Classical Microlocal Analysis in the Space of Hyperfunctions
The book develops "Classical Microlocal Analysis" in the spaces of hyperfunctions and microfunctions, which makes it possible to apply the methods in the distribution category to the studies on partial differential equations in the hyperfunction category.
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1737
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Seiichiro Wakabayashi
Classical Microlocal Analysis in the Space of Hyperfunctions
Springer
Author Seiichiro Wakabayashi Institute of Mathematics University of Tsukuba Tsukuba-shi, Ibaraki 305-8571, Japan E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Wakabayashi, Seiichiro: Classical microloca analysis in the space of hyperfunctions / Seiichiro Wakabayashi. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in mathematics; 1737) ISBN 3-540-67603-1
Mathematics Subject Classification (2000): 35-02, 35S05, 35S30, 35A27, 35A20, 35A07, 35HIO, 35A21 ISSN 0075-8434 ISBN 3-540-67603-1 Springer-Verlag Berlin Heidelberg New York 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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. Springer-Verlag is a company in the BertelsmannSpringer publishing group. © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10724347 4113143/du
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Preface Many author have studied the theory of hyperfunctions from the viewpoint of "Algebraic Analysis," which is not necessarily accessible to us, studying partial differential equations ( P.D.E.) in the framework of distributions. The treatment there is considably different from ours. Although we think that it is natural to work in the space of hyperfunctions for the purpose of studying P.D.E. with analytic coefficients, we do not think that "Algebraic Analysis" is indispensable for this purpose. We want to apply various methods in the framework of distributions to the studies on P.D.E. with analytic coefficients. In so doing the major difficulty is not to be able to use the "cutoff" technique. For there is obviously no nontrivial real analytic function with compact support. We shall use here "cutoff" operators ( pseudodifferential operators) instead of "cutoff" functions, which map real analytic functions and hyperfunctions to real analytic functions and hyperfunctions, respectively. In this lecture notes we attempt to establish "Classical Microlocal An
