Pseudo-differentia l Operator s an d the Nash-Mose r Theore m

http://dx.doi.org/10.1090/gsm/082

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Pseudo-differentia l Operator s an d the Nash-Mose r Theore m

Serg e Alinha c Patric k Gerar d

Translated by Stephen S. Wilson

Graduate Studies

in Mathematics

Volume 82

| l | | ^ | | | American Mathematical Society ^W-J^ Providence, Rhode Island

Editorial Board

David Cox Walter Craig

Nikolai Ivanov Steven G. Krantz

David Saltman (Chair)

This work was originally published in French by E D P Science, Par is , under t h e t i t le "Opera teurs pseudo-different iels et theoreme de Nash-Moser" © 1991 Inter Ed i t ions and Edi t ions de C N R S . T h e present t rans la t ion was created under license for t he American Mathemat ica l Society and is published by permission.

Trans la ted by Stephen S. Wilson

2000 Mathematics Subject Classification. P r i m a r y 35-02; Secondary 35Sxx, 47G30, 47N20.

For addi t ional information and u p d a t e s on th is book, visit w w w . a m s . o r g / b o o k p a g e s / g s m - 8 2

Library of Congress Cataloging-in-Publicat ion D a t a

Alinhac, S. (Serge) [Operateurs pseudo-differentiels et theoreme de Nash-Moser. English] Pseudo-differential operators and the Nash-Moser theorem / Serge Alinhac, Patrick Gerard ;

translated by Stephen S. Wilson. p. cm.

Includes bibliographical references and index. ISBN 978-0-8218-3454-1 (alk. paper) 1. Pseudodifferential operators. 2. Implicit functions. I. Gerard, Patrick, 1961- II. Title.

QA329.7.A4513 2007 515/.7242—dc22 2006047985

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© 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

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10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07

Contents

Preface to the English edition vii

General introduction 1

Chapter 0. Notation and review of distribution theory 5

§1. Spaces of differentiate functions and differential operators 5

§2. Distributions on an open set of E n 6

§3. Convolution 8

§4. Kernels 9

§5. Fourier analysis on Mn 10

Chapter I. Pseudo-differential operators 15

§1. Introduction 15

§2. Symbols 20

§3. Pseudo-differential operators in S and Sf 24

§4. Composition of operators 28

§5. Action of pseudo-differential operators and Sobolev spaces 29

§6. Operators in an open subset of Rn 34

§7. Operators on a manifold 37

§8. Appendix 41

Commentary on Chapter I 49

Exercises for Chapter I 50

Chapter II. Nonlinear dyadic analysis, microlocal analysis, energy estimates 77

VI Contents

§A. Nonlinear dyadic analysis 77

§B. Microlocal analysis: wave front set and pseudo-differential

operators 89

§C. Energy estimates 98

Commentary on Chapter II 106

Exercises for Chapter II 107

Chapter III. Implicit function theorems 121

§A. Implicit function theorem and elliptic problems 121

§B. Two examples of the use of the fixed-point method 128

§C. Nash-Moser theorem 135

Commentary on Chapter III 153

Exercises for Chapter III 154

Bibliography 161

Main notation introduced 165

Index 167

Preface to the English edition

We are happy to welcome the English translation of our book, which origi­nally appeared under the title 'Operateurs pseudo-differentiels et theoreme de Nash-Moser' in 1991 (InterEditions/Editions du CNRS, Paris).

Though the world of partial differential equations has changed a lot during these years, we think that the elementary presentation of the subjects touched upon in our book is still up to date and can be useful; thus, we made no changes, except for correcting some misprints. On the other hand, several remarkable books on partial differential equations have appeared since: though their scopes largely exceed that of our book, we thought it relevant to mention them in our bibliography.

Finally, we wish to thank the translator, Dr. Stephen S. Wilson, and the editorial board of the AMS, who worked to produce this new edition of our work.

Orsay, November 2006 Serge Alinhac and Patrick Gerard

Translator's note

The numbering system I have used in my translation is essentially that employed by the authors in the original French edition so that the actual equation numbers etc. are the same in both versions. I did, however, make certain changes to the cross-referencing system: for example, to remove ambiguity, outside of Chapter II Exercise A.l of that chapter may be referred to here as Exercise II.A. 1 although within Chapter II it is referred to as Exercise A.l.

Cheltenham, February 2007 Stephen S. Wilson

vn

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Bibliography

Works to be read prior to this book

[CP] J. Chazarain and A. Piriou, Introduction a la theorie des equations aux derivees partielles lineaires, Gauthier-Villars, 1981

[R] W. Rudin, Functional Analysis, McGraw-Hill, 1973

[S] L. Schwartz, Methodes mathematiques pour les sciences physiques, Hermann, Paris, 1965

[Tl] M. E. Taylor, Partial Differential Equations. 1: Basic Theory, Applied Mathematical Sciences 115, Springer-Verlag, Berlin, 1996

[Z] C. Zuily, Elements de distributions et dyequations aux derivees partielles. Cours et problemes resolus, Dunod, Paris, 2002

Works accessible at the level of this book

[CM] R. R. Coifman and Y. Meyer, Au-dela des operateurs pseudo-differentiels, Asterisque, vol. 57, SMF, Paris, 1978

[Ha] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 65-222

[HI] L. Hormander, The analysis of linear partial differential operators, vol. 1, Springer, 1983

[H2] , On the existence and the regularity of linear pseudo-differential equations, En-seign. Math. 18 (1971)

[H3] , The analysis of linear partial differential operators, vol. 3, §18.1, Springer, 1985

[Mai] A. J. Majda, Compressible fluid flow and systems of conservation laws in several variables, Appl. Math. Sci., vol. 53, Springer, 1984

[Mo] J. Moser, A rapidly convergent iteration method and nonlinear differential equations, Ann. Sc. Norm. Super. Pisa CI. Sci. 20 (1966), 499-535

[N] L. Nirenberg, Lectures on linear partial differential equations, Regional Conference Series in Mathematics, vol. 17, AMS, Providence, R.I., 1973

161

162 Bibliography

[Ra] J. Rauch, Singularities of solutions to semilinear wave equations, J. Math. Pures Appl. 58 (1979), 299-308

[Sp] M. Spivak, Calculus on manifolds. A modern approach to classical theorems of ad­vanced calculus, Benjamin, 1965

[T] M. E. Taylor, Pseudodifferential operators, Princeton, 1981

Complementary works at the research level

[A] S. Alinhac, Paracomposition et operateurs paradifferentiels, Comm. Partial Differential Equations 11 (1986), 87-121

[Au] T. Aubin, Inegalites ^interpolation, Bull. Sci. Math. 105 (1981), 229-234

[Bl] J.-M. Bony, Calcul symbolique et propagation des singularites pour les equations aux

derivees partielles non lineaires, Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), 209-246

[B2] , Second microlocalization and propagation of singularities for semilinear hyper­bolic equations, Workshop and symposium on hyperbolic equations and related topics, Kokota and Kyoto, 1984

[Bt] J.-B. Bost, Tores invariants des systemes dynamiques hamiltoniens, Seminaire Bour-baki 1984-85, no. 639, Asterisque, vol. 133-134

[DJ] G. David and J.-L. Journe, A boundedness criterion for generalized Calderon-Zygmund operators, Ann. Math. 120 (1984), 371-397

[G] M. Giinther, On the perturbation problem associated to isometric embeddings of Rie-mannian manifolds, Ann. Global Anal. Geom. 7(1) (1989), 69-77

[H4] L. Hormander, Fourier integral operators I, Acta Math. 127 (1971), 79-183

[H5] , The analysis of linear partial differential operators, vols. 3 and 4, Springer, 1985

[H6] , The boundary problem of physical geodesy, Arch. Ration. Mech. Anal. 62 (1976), 1-52

[H7] , On the Nash-Moser implicit function theorem, Ann. Acad. Sci. Fenn. Math. Series A.I. Mathematics 10 (1985), 255-259

[H8] , Pseudo-differential operators of type 1,1, Commun. Partial Differential Equa­tions 13 (1988), 1085-1111

[H9] , The Nash-Moser theorem and paradifferential operators, Analysis, et cetera, pp. 429-449, Academic Press, Boston, MA, 1990

[Hw] I. L. Hwang, The L2 boundedness of pseudodifferential operators, Trans. Amer. Math. Soc. 302 (1987), 55-76

[L] G. Lebeau, Deuxieme microlocalisation sur les sous-varietes isotropes, Ann. Inst. Fourier (Grenoble) 35 (1985), 145-216

[Ma2] A. J. Majda, The stability and the existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 , no. 275 and 43, no. 281 (1983)

[M] Y. Meyer, Remarque sur un theoreme de J.-M. Bony, Suppl. Rend. Circ. Mat. Palermo (2), Atti del Seminario di Analisi Armonica, Pisa, no. 1, (1981)

[Na] J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math. 63 (1956), 20-63

[Sj] J. Sjostrand, Singularites analytiques microlocales, Asterisque, vol. 95, SMF, Paris, 1982

Bibliography 163

[T2] M. E. Taylor, Partial Differential Equations. 2: Qualitative Studies of Linear Equa­tions, Applied Mathematical Sciences 116, Springer-Ver lag, New York, 1996

[T3] , Partial Differential Equations. 3: Nonlinear Equations, Applied Mathematical Sciences 117, Springer-Verlag, Berlin, 1996

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Main notation introduced

a, \a\,a\,da,Da,£a, 6 P,Pm, 6 C*(fi),C°°(fi), 5

c0°°(?)> 6

Cfc(ft), 5 <v>, 6 ( v ) , 12 supp «, sing supp M, 7 u * v, 8 P'(fi),£ /(fi), 6,8 S,S',u,P, 10,11 A, 16 M , 29 • , 94

S M a l ^ , 20,21

S%, 52 a(s,£>), Op (a), a*, 24,25,28 a#6, 28 T*M,7r,a,<7,{/,<?}, 38,39,40 / v (a) , 42 Up, Spu, 78 Nlo, IMU \H'i, Ca, 79, 80 C,1, 108 \\o,\\8,H

B, 30,31

Z(u),Zx(u),WF(u), 89,90 WF' , 92 WF,, 113

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Index

adjoint, 26 almost orthogonality, 78 asymptotic sum, 22

bicharacteristic (curve), 116

Cauchy problem, 93 change of variables, 37 characteristic (surface), 111 commutator, 40 continuity, L2 , 30 convexity inequalities, 82 cotangent bundle, 29

density on a manifold, 74 Dirichlet problem, 124

elliptic (symbol, operator), 33 energy (inequality), 98

1-form, canonical, 39 2-form, symplectic, 39 factorization (of an operator), 103 Fourier distribution, 91 Fourier transform, 11

Garding inequality, 31 Gagliardo-Nirenberg inequality, 86

Hamiltonian (vector field), 40 Holder spaces, 80 Hormander's theorem, 102 hyperbolic (operator, system), 103

implicit function theorem, 121, 122 interpolation, 108

isometric embedding, 133

kernels, 9

Laplacian, 16

Littlewood-Paley decomposition, 77 local inversion theorem, 121 local symbol, 35

Meyer multiplier, 87 Morse's lemma, 69

Newton schenie, 146 non-stationary phase, 41

operator on a manifold, 37 oscillatory integral, 41

parametrix, 17 paraproduct, 86 partition of unity, 7, 36 Poisson bracket, 29 principal symbol, 27 propagation of singularities, 101 properly supported operator, 36 pseudo-differential operator, 15, 24 pseudo-local property, 19

quasilinear system, 128

Rauch's lemma, 113 regularization operator, 83

singular integral, 60 small divisors, 138 Sobolev space, 31

168 Index

spectrum, 85 stationary phase, 68 sub-elliptic operator, 67 symbol, 15 symbolic calculus, 16, 28 symmetric system, 128

trace, 92 transmission property, 68

wave equation, 94 wave front set, 89

tame (mapping, estimate), 137 Zygmund class, 108