Progress in Mathematics Volume 154

Series Editors

H. Bass J. Oesterle A. Weinstein

Albrecht B6ttcher Yuri 1. Karlovich

Carleson Curves, Muckenhoupt Weights, and lbeplitz Operators

Springer Base} AG

Authors:

Albrecht B6ttcher Fakultăt fiir Mathematik TUChemnitz D-09107 Chemnitz Germany e-mail: aboettch@mathematik.tu-chemnitz.de

Yuri 1. Karlovich Ukrainian Academy of Sciences Marine Hydrophysical Institute Hydroacoustic Department Preobrazhenskaya Street 3 270 100 Odessa Ukraine e-mail: karlovic@math.ist.utl.pt

1991 Mathematics Subject Classification 47B35, 45P05

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data

Bottcher, Albrecht: Carleson curves, Muckenhaupt weights, and Toeplitz operators / Albrecht B6ttcher ; Yuri 1. Karlovich. - Basel ; Boston ; Berlin Birkhăuser, 1997

(Progress in mathematics ; VoI. 154) ISBN 978-3-0348-9828-7 ISBN 978-3-0348-8922-3 (eBook) DOI 10.1007/978-3-0348-8922-3

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.

© 1997 Springer Basel AG Originally published by Birkhăuser Verlag in 1997 Softcover reprint of the hardcover 1 st edition 1997 Printed on acid-free paper produced of chlorine-free pulp. TCF 00

ISBN 978-3-0348-9828-7

987654321

• D'ESTVDIS )C

I ~ BARCELONA

Fernando Sunyer i Balaguer 1912-1967

* * * This book has been awarded the Ferran Sunyer i Balaguer 1997 prize.

Each year, in honor of the memory of Ferran Sunyer i Balaguer, the Institut d'Estudis Catalans awards an international research prize for a mathematical monograph of expository nature. The prize-winning monographs are published in this series. Details about the prize can be found at

http://crm.es/info/ffsb.htm

Previous winners include

- Alexander Lubotzky Discrete Groups, Expanding Graphs and Invariant Measures (vol. 125)

- Klaus Schmidt Dynamical Systems of Algebraic Origin (vol. 128)

- M. Ram Murty fj V. Kumar Murty N on-vanishing of L-functions and Applications (vol. 157)

Fernando Sunyer i Balaguer 1912-1967

Born in Figueras (Gerona) with an almost fully incapacitating physical disability, Fernando Sunyer i Balaguer was confined for all his life to a wheelchair he could not move himself, and was thus constantly dependent on the care of others. His father died when Don Fernando was two years old, leaving his mother, Dona An­gela Balaguer, alone with the heavy burden of nursing her son. They subsequently moved in with Fernando's maternal grandmother and his cousins Maria, Ange­les, and Fernando. Later, this exemplary family, which provided the environment of overflowing kindness in which our famous mathematician grew up, moved to Barcelona.

As the physician thought it advisable to keep the sickly boy away from all sorts of possible strain, such as education and teachers, Fernando was left with the option to learn either by himself or through his mother's lessons which, thanks to her love and understanding, were considered harmless to his health. Without a doubt, this education was strongly influenced by his living together with cousins who were to him much more than cousins for all his life. After a period of in­tense reading, arousing a first interest in astronomy and physics, his passion for mathematics emerged and dominated his further life.

In 1938, he communicated his first results to Prof. J. Hadamard of the Academy of Sciences in Paris, who published one of his papers in the Academy's "Comptes Rendus" and encouraged him to proceed in his selected course of inves­tigation. From this moment, Fernando Sunyer i Balaguer maintained a constant interchange with the French analytical school, in particular with Mandelbrojt and his students. In the following years, his results were published regularly. The lim­ited space here does not, unfortunately, allow for a critical analysis of his scientific achievements. In the mathematical community his work, for which he attained international recognition, is well known.

Don Fernando's physical handicap did not allow him to write down any of his papers by himself. He dictated them to his mother until her death in 1955, and when, after a period of grief and desperation, he resumed research with new vigor, his cousins took care of the writing. His working power, paired with exceptional talents, produced a number of results which were eventually recognized for their high scientific value and for which he was awarded various prizes. These honours not withstanding, it was difficult for him to reach the social and professional posi­tion corresponding to his scientific achievements. At times, his economic situation was not the most comfortable either. It wasn't until the 9th of December 1967, 18 days prior his death, that his confirmation as a scientific member was made public by the Division de Ciencias, Medicas y de N aturaleza of the Council. Furthermore, he was elected only as "de entrada" , in contrast to class membership.

Due to his physical constraints, the academic degrees for his official studies were granted rather belatedly. By the time he was given the Bachelor degree, he had already been honoured by several universities! In 1960 he finished his Master's

Fernando Sunyer i Balaguer 1912-1967 Vll

degree and was awarded the doctorate after the requisite period of two years as a student. Although he had been a part-time employee of the Mathematical Seminar since 1948, he was not allowed to become a full member of the scientific staff until 1962. This despite his actually heading the department rather than just being a staff member.

His own papers regularly appeared in the journals of the Barcelona Seminar, Collectanea Mathematica, to which he was also an eminent reviewer and advisor. On several occasions, he was consulted by the Proceedings of the American Society of Mathematics as an advisor. He always participated in and supported guest lectures in Barcelona, many of them having been prepared or promoted by him. On the occasion of a conference in 1966, H. Mascart of Toulouse publicly pronounced his feeling of being honoured by the presence of F. Sunyer Balaguer, "the first, by far, of Spanish mathematicians" .

At all times, Sunyer Balaguer felt a strong attachment to the scientific ac­tivities of his country and modestly accepted the limitations resulting from his attitude, resisting several calls from abroad, in particular from France and some institutions in the USA. In 1963 he was contracted by the US Navy, and in the following years he earned much respect for the results of his investigations. "His value to the prestige of the Spanish scientific community was outstanding and his work in mathematics of a steady excellence that makes his loss difficult to accept" (letter of condolence from T.E. Owen, Rear Admiral of the US Navy).

Twice, Sunyer Balaguer was approached by young foreign students who want­ed to write their thesis under his supervision, but he had to decline because he was unable to raise the necessary scholarship money. Many times he reviewed doctoral theses for Indian universities, on one occasion as the president of a distinguished international board. The circumstances under which Sunyer attained his scientific achievements also testify to his remarkable human qualities. Indeed, his manner was friendly and his way of conversation reflected his gift for friendship as well as enjoyment of life and work which went far beyond a mere acceptance of the situation into which he had been born. His opinions were as firm as they were cautious, and at the same time he had a deep respect for the opinion and work of others. Though modest by nature, he achieved due credit for his work, but his petitions were free of any trace of exaggeration or undue self-importance. The most surprising of his qualities was, above all, his absolute lack of preoccupation with his physical condition, which can largely be ascribed to the sensible education given by his mother and can be seen as an indicator of the integration of the disabled into our society.

On December 27, 1967, still fully active, Ferran Sunyer Balaguer unexpectedly passed away. The memory of his remarkable personality is a constant source of stimulation for our own efforts.

Translated from Juan Auge: Fernando Sunyer Balaguer. Gazeta Matematica, l.a Serie - Torno XX - Nums. 3 y 4, 1968, where a complete bibliography can be found.

Contents

Preface .................................................................. XllI

1 Carleson curves

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Definitions and examples ........................................... . Growth of the argument ............................................ . Seifullayev bounds ................................................. . Submultiplicative functions ......................................... . The W transform .................................................. . Spirality indices .................................................... . Notes and comments

2 Muckenhoupt weights

1 8

11 13 15 18 26

2.1 Definitions.......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Power weights.. ........... ......................... ...... ........... 30 2.3 The logarithm of a Muckenhoupt weight.... ............. .... ........ 32 2.4 Symmetric and periodic reproduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Portions versus arcs ................................................. 39 2.6 The maximal operator .............................................. 44 2.7 The reverse Holder inequality ....................................... 48 2.8 Stability of Muckenhoupt weights ................................... 56 2.9 Muckenhoupt condition and W transform............................ 59 2.10 Oscillating weights .................................................. 66 2.11 Notes and comments..................... .......... ................. 68

3 Interaction between curve and weight

3.1 3.2 3.3

Moduli of complex powers .......................................... . U and V transforms ................................................ . Muckenhoupt condition and U transform

IX

71 73 79

x Contents

3.4 Indicator set and U transform ....................................... 84 3.5 Indicator functions .................................................. 90 3.6 Indices of powerlikeness ............................................. 97 3.7 Shape of the indicator functions ..................................... 101 3.8 Indicator functions of prescribed shape .............................. 105 3.9 Notes and comments................................................ 114

4 Boundedness of the Cauchy singular integral

4.1 The Cauchy singular integral ........................................ 117 4.2 Necessary conditions for boundedness ............................... 123 4.3 Special curves and weights .......................................... 130 4.4 Brief survey of results on general curves and weights ................. 137 4.5 Composing curves and weights ...................................... 139 4.6 Notes and comments................................................ 144

5 Weighted norm inequalities

5.1 Again the maximal operator......................................... 145 5.2 The Calder6n-Zygmund decomposition .............................. 149 5.3 Cotlar's inequality .................................................. 151 5.4 Good A inequalities ................................................. 157 5.5 Modified maximal operators......................................... 160 5.6 The maximal singular integral operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165 5.7 Lipschitz curves ..................................................... 171 5.8 Measures in the plane ............................................... 184 5.9 Cotlar's inequality in the plane. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. 187 5.10 Maximal singular integrals in the plane. . . . . . . . . . . . . . . . .. . . . . . . . . . . .. 190 5.11 Approximation by Lipschitz curves .................................. 194 5.12 Completing the puzzle .............................................. 198 5.13 Notes and comments................................................ 199

6 General properties of Toeplitz operators

6.1 Smirnov classes ..................................................... 204 6.2 Weighted Hardy spaces.............................................. 207 6.3 Fredholm operators ................................................. 211 6.4 Toeplitz operators ................................................... 213 6.5 Adjoints................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 215 6.6 Two basic theorems ................................................. 217 6.7 Hankel operators .................................................... 219 6.8 Continuous symbols................................................. 221

Contents xi

6.9 Classical Toeplitz matrices 222 6.10 Separation of discontinuities. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . .. . . .. 225 6.11 Localization......................................................... 226 6.12 Wiener-Hopf factorization. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 228 6.13 Notes and comments................................................ 231

7 Piecewise continuous symbols

7.1 Local representatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . .. 235 7.2 Fredholm criterion .................................................. 240 7.3 Leaves and essential spectrum ....................................... 242 7.4 Metamorphosis of leaves ............................................. 243 7.5 Logarithmic leaves .................................................. 247 7.6 Generalleaves...................................................... 252 7.7 Index and spectrum ................................................. 258 7.8 Semi-Fredholmness............................................. . . . .. 261 7.9 Notes and comments 265

8 Banach algebras

8.1 General theorems ................................................... 267 8.2 Operators of local type. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 270 8.3 Algebras generated by idempotents .................................. 273 8.4 An N projections theorem ........................................... 275 8.5 Algebras associated with Jordan curves. . . . . . . . . . . . . .. . . . . . . .. . . . . . .. 288 8.6 Notes and comments 297

9 Composed curves

9.1 Extending Carleson stars. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . .. 301 9.2 Extending Muckenhoupt weights .................................... 304 9.3 Operators on flowers ................................................ 316 9.4 Local algebras ...................................................... 320 9.5 Symbol calculus..................................................... 326 9.6 Essential spectrum of the Cauchy singular integral. . . . . . . . . . . . .. . . . .. 329 9.7 Notes and comments 332

10 Further results

10.1 Matrix case ......................................................... 337 10.2 Index formulas...................................................... 340 10.3 Kernel and cokernel dimensions ..................................... 345 lOA Spectrum of the Cauchy singular integral ............................ 346

xii Contents

10.5 Orlicz spaces ........................................................ 349 10.6 Mellin techniques ................................................... 355 10.7 Wiener-Hopf integral operators...................................... 367 10.8 Zero-order pseudo differential operators .............................. 372 10.9 Conformal welding and Haseman's problem.......................... 374 10.10 Notes and comments ............................................... 379

Bibliography............................................................. 381

Index.................................................................... 392

Preface

This book is a reasonably self-contained introduction to the spectral theory of Toeplitz operators with piecewise continuous symbols and of singular integral oper­ators with piecewise continuous coefficients on Carleson curves with Muckenhoupt weights. For piecewise Lyapunov curves with power weights, the corresponding theory was accomplished by Gohberg and Krupnik in the seventies. Only in the eighties, after a long development and by the efforts of many mathematicians, did it become clear that the Cauchy singular integral operator Sr is bounded on the weighted Lebesgue space LP(r, w) (1 < p < (0) if and only if r is a Carleson curve and w is a Muckenhoupt weight. Extending the Gohberg-Krupnik theory to this more (and even, in a sense, "most") general setting would have been a thankless job had it turned out that by means of refined techniques the results for piecewise Lyapunov curves and power weights could essentially be carried over to Carles on curves and Muckenhoupt weights. However, in recent times it was discovered that general Carleson curves and general Muckenhoupt weights yield qualitatively new phenomena in the spectra of Toeplitz and singular integral operators. The result­ing spectral theory is surprisingly rich and extremely beautiful. It is the subject of this book.

To get an idea of what is going on, let us consider the essential spectrum of the Cauchy singular integral operator Sr on LP (r, w) in the case where r is a bounded simple arc. If r is piecewise Lyapunov and w is a power weight, then the essential spectrum consists of two circular arcs between -1 and 1. We will show that these circular arcs metamorphose into logarithmic double spirals for more complicated curves and that in the case of general Carleson curves these double spirals may blow up to heavy sets whose boundaries are nevertheless comprised of pieces of logarithmic spirals. Proper (i.e. non-powerlike) Muckenhoupt weights may further thicken the spectrum: until some point the weights are unable to destroy the circular arcs and logarithmic spirals in the spectrum, but beyond this point some kind of interference between the curve and the weight results in a complete disappearance of spirality and the emergence of so-called leaves. In other words: when considering boundedness of Sr, it is only workers in the field who know of the precipice between Lyapunov curves and Carleson curves or between

xiii

xiv Preface

power weights and Muckenhoupt weights - when looking at the spectrum of Sr, everyone can see this precipice.

The problem of finding the spectrum of the Cauchy singular integral operator is, in a sense, equivalent to describing the spectrum of Toeplitz operators with piecewise continuous symbols on weighted Hardy spaces L~(r, w) over Jordan curves r. The language of Toeplitz operators is more convenient for our purposes, and therefore it is Toeplitz operators which will play the dominant part in this book. Having identified the local spectra of Toeplitz operators, we will employ local principles, an appropriate N projections theorem, and results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights in order to construct a symbol calculus for Banach algebras of singular integral operators over composed curves.

The table of contents provides an overall view of what this book is all about. We merely want to add the following remarks.

The first three chapters are an introduction to Carleson curves and Muck­enhoupt weights. Various results of these chapters are well known, but a series of concepts, methods, and results are new and are dictated by the needs of the spectral theory of Toeplitz and singular integral operators. In particular, the use of submultiplicative functions and their indices in order to characterize Mucken­houpt weights seems to be a novelty. Here, we also introduce the notions of the indicator set and of the indicator functions, which contain just the information hidden in the curve and the weight that is of relevance in the spectral theory. The spirality indices of a curve and the indices of power likeness of a weight are important parameters of the indicator functions.

In Chapters 4 and 5 we give a detailed proof of the theorem stating that the Cauchy singular integral operator is bounded on LP (r, w) (1 < p < 00) if and only if r is a Carleson curve and w is a Muckenhoupt weight.

Chapter 6 contains some background material on Toeplitz operators and exhibits two basic techniques for tackling them: localization and Wiener-Hopf fac­torization. Chapter 7 is the high point of the book. In this chapter we completely describe the essential spectrum and the spectrum of Toeplitz operators with piece­wise continuous symbols. In a sense, Chapters 1 to 6 serve to prepare for Chapter 7, while Chapters 8 to 10 are the harvest from Chapter 7.

Harvest needs harvesting machines. The central result of Chapter 8 is an N projections theorem, whose N = 2 version allows us to establish a symbol calculus for algebras of singular integral operators over Jordan curves. In Chapter 9 we employ the machinery of geometric function theory in order to deal with certain problems of extending Carleson curves and Muckenhoupt weights. Thereafter, we can use the results of Chapters 7 and 8 (including the N projections theorem in its full strength) to construct a symbol calculus for singular integral operators over composed curves. Chapter 10 records some further results, which could not be incorporated into the main text for lack of space.

Preface xv

In the late seventies, the spectral theory of Toeplitz operators with piecewise continuous symbols was considered as round and complete. In 1990, Spitkovsky surprised the community with the spectacular discovery that in the case of Lya­punov curves with arbitrary Muckenhoupt weights the circular arcs metamorphose into horns, and again it seemed then that there remained nothing to say. We now know the spectra of Toeplitz operators with piecewise continuous symbols in the case of arbitrary Carleson curves and arbitrary Muckenhoupt weights. Is this the end of the story ? Our experience tells us that the answer to such a question must be NO. Also notice that consideration of operators with oscillating symbols or passage to higher dimensions are among the challenges of the future.

Part of the book is heavily based on results obtained only in the last three years. Thus, we are aware of the fact that several things certainly can and will be done better. We nevertheless hope that we succeeded to convey to the reader an idea of the fascinating beauty of the spectral theory of Toeplitz and singular integral operators and the mathematics behind it.

Acknowledgements. This book was written during a stay of the second author at the Technical University Chemnitz from 1993 to 1996. We are deeply indebted to the

Alfried Krupp Foundation

for supporting our joint work over these years through funds from a Forderpreis fur junge Hochschullehrer. Without the support by the Krupp Foundation, this book would not exist.

We also wish to express our sincere gratitude to Sylvia Bottcher for the energy and patience she devoted to the production of the 1\\1EX masters of this book and to Alexei Yu. Karlovich for proof-reading the entire manuscript with untiring enthusiasm and for suggesting a large number of improvements.

Chemnitz, April 1997 The authors