Pseudo Differential Operators
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Lecture Notes in Mathematics Edited by A Dold and B. Eckmann
416 Michael Taylor
Pseudo Differential Operators
Springer-Verlag Berlin· Heidelberg· New York 1974
Dr. Michael E. Taylor University of Michigan Ann Arbor, Ml 481 04/USA
Library of Congress Cataloging in Publication Data
Taylor, Michael Eugene, 1946rseudo differential operators. (Lecture notes in mathematics ; 416) Bibliography: p. Includes index. 1. Differential equations, Partial. 2. Pseudodifferential operators. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 416. 74-23846 QA3.I28 no. 416 [QA374] 510'.8s [515'.724]
AMS Subject Classifications (1970): 35-02, 35S05 ISBN 3-540-06961-5 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-06961-5 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 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
TABLE OF CONTENTS Introduction
l
Singular Integral Operators on the circle
4
1.
The algebra of singular integral operators
6
2.
The oblique derivative problem
10
3.
C* algebras and singular integral operators
14
Chapter I.
Chapter II.
Pseudo Differential Operators
19
1.
The Fourier integral representation
19
2.
The pseudo local property
22
3.
Asymptotic expansions of a symbol
24
4.
Adjoints and products
31
5.
Coordinate changes, operators on a manifold
33
6.
Continuity on
37
7.
Families of pseudo differential operators
41
8.
Garding's inequality
44
Chapter!!!.
Hs
Elliptic and Hypoelliptic Operators
45
1.
Elliptic operators
2.
Hypoelliptic operators with constant strength 48
3.
References to further work
Chapter IV.
The Initial Value Problem. Hyperbolic Operators
45
57 58
1.
Reduction to a first order system
59
2.
Symmetric hyperbolic systems_
62
3.
Strictly hyperbolic equations
66
4.
Finite propagation speed; finite domain of dependence
72
IV
Chapter
5.
The vibrating membrane problem
76
6.
Parabolic evolution equations
79
7.
References to further work
82
v.
Elliptic Boundary Value Problems; Petrowsky Parabolic Operators
84
1.
A priori estimates and regularity theorems
91
2.
Closed range and Fredholm properties
98
3.
Regular boundary value problems
107
4.
A subelliptic estimate; the oblique derivative problem
115
5.
References to further work
119
Chapter VI.
Propagation of Singularities; Wave Front Sets
120
l.
The wave front set of a distribution
120
2.
Propagation of singularities; the Hamilton flow
125
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