inversion

global, 580Fourier inversion formula

Euclidean, 225on S ′(Rn), 238on S(Zn), C∞(Tn), 301

Fourier serieson L2(Tn), 302on a group, 475

Fourier series operator, 393, 407

Fourier transformf(l)mn, on SU(2), 632and rotations, 227Euclidean, 222inverse, on S ′(G), 545inverse, on Lp(G), 548matrix, 533multiplication formula, 226of Gaussians, 226of tempered distributions, 233on D′(G), 545on L1(G), 548on Lp(G), 548on D′(Tn), 305on a group, 475on group G, 531toroidal, periodic, 301

Frechet space, 87Fredholm

integral equations, 44operator, 381

freezing principle, 288Frobenius reciprocity theorem, 488Fubini theorem, 187Fubini–Tonelli theorem, 186function, 12M-measurable, Borel measurable,

Lebesgue measurable, 135Holder continuous, 306harmonic, 293holomorphic, 217negative part of, 141periodic, 300positive part of, 141simple, 141test, 88weakly holomorphic, 204

functionalHaar, 454–460linear, 82positive, 175, 453positive, in C∗-algebra, 216

Index 701

functional calculus at the normalelement, 216

Gelfandtheorem, 1939, 203theorem, 1940, 208, 210theory, 207topology, 209, 517transform, 209, 517

Gelfand–Beurling spectral radiusformula, 205

Gelfand–Mazur theorem, 205Gelfand–Naimark theorem, 214

commutative, 215graph, 99group, 430

SU(2), 599SO(2), 596SO(3), 596Sp(n, C), 606Sp(n), 606U(1), 595unitary, U(H), 438action, 436affine, 431centre of, 432commutative, Abelian, 431, 491compact, 451finite, 431general linear GL(n, R),

GL(n, C), 433homomorphism, 435infinitesimal, 498isomorphism, 435Lie, 491linear Lie group, 491locally compact, 451orthogonal O(n), 433permutation, 432product, 431representation of, 439special linear SL(n, K), 501special orthogonal SO(n), 433special unitary SU(n), 433

symmetric, 431topological, 445transformation, right, left, 668unitary U(n), 433, 438

Holder’s inequality, 153, 230converse of, 173discrete, 319for Schatten classes, 113general, 231

Haarfunctional, 454–460integral, 454measure, 454

Haar integralon SO(3), 599on SU(2), 605

Hadamard’s principal value, 238Hahn decomposition, 161Hahn–Banach theorem, 96

in locally convex spaces, 88Hamel basis, 80Hausdorff

maximal principle, 18, 73space, 53total boundedness theorem, 86

Hausdorff–Young inequality, 236,304on G and G, 548

Heaviside function, 239Heine–Borel property, 90Heine–Borel theorem, 59Hilbert

duality, 308space, 103

Hilbert–Schmidt, 559, 662operators, 112spectral theorem, 109

homeomorphism, 48homomorphism, 195, 430

continuous, 448differential, 502space HOM(G1, G2), 448

702 Index

Hopf algebra, 520“everyone with the antipode”

diagram, 521and C∗-algebra, 524antipode, 520co-multiplication and unit

diagram, 521for compact group, 523for finite group, 522multiplication and

co-multiplication diagram, 521multiplication and co-unit

diagram, 521tensor product, 520

Hopf fibration, 673hyperbolic equations, 410hypoellipticity, 384, 385

idealspanned by a set, 193two-sided, maximal, proper, 193

index, 388of Fredholm operator, 381

index sets, 11induced

family, 14induced representation space

IndGφH, 484

injection, 13inner product, 103

on V ⊗W , 84integral, 143

Haar, 454Lebesgue, 143Pettis, weak, 90, 482Riemann, 151

integrationdiscrete, 314

interior, 36interpolation theorems, 385invariant

vector fields, 500isometry, 443

isomorphism, 195, 430canonical, 308intertwining, 531isometric, 48

isotropy subgroup Gq, 437

Jacobi identity, 361, 498for Gaussians, 361

Jordan decomposition, 159, 160

kernelof a linear operator, 82

kernel, null space, 430, 435kernels la, la(x), rA, rA(x), 583Killing form, 509Krein–Milman theorem, 89Krull’s theorem, 193Kuratowski’s closure axioms, 37

Laplace operator, 225on SU(2), 611, 625on a group, 512, 534on a group, symbol of, 554

law of trichotomy, 21Lebesgue

Lp(μ)-norm, 152Lp(μ)-spaces, 154–B.Levi monotone convergence

theorem, 144conjugate, 153covering lemma, 61decomposition of measures, 168differentiation theorem, 252, 253dominated convergence theorem,

150, 222integral, 143measurable function, 135measure, 128measure, translation and rotation

invariance, 129measurelet, 117non-measurable sets, 133outer measure, 117space Lp(μG) on a group, 460

left quotient H\G, 434

Index 703

Leibniz formulaasymptotic, 571discrete, 311Euclidean, 249on an algebra, 499

LF-space, 88Lie

algebra, 498algebra sl(n, K), 501algebra homomorphism, 498algebra of a Lie group, 499algebra, canonical mapping, 507algebra, semisimple, 509algebras gl(n, K), o(n), so(n),

u(n), su(n), 500group, 491group, dimension of, 499group, exponential coordinates,

500group, linear, 491group, semisimple, 509subalgebra, 498

lifting of operators, 673linear operator

bounded, 94compact, 95norm of, 94

Liouville’s theoremfor harmonic functions, 293for holomorphic functions, 217

Littlewood’s principles, 142locality, 265, 382logarithm of a matrix, 495Luzin’s theorem, 141

manifold, 65, 417closed, 419differentiable, 66orientable, 418paracompact, 423

mapping, 12continuous, uniformly continuous,

Lipschitz continuous, 49measurable, 134

Marcinkiewicz’ interpolationtheorem, 256

maximum, minimum, supremum,infimum, 15

measure, 121absolutely continuous, 163action-invariant on G/H, 463Caratheodory–Hahn extension of,

123Haar, 454Hahn decomposition of, 161Jordan decomposition of, 159Lebesgue, 128Lebesgue decomposition of, 168outer, 116probability, 122product of, 181Radon–Nikodym derivative of,

162sampling, 455semifinite, 158signed, 158variations of, positive, negative,

total, 159measure space, 122

σ-finite, 164Borel, 122complete, 122finite, 122

measurelet, 116measures

mutually singular, 161metric, 26

discrete, 27Euclidean, 27interior, closure, boundary, 35subspace, 27sup-metric, d∞, 27

metric spacecomplete, 40sequentially compact, 58totally bounded, 61

metricsequivalent, 49

704 Index

Lipschitz equivalent, 33Minkowski’s functional, 87Minkowski’s inequality, 153, 231

for integrals, 188mollifier, 251monoid, 456Montel space, 90multi-indices, 223multiplication of distributions, 238

Napier’s constant, 40neighbourhood, 29, 36net, 77

Cauchy, 86neutral element, inverse, 431norm, 92

equivalent, 93operator, 94trace, 111

normaldivisors, 432element, 216, 517subgroup, 432

nuclear space, 92numbers, 10

algebraic, 24Stirling, 321

open mapping, 98open mapping theorem, 98operator

Dα, 224Dα

x , D(α)x , 327

(−Dy)(α), 394adjoint, 107classical, 355compact, 193intertwining, 441left-invariant, right-invariant, 551linear, 82order of, 414, 420properly supported, 280self-adjoint, 107

operator norm, 94

operatorsHilbert–Schmidt, 112trace class, 111

operators ∂+, ∂−, ∂0, 611applied to tl, 628symbols of, 634in Euler angles, 611

orderpartial, 15total, linear, 15

orthogonal projection, 105outer measure, 116

Borel regular, 124metric, 125product of, 181

parallelogram law, 105parametrix, 287, 378, 380

on a group, 577Parseval’s identity

on Rn, 236on a group, 475, 476, 538

partition of unity, 56path, 67Pauli matrices, 607Peetre’s

inequalities, 321theorem, 266

periodicSchwartz kernel, 336Taylor expansion, 328

periodic integral operator, 387periodicity, 300periodisation, 360

compactly supportedperturbations, 366

of operators, 363of symbols, 365

Peter–Weyl theorem, 470for Tn, 471left, 471

Pettis integral, 90, 482Plancherel’s identity, 302

in Hilbert space, 110

Index 705

on Rn, 236on a group, 475, 476

pointaccumulation, 36, 52fixed, 43isolated, 36

Poisson summation formula, 361polynomial

discrete, 313trigonometric, TrigPol(G), 474trigonometric, on Tn, 305

Pontryagin duality, 469power set P(X), 10preimage, 30preimage, image, 13principal symbol, 352principle

convergence, 234Littlewood, 142uniqueness, 234

productof measures, outer measures, 181topology, 34, 40, 72

product group, 431projection PG/H , 462pseudo-differential operator

local, 414on a manifold, 418periodic, continuity of, 343toroidal, 338

pseudo-differential operatorsΨm(G), 573Ψm(M), 418Ψm(SU(2)), Ψm(SU(2)), 528on a group, 553

pseudolocality, 265, 383pushforwards, φ–, 556Pythagoras’ theorem, 103

quantizationon Rn, 276on a group, 552operator-valued, 583toroidal, 336

quantum numbers, 630, 661quaternions, 603, 660quotient

algebra, 194, 199left H\G, 434right G/H, 433topology, 69, 199topology on G/H, 447vector space, 82

Radon–Nikodym derivative,theorem, 162, 164

relation, 12Rellich’s theorem, 295representation

tl on SU(2), 617regular, left, right, 551, 580adjoint, of a Lie algebra, 506adjoint, of a Lie group, 505cyclic, 450decomposition of, 468dimension dim (ξ), 530dimension of, 439direct sum of, 450equivalent, 441induced, 484irreducible, 440matrix, 469multiplicity of, 488of a group, 439regular, left, right, πL, πR, 439,

470restricted, 440space IndG

φH, 484space Rep(G), 530strongly continuous, 449topologically irreducible, 450unitary, 439unitary matrix, 439

resolvent mapping, 204restriction, 13Riemann integral, sums, 151Riemann–Lebesgue lemma, 223

706 Index

Rieszalmost orthogonality lemma, 95representation theorem, 107topological representation

theorem, 175, 177Riesz–Thorin interpolation theorem,

158right quotient G/H, 433right transformation group, 668Russell’s paradox, 11

scaling, 241Schatten class, 113Schroder–Bernstein theorem, 21Schrodinger equation, 412Schur’s lemma, 284, 443Schwartz kernel, 92, 340, 350, 550,

592periodic, 336

Schwartz kernel theorem, 92Schwartz space, 87, 224S(Rn), 224S(Zn), 300

Schwartz’ impossibility result, 238semigroup, 456seminorm, 92separability, topological, 36separating points, 55, 75sequence, 29

Cauchy, 40generalised, 77

sequential density of functions, 240set

directed, 77sets, 9, 10

ψ-measurable, 118balanced, 80Borel, 119bounded, 28compact, 49convex, 80elementary, 116extreme, 89Lebesgue non-measurable, 133

linearly independent, 80open, 29open, closed, 31well-ordered, 16

singular support, 243, 245, 383small sets property, 85small subgroups, 496smooth mapping, 417smoothing operators, 266smoothing periodic ΨDOs, 347Sobolev spaces, 293

Hs(G), 535, 542Lp

k(G), 550Lp

k(Ω), 247Lp

s(Tn), 375

biperiodic, 309localisation, Lp

k(Ω)loc, 248on manifolds, Hs(M), 419toroidal, Hs(Tn), 307

spaceC∞(Tn × Zn), 338Ck(M), C∞(M), 417C∞0 (Ω), 239L2(Tn), 302L2(G), 477Lp

loc(Rn), 241

Lp(G), 546Lp(Rn), 221Lp(Rn), interpolation, 232Lp(Tn), 304Pol1(SU(2)), 657K-vector, 79Ψm(M), 418D′(G), 534, 591D′(M), 419D(G), 591D(M), 419S ′(G), 543M(G), 537S(G), 539Diff(M), 419barreled, 90base, 668

Index 707

homogeneous, 669measure, 122metric, 26quotient, 14simply-connected, 504topological, 31total, 668

span, 80spectral radius formula, 205spectrum

of an algebra element, σ(x), 192of an algebra, Spec(A), 517of an operator, σ(A), 82

Stein–Weiss interpolation, 549Stirling numbers, 321

recursion formulae, 322Stone–Weierstrass theorem, 63subalgebra

involutive, 63subcover, 49subgroup, 432

trivial, 432subnet, 77subspace

compact, 50invariant, 440metric, 27, 34trivial, 80, 440vector, 80

subspacesorthogonal, 103

sum, infinite sum, 115summation by parts, 313summation on SU(2), 630support, 55, 243surjection, 13Sweedler’s example, 526symbol

classical, 283, 355elliptic, 289Euclidean, σA, 262homogeneous, 283of periodic ΨDO, 357on a group, 552

operator-valued, left, right, 583principal, 352toroidal, 335, 336

symbol classSm

ρ,δ(Rn × Rn), 261

Smρ,δ(T

n × Rn), 338Sm

ρ,δ(Tn × Zn), 338

Sm(S3), 661Sm(SU(2)), 656Σm(G), 575Σm(SU(2)), 633Σm

0 (SU(2)), Σmk (SU(2)), 633

Taylor expansionbiperiodic, 328discrete, 315on a group, 561periodic, 327, 328

tempered distributionsS ′(Rn), 233S ′(Zn), 300

tensor productalgebra, 196, 518algebraic, 84Banach, injective, 91Banach, projective, 91Hopf algebra, 520injective, 91of operators, 84of spaces, 83projective, 90spaces, dual of, 84

test functions, 88, 300Tietze’s extension theorem, 71Tihonov’s theorem, 73topological

algebra, 196equivalence, 48group, 445interior, closure, boundary, 36property, 48vector space, 85zero divisor, 203

708 Index

topological approximationof Lebesgue measurable sets, 131of measurable sets, 126

topological spacecompact, 49complete, 86completion of, 86connected, disconnected, 66Hausdorff, 53locally compact, 49locally convex, 87paracompact, 423path-connected, 67separable, 36totally bounded, 86

topology, 31F-induced, 71base of, 39co-induced, 69discrete, 75induced, 48injective tensor, 91metric, 36, 39metric, canonical, 32metric, comparison, 32metrisable, 74norm, 94on R2, 34product, 34, 40, 71projective tensor product,

π-topology, 90quotient, 69, 199quotient on G/H, 447relative, 33second countable, 39strong operator, 449subbase, subbasis, 39weak, 88, 109weak∗, 88, 99

toroidalamplitude, 340

torus, 299inflated, 362

tower, 19

trace, trace class, trace norm, 111transfinite induction, 17

mathematical induction, 17transpose, 279transposed operator

on a group, 570, 591triangle inequality, 26, 104trigonometric polynomials

TrigPol(G), 474TrigPol(Tn), 305

uncertainty principle, 241uniform boundedness principle, 97unit, 191unital algebra, 191unitary dual G, 468, 530universal enveloping algebra, 507

as Hopf algebra, 526universality

of enveloping algebra, 507of permutation groups, 436of unitary groups, 491

Urysohn’s lemma, 55smooth, 254

vector space, 79Banach, 94dimension of, 81Frechet, 87Hilbert, 103inner-product, 103LF-, 88locally convex, 87Montel, 90normed, 92nuclear, 92quotient, 82topological, 85

vectorslinearly independent, 80orthogonal, orthonormal, 103

Vitali’s convergence theorem, 156

Index 709

wave front set, 390weak derivative, 246weak topology, 88, 109weak type (p, p), 256weak∗-topology, 88, 99Weierstrass theorem, 62Well-ordering principle, 17, 25Whitney’s embedding theorem, 418

Young’s inequality, 187discrete, 320for convolutions, 231

general, 232on Rn, 229

Zermelo–Fraenkel axioms, 26Zorn’s lemma, 19

pseudo-differential operators and symmetries: background … · 2017-06-14 · pseudo-differential...

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