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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
Printed in the United States of America
All rights reserved except those granted to the United States Government
This book, or parts thereof, may not be reproduced in any form without the permission of the publisher, except as indicated on the page containing information on
Copying and Reprinting at the back of this volume.
The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
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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