Classics in Mathematics Lars Hormander The Analysis of Linear Partial

Differential Operators I

Born on January 24,1931, on the southern coast of Sweden, Lars Hormander did his secondary schooling as well .. as his undergradu­ate and doctoral studies in Lund. His principal teacher and adviser at the University of Lund was Marcel Riesz until he retired, then Lars Gărding. In 1956 he worked in the USA, at the universities of Chicago, Kansas, Minnesota and New York, before returning to a chair at the University of Stockholm. He remained a frequent visitor to the US, particularly to Stanford and was Professor at the IAS, Princeton from 1964 to 1968. In 1968 he accepted a chair at the University of Lund, Sweden, where, today, he is Emeritus Professor.

Hormander's lifetime work has been devoted to the study of partial differential equations and its applications in complex analysis. In 1962 he was awarded the Fields Medal for his contributions to ţhe general theory oflinear par,tial differential opera tors. His book · Linear Partial Differential Operators published 1963 by Springer in the Grundlehren series was the first major account of this theory. His four volume text The Analysis of Linear Partial Differential Opera tors, published in the same series 20 years later, illustrates the vast expansion of the subject in that period.

Lars Hormander

The Analysis of Linear Partial Differential Operators I

Distribution Theory and Fourier Analysis

Reprint of the 2nd Edition 1990

Springer

Lars Hormander Department of Mathematics University of Lund Box 118

SE-22100 Lund, Sweden e-mail: lvh@maths.lth.se

Originally published as Vol. 256 in the series: Grundlehren dermathematischen Wissenschaften in 1983 and 1990, and thereafter as a Springer Study Edition in 1990

Library of Congress Cataloging.in·Publication Data

HOrmander, Lars. The analysis of linear partial differential operato", I L. HOrmander.

p. cm. - (Classics in mathematics, ISSN 1431-0821) Originally published: 2nd ed. Berlin; Now Yark : Springer-Verlag, c1990- . Includes bibliographical ref"""""", and index ... Contents: 1. Distribution theoIy and Fourier analysis ISBN-13: 978-3-540-00662-6

1. Differential equations. Partial 2. Partial differential operators. l Title. II. Series.

QA377.H5782003 515',7242-dc21

2003050516

Mathematics Subject Classification (2000): 46F, 46E, 26A, 42A, 35A, 35J, 35L

ISSN 1431-0821 ISBN-13: 978-3-540-00662-6 DOl: 10.1007/978-3-642-61497-2

e-ISBN-13: 978-3-642-61497-2

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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

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Printed on acid-free paper 41/3142ck-543210

Lars Ho-rmander

The Analysis of Linear Partial Differential Operators. I Distribution Theory and Fourier Analysis

Second Edition

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Lars Hormander Lunds Universitet Matematiska Institutionen Box 118 S~22100 Lund Sweden

The hard cover edition of this book has been published as: Grundlehren der mathematischen Wissenschaften, Band 256, with the ISBN 3-540-52345-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52345-6 Springer-Verlag New York Berlin Heidelberg

With 5 Figures

Mathematics Subject Classification (1980): 46F, 46E, 26A, 42A, 35A, 35E, 351, 35L

ISBN 978-3-540-52343-7 Springer-Verlag Berlin Heidelberg New York ISBN 978-0-387-52343-9 Springer-Verlag New York Berlin Heidelberg

Library of Congress Catalog Card Number 89-26134

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. reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

©Springer-Verlag Berlin Heidelberg 1983, 1990

Typesetting: Universitli.tsdruckerei H. StUrtz AG, Wiirzburg 2141/3111 -54321- Printed on acid-free paper

Preface to the Second Edition

The main change in this edition is the inclusion of exercises with answers and hints. This is meant to emphasize that this volume has been written as a general course in modern analysis on a graduate student level and not only as the beginning of a specialized course in partial differen­tial equations. In particular, it could also serve as an introduction to harmonic analysis. Exercises are given primarily to the sections of gen­eral interest; there are none to the last two chapters.

Most of the exercises are just routine problems meant to give some familiarity with standard use of the tools introduced in the text. Others are extensions of the theory presented there. As a rule rather complete though brief solutions are then given in the answers and hints.

To a large extent the exercises have been taken over from courses or examinations given by Anders Melin or myself at the University of Lund. I am grateful to Anders Melin for letting me use the problems originating from him and for numerous valuable comments on this collection.

As in the revised printing of Volume II, a number of minor flaws have also been corrected in this edition. Many of these have been called to my attention by the Russian translators of the first edition, and I wish to thank them for our excellent collaboration.

Lund, October 1989 Lars Hormander

Preface

In 1963 my book entitled "Linear partial differential operators" was published in the Grundlehren series. Some parts of it have aged well but others have been made obsolete for quite some time by techniques using pseudo-differential and Fourier integral operators. The rapid de­velopment has made it difficult to bring the book up to date. Howev­er, the new methods seem to have matured enough now to make an attempt worth while.

The progress in the theory of linear partial differential equations during the past 30 years owes much to the theory of distributions created by Laurent Schwartz at the end of the 1940's. It summed up a great deal of the experience accumulated in the study of partial differ­ential equations up to that time, and it has provided an ideal frame­work for later developments. "Linear partial differential operators" be­gan with a brief summary of distribution theory for this was still un­familiar to many analysts 20 years ago. The presentation then pro­ceeded directly to the most general results available on partial differ­ential operators. Thus the reader was expected to have some prior fa­miliarity with the classical theory although it was not appealed to ex­plicitly. Today it may no longer be necessary to include basic distribu­tion theory but it does not seem reasonable to assume a classical background in the theory of partial differential equations since mod­ern treatments are rare. Now the techniques developed in the study of singularities of solutions of differential equations make it possible to regard a fair amount of this material as consequences of extensions of distribution theory. Rather than omitting distribution theory I have therefore decided to make the first volume of this book a greatly ex­panded treatment of it. The title has been modified so that it indicates the general analytical contents of this volume. Special emphasis is put on Fourier analysis, particularly results related to the stationary phase method and Fourier analysis of singularities. The theory is illustrated throughout with examples taken from the theory of partial differential equations. These scattered examples should give a sufficient know­ledge of the classical theory to serve as an introduction to the system-

VIII Preface

atic study in the later volumes. Volume I should also be a useful in­troduction to harmonic analysis. A chapter on hyperfunctions at the end is intended to give an introduction in the spirit of Schwartz distri­butions to this subject and to the analytic theory of partial differential equations. The great progress in this area due primarily to the school of Sato is beyond the scope of this book, however.

The second and the third volumes will be devoted to the theory of differential equations with constant and with variable coefficients re­spectively. Their prefaces will describe their contents in greater detail. Volume II will appear almost simultaneously with Volume I, and Volume III will hopefully be published not much more than two years later.

In a work of this kind it is not easy to provide adequate refer­ences. Many ideas and methods have evolved slowly for centuries, and it is a task for a historian of mathematics to uncover the development completely. Also the more recent history provides of course consider­able difficulties in establishing priorities correctly, and these problems tend to be emotionally charged. All this makes it tempting to omit ref­erences altogether. However, rather than doing so I have chosen to give at the end of each chapter a number of references indicating re­cent sources for the material presented or closely related topics. Some references to the earlier literature are also given. I hope this will be helpful to the reader interested in examining the background of the results presented, and I also hope to be informed when my references are found quite inadequate so that they can be improved in a later edition.

Many colleagues and students have helped to improve this book, and I should like to thank them all. The discussion of the analytic wave front sets owes much to remarks by Louis Boutet de Monvel, Pierre Schapira and Johannes Sjostrand. A large part of the manuscript was read and commented on by Anders Melin and Ragnar Sigurdsson in Lund, and Professor Wang Rou-hwai of Jilin University has read a large part of the proofs. The detailed and constructive criticism given by the participants in a seminar on the book conducted by Gerd Grubb at the University of Copenhagen has been a very great help. Niels JI1lrgen Kokholm took very active part in the seminar and has also read all the proofs. In doing so he has found a number of mis­takes and suggested many improvements. His help has been invalu­able to me.

Finally, I wish to express my gratitude to the Springer Verlag for encouraging me over a period of years to undertake this project and for first rate and patient technical help in its execution.

Lund, January 1983 Lars Hormander

Contents

Introduction . . . . . .

Chapter I. Test Functions 5

Summary ...... 5 1.1. A review of Differential Calculus 5 1.2. Existence of Test Functions 14 1.3. Convolution . . . . . . . . . 16 1.4. Cutoff Functions and Partitions of Unity 25 Notes . . . . . . . . . . . . . . . . . 31

Chapter II. Definition and Basic Properties of Distributions 33

Summary ..... 33 2.1. Basic Definitions . . . . . . . . . 33 2.2. Localization . . . . . . . . . . . 41 2.3. Distributions with Compact Support 44 Notes . . . . . . . . . . . . . . . . 52

Chapter III. Differentiation and Multiplication by Functions 54

Summary .......... 54 3.1. Definition and Examples 54 3.2. Homogeneous Distributions 68 3.3. Some Fundamental Solutions 79 3.4. Evaluation of Some Integrals 84 Notes . . . . . . . 86

Chapter IV. Convolution 87

Summary ..... 87 4.1. Convolution with a Smooth Function 88 4.2. Convolution of Distributions . . . 100 4.3. The Theorem of Supports 105 4.4. The Role of Fundamental Solutions 109

X Contents

4.5. Basic I! Estimates for Convolutions 116 Notes . . . . . . . . . . . . . . . 124

Chapter V. Distributions in Product Spaces 126

Summary ....... 126 5.1. Tensor Products 126 5.2. The Kernel Theorem 128 Notes . . . . . . . . . 132

Chapter VI. Composition with Smooth Maps 133

Summary .......... 133 6.1. Definitions . . . . . . . . 133 6.2. Some Fundamental Solutions 137 6.3. Distributions on a Manifold 142 6.4. The Tangent and Cotangent Bundles 146 Notes . . . . . . . . . . . . . . . 156

Chapter VII. The Fourier Transformation 158

Summary . . . . . . . . . . . . . 158 7.1. The Fourier Transformation in !/' and in !/" 159 7.2. Poisson's Summation Formula and Periodic Distributions 177 7.3. The Fourier-Laplace Transformation in If' 181 7.4. More General Fourier-Laplace Transforms 191 7.5. The Malgrange Preparation Theorem . . 195 7.6. Fourier Transforms of Gaussian Functions 205 7.7. The Method of Stationary Phase 215 7.8. Oscillatory Integrals 236 7.9. H(S)' I! and HOlder Estimates 240 Notes . . . . . . . . . . . . . 248

Chapter VIII. Spectral Analysis of Singularities 251

Summary ............... 251 8.1. The Wave Front Set ........ 252 8.2. A Review of Operations with Distributions 261 8.3. The Wave Front Set of Solutions of Partial Differential

Equations. . . . . . . . . . . . . . 271 8.4. The Wave Front Set with Respect to CL . . . . • 280 8.5. Rules of Computation for WFL • . . • . • . • • 296 8.6. WFL for Solutions of Partial Differential Equations 305 8.7. Microhyperbolicity 317 Notes . . . . . . . . . . . . . . . . . . . . . . 322

Contents XI

Chapter IX. Hyperfunctions 325

Summary ....... 325 9.1. Analytic Functionals 326 9.2. General Hyperfunctions 335 9.3. The Analytic Wave Front Set of a Hyperfunction 338 9.4. The Analytic Cauchy Problem . . . . . . . . 346 9.5. Hyperfunction Solutions of Partial Differential Equations 353 9.6. The Analytic Wave Front Set and the Support 358 Notes . . . . . . . . . . 368

Exercises 371

Answers and Hints to All the Exercises. 394

Bibliography. 419

Index .... 437

Index of Notation. 439