Lecture Notes in Mathematics a Collection of Informal Reports and Seminars Edited by A

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich

287

Hyperfunctions and Pseudo-Differential Equations Proceedings of a Conference at Katata, 1971

Edited by Hikosaburo Komatsu, University of Tokyo, Tokyo/Japan

Springer-Verlag Berlin. Heidelberg New York 1973 AMS Subject Classifications 1970): 35 A 05, 35 A 20, 35 D 05, 35 D 10, 35 G 05, 35 N t0, 35 S 05, 46F 15

ISBN 3-540-06218-1 Springer-Verlag Berlin- Heidelberg" New York ISBN 0-387-06218-1 Springer-Verlag New York • Heidelberg • Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin . Heidelberg 1973. Library of Congress Catalog Card Number 72-88782. Printed in Germany. Offsetdruck: Jutius Beltz, Hemsbach/Bergstr. Dedicated to the memory of the late professor

Andre MARTINEAU, who had originally planned to attend this conference. He appreciated the importance of hyperfunctions for the first time and has made the most profound contributions to the theory of hyperfunctions. LIST OF PARTICIPANTS

Y. AKIZUKI (Gunma University)

H. HIRONAKA (Harvard University)

A. KANEKO (University of Tokyo)

M. KASHIWARA (RIMS, Kyoto University)

T. KAWAI (RIMS, Kyoto University)

H. KO~TSU (University of Tokyo)

T. KOTAKE (Tohoku University)

J. LERAY (Coll@ge de France)

M. MATSUMIfRA (Fac. Eng., Kyoto University)

S. MATSUURA (RIMS, Kyoto University)

T. MATUMOTO (Kyoto University)

T. MIWA (University of Tokyo)

S. MIZOHATA (Kyoto University)

M. MORIMOTO (University of Tokyo)

Y. NAMIKAWA (Nagoya University)

I. NARUKI (RIMS, Kyoto University)

Y. OHYA (Fac° Eng., Kyoto University)

T. OSHIMA (University of Tokyo)

H. SATO (University of Tokyo)

M. SATO (RIMS, Kyoto University)

P. SCHAPIRA (Universit~ de Paris)

T. SHIROTA (Hokkaido University)

H. SUZUKI (Tokyo University of Education)

M. YAMAGUTI (Kyoto University) TABLE OF CONTENTS

PART I

Preface Part I ......

CONFERENCE AT KATATA

Hikosaburo KONLATSU: An introduction to the theory of hyperfunctions ...... 3

Mitsuo MORiMOTO: Edge of the wedge theorem and hyperfunction . 41

Jean-Michel BONY et Pierre SCHAPIRA: Solutions hyperfonctions du probl@me de Cauehy ...... 82

Takahiro KAWAI: On the global existence of real analytic solutions of linear differential equations ...... 99

Akira KANEK0: Fundamental principle and extension of solutions of linear differential equations with constant coefficients 122

CONFERENCE AT RIMS

Sunao ~UCHI: On abstract Cauchy problems in the sense of hyperfunctions ...... 135

Shinichi KOTANI and Yasunori OKABE: On a Markovian property of stationary Gaussian processes with a multi-dimensional parameter ...... 153

Hikosaburo KOMATSU: Ultradistributions and hyperfunctions .... 164

APPENDICES

Hikosaburo KOMATSU: Hyperfunctions and linear partial differential equations ...... 180

Hikosaburo KONATSU: Relative cohomology of sheaves of solutions of differential equations ...... 192

PART II

Preface Part II...... 264

Mikio SATO~ Takahiro KAWAI and Masaki KASHIWARA: Microfunctions and Pseudo-differential Equations

CHAPTER I. Theory of Microfunctions ...... 265 VI

1. Construction of the sheaf of microfunctions ...... 265 1.1. Hyperfunctions ...... 265 1.2. Real monoidal transformation and real comonoidal transformation ...... 266 1.3. Definition of microfunctions ...... 273 1.4. Sheaves on sphere bundle and on cosphere bundle .... 277 1.5. Fundamental diagram on C ...... 282

2. Several operations on hyperfunctions and microfunctions. . . 286 2.1. Linear differential operators ...... 286 2.2. Substitution ...... 287 2.3. Integration along fibers ...... 294 2.4. Products ...... 296 2.5. Micro-local operators ...... 299 2.6. Complex conjugation ...... 302

3. Techniques for construction of hyperfunctions and microfunctions ...... 302 3.1. Real analytic functions of positive type ...... 303 3.2. Boundary values of hyperfuctions with holomorphic para- meters and examples ...... 307

CHAPTER II. Foundation of the Theory of Pseudo-differential Equations ...... 315

I. Definition of pseudo-differential operators ...... 315 1.1. Definition of pseudo-differential operators ...... 315 1.2. Operations on holomorphic microfunctions ...... 324 1.3. Sheaf of pseudo-differential operators ...... 329 1.4. Concrete expression of holomorphic microfunctions . . . 332 1.5. Adjoints, composites and coordinate transformations . 344

2. Fundamental properties of pseudo-differential operators. . . 356 2.1. Theorems on ellipticity and the equivalence of pseudo-differential operators ...... 356 2.2. Theorems on division of pseudo-differential operators . 365

3. Algebraic properties of the sheaf of pseudo-differential operators...... 384 3.1. Pseudo-differential operators with holomorphic para- meters ...... 384 3.2. Properties of the Ring of formal pseudo-differential operators ...... 385 3.3. Contact structure and quantized contact transforms. . . 391 3.4. Faithful flatness ...... 400 3.5. Operations on systems of pseudo-differential equations. 406 VII

4. Maximally overdetermined systems ...... 419 4.1. Definition of maximally overdetermined systems ..... 419 4.2. Invariants of maximally overdetermined systems ..... 419 4.3. Quantized contact transforms - general case -- . .... 427

5. Structure theorem for systems of pseudo-differential equations in the complex domain ...... 429 5.1. Structure theorem for systems of pseudo-differential equations with simple characteristics ...... 429 5.2. Equivalence of pseudo-differential operators with constant multiple characteristics ...... 434 5.3. Structure theorem for regular systems of pseudo- differential equations ...... 448

CHAPTER III. Structure of Systems of Pseudo-differential Equations ...... 457

I. Realification of holomorphic microfunctions ...... 457 1.1. Realification of holomorphic hyperfunctions ...... 457 1.2. Realifieation of holomorphic microfunctions ...... 462 1.3. Real "quantized" contact transforms ...... 467

2. Structure theorems for systems of pseudo-differential equations in the real domain ...... 469 2.1. Structure theorem I -- partial de Rham type --...... 470 2.2. Structure theorem II -- partial Cauchy Riemann type --. . 479 2.3. Structure theorem III -- Lewy-Mizohata type -...... 496 2.4. Structure theorem IV -- general ease -- ...... 520

Bibliographie ...... 524