PSEUDODIFFERENTIAL OPERATORS AND APPLICATIONS

http://dx.doi.org/10.1090/pspum/043

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

Volume 43

PSEUDODIFFERENTIAL OPERATORS AND APPLICATIONS

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 43

PROCEEDINGS OF THE SYMPOSIUM ON PSEUDODIFFERENTIAL OPERATORS AND FOURIER INTEGRAL OPERATORS WITH APPLICATIONS TO

PARTIAL DIFFERENTIAL EQUATIONS HELD AT THE UNIVERSITY OF NOTRE DAME

NOTRE DAME, INDIANA APRIL 2-5, 1984

EDITED BY

FRANCOIS TREVES

Prepared by the American Mathematical Society with partial support from National Science Foundation grant DMS-8318439

1980 Mathematics Subject Classification. 22E30, 32F99, 35-XX, 42B20, 43A80, 47-02, 47D30, 47G05, 58GXX, 81C12, 85B40, 85P05.

Library of Congress Cataloging in Publication Data

Symposium on Pseudodifferential Operators & Fourier Integral Operators With Applications to Partial Differential Equations (1984: University of Notre Dame) Pseudodifferential operators and applications. (Proceedings of symposia in pure mathematics; v. 43) Proceedings of a symposium held at the University of Notre Dame,

Apr. 2 -5 , 1984. Bibliography: p. 1. Pseudodifferential operators—Congresses. 2, Fourier integral operators—

Congresses. 3. Differential equaitons, Partial—Congresses. I. Treves, Francois, 1930— . IL American Mathematical Society. Ill, Title. IV. Series. QA329.7.S96 1984 515.7'242 85-1419 ISBN 0-8218-1469-9

Copyright ©1985 by the American Mathematical Society

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Table of Contents

Preface vii

Propagation of local analyticity for the Euler equation By S. ALINHAC and G. METIVIER 1

A functional calculus for a class of pseudodifferential operators with singular symbols

By J. L. ANTONIANO and G. A. UHLMANN 5

Uniqueness in a class of nonlinear Cauchy problems By M. S. BAOUENDI.. . . 17

Propagation of smoothness for nonlinear second-order strictly hyperbolic differential equations

By MICHAEL BEALS 21

Multidimensional inverse scatterings and nonlinear partial differential equations

By R. BEALS and R. R. COIFMAN 45

Nonlinear harmonic analysis and analytic dependence By R. R. COIFMAN and YVES M E Y E R 71

On some C*-algebras and Frechet*-algebras of pseudodifferential operators By H. O. CORDES 79

Boundary-value problems for second-order elliptic equations in domains with corners

By G. ESKIN 105

Imbedding C n in Hn

By P. C. GREINER. 133

On some results of Gelfand in integral geometry By VICTOR GUILLEMIN 149

The propagation of singularities for solutions of the Dirichlet problem By LARS HORMANDER 157

Application of the microfocal theory of sheaves to the study of Ox By M. KASHIWARA and P. SCHAPIRA 167

Recent progress on boundary-value problems on Lipschitz domains By C. E. KENIG 175

Estimates for 8& on compact pseudoconvex CR manifolds By J. J. KOHN 207

v

VI CONTENTS

Real analysis and operator theory By YVES M E Y E R 219

Integrability and holomorphic extendibility for rigid CR structures By L. P. ROTHSCHILD 237

Multiple wells and tunneling By JOHANNES SJOSTRAND 241

The real analytic and Gevrey regularity of the heat kernel for D& By N. K. STANTON and D. S. TARTAKOFF 247

Fefferman-Phong inequalities in diffraction theory By M. E. TAYLOR. 261

Preface

This volume gathers a number of papers devoted to microlocal analysis, that is to say, to the precise application of Fourier transforms to analysis on manifolds. Mostly these articles are the written versions of lectures given by their authors at the AMS Symposium held at the University of Notre Dame from April 2 to April 5, 1984. In some instances the version is an abbreviated one; in one or two cases the content of the written paper differs from that of the lecture delivered, but covers a related topic.

Microlocalization is the most powerful tool of linear analysis to have emerged since distribution theory; it is the natural continuation of the latter. Its core is the theory of pseudodifferential and Fourier integral operators. Its central tenet is that the natural context for the study of PDE is the cotangent bundle—a claim that would not have excessively surprised the mathematicians of the nineteenth century or the physicists of the twentieth. In fact, physics has anticipated much of the new developments in the direction of microlocal analysis, often with less rigor and greater daring.

The successes of microlocalization have been truly splendid. They have made the analytic foundation of the Atyah-Singer formula simple, helped us penetrate deeper into the difficult study of uniqueness of solutions, and helped us under­stand why and when boundary problems like the oblique derivative or the 3-Neumann problem could be solved. It has allowed us to assert the existence or nonexistence of solutions to large classes of linear PDE and apply the vivid language of geometrical optics to describe their singularities. That same blend of geometrical optics and hard analysis has led to a greatly improved mathematical theory of diffraction. And hyperfunction theory has revitalized the study of overdetermined systems in the analytic category. Recently, the range of microlo­cal analysis has widened even more: Its advancing edge now tackles inverse scattering, the analysis of tunneling, the study of Cauchy-Riemann structures. It is even beginning to percolate into that most forbidding realm—nonlinear PDE. Although the truly severe nonlinearities, particularity those connected with shocks, seem at present out of reach, there is reason to believe that the future (and the near future, at that) will witness good progress in this direction.

Each of the topics I have just alluded to were discussed by the lecturers at the Symposium in Notre Dame; most of them, and many more, are studied in the articles in this volume. They bear testimony to the vitality and scope of the Fourier transform method. _ ^ v

FRANCOIS TREVES

vii

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