1983 • 369 Pages • 15.16 MB • English

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iMi^wnwapiis and Studies m Matriematies 19 t.:oc.jl> \i ( - ci s r t i j Irn I f ']T'r^ Tr^/T^'lT ll’fiJ) JiMi&xU;;UMl& clicili ill ji:-':Qc:^3^ BENT E PETEISEI

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Introduction to the Fourier transform & pseudo-differential operators

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Main Editors A. Jeffrey, University of Newcastle-upon-Tyne R. G. Douglas, State University of New York at Stony Brook Editorial Board F. F. Bonsall, University of Edinburgh H. Brezis, Universite de Paris G. Fichera, Universita di Roma R. P. Gilbert, University of Delaware K. Kirchgassner, Universitat Stuttgart R. E. Meyer, University of Wisconsin-Madison J. Nitsche, Universitat Freiburg L. E. Payne, Cornell University G. F. Roach, University of Strathclyde I. N. Stewart, University of Warwick S. J. Taylor, University of Liverpool

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Introduction to the Fourier Transform & Pseudo -differential Operators Bent E . Petersen Oregon State University j t Pitman Advanced Publishing Program Boston • London • Melbourne

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PITMAN PUBLISHING LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto First published 1983 © Bent E. Petersen 1983 AMS Subject Classifications: 42B10, 53S05, 47G05 British Library Cataloguing in Publication Data Petersen, Bent E. Introduction to the Fourier transform & pseudo differential operators.— (Monographs and studies in mathematics; 19) 1. Fourier transforms I. Title II. Series 515.T23 QA403.5 ISBN 0-273-08600-6 Library of Congress Cataloging in Publication Data Petersen, Bent E. Introduction to the Fourier transform and pseudo differential operators. (Monographs and studies in mathematics; 19) Bibliography; p. Includes index. 1. Pseudo-differential operators; 2. Fourier transformation. I. Title II. Series QA 329.7.P47 1983 515.T242 83-8083 ISBN 0-273-08600-6 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers. Printed in Northern Ireland at The Universities Press (Belfast) Ltd.

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Contents Preface ix Acknowledgment x Chapter 1 Theory of distributions 1 1. Introduction to Chapter 1 1 2. Convolution of LF functions 4 3. Regularization 6 4. Complex Borel measures 8 5. Partitions of unity 12 6. Integration by parts 13 7. Distributions 15 8. Restriction and support 21 9. Differentiation of distributions 23 10. Fundamental solutions of the Laplacian 25 11. The Newtonian potential in [R” 28 12. Leibniz’ formula. Classical derivatives 30 13. Distributions with point support 31 14. Weak derivatives and integration by parts 37 15. Distribution valued holomorphic functions 38 16. Boundary values of holomorphic functions 43 17. Operations on distributions 46 18. Convolution of distributions 47 19. Oscillatory integrals 58

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VI CONTENTS Chapter 2 The Fourier transform 65 1. Introduction to Chapter 2 65 2. Fourier transform. O theory 67 3. V inversion theory 71 4. Fourier transform. theory 78 5. Temperate distributions and 6^ 82 6. Convolution of temperate distributions. The space 0^ 87 7. The Fourier transform on Sf'. The exchange formulae 91 8. The Fourier transform on The Paley-Wiener theorem 95 9. Operators defined by the Fourier transform 106 10. Homogeneous distributions 119 11. Periodic distributions and Fourier series 122 12. Laplace transform 128 13. The wave front set of a distribution 145 Chapter 3 Pseudo-differential operators 161 1. Introduction to Chapter 3 161 2. Pseudo-differential operators 166 3. Smoothing operators and properly supported operators 178 4. Operators of the form a'(X, - iD , X) 185 5. Transpose and composition 192 6. Classical pseudo-differential operators 198 7. Invariance of pseudo-differential operators 204 8. The pseudo-local property 208 9. Characteristics. The regularity theorem 212 Chapter 4 Hflbert space methods 225 1. Introduction to Chapter 4 225 2. Sobolev spaces 228

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CONTENTS Vll 3. Operators on Sobolev spaces 237 4. continuity of pseudo-differential operators 247 5. Local Sobolev spaces 255 6. The wave front set 259 7. Subellipticity and local existence 265 8. Appendix. The Seidenberg-Tarski theorem 293 Chapter 5 Garding’s inequality 305 1. Introduction to Chapter 5 305 2. The spaces 306 3. The Dirichlet problem 313 4. Discussion of Garding’s inequality 322 5. Generalized Dirichlet forms 325 6. The Friedrichs’ symmetrization 329 7. Propagation of singularities 335 Bibliography 346 Index 353

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Preface This book is an introduction to the Fourier transform and to the theory of pseudo-differential operators. As a text it is intended to be used at the second year graduate level. However, care has been taken to keep the text reasonably accessible. Thus large parts of it may profitably be used as a supplement for a first year course in functional analysis. Chapter 1 presents Schwartz’ theory of distributions. It is possible to cover much ground without invoking the theory of topological vector spaces. One then misses, however, the chance to illustrate some of the basic theorems of functional analysis. Moreover, eventually the functional analysis becomes well-nigh indispensable. Therefore in Section 7, prior to considering distributions, we give a very brief introduction to the theory of locally convex spaces. The reader familiar with Banach spaces should have little difficulty making the transition to the more general setting. In Chapter 2 we continue the study of distributions and develop the theory of the Fourier transform on the space of temperate distributions. The remainder of the book is given over to pseudo-differential operators. In Chapter 3 we construct an operational calculus for the (non-commuting) operators of multiplication by the coordinate functions and differentiation. The Fourier transform is the major tool here. The resulting operators are the pseudo-differential operators. Chapter 4 con cerns the continuity of pseudo-differential operators on Sobolev spaces. In Chapter 5 we first discuss the Dirichlet problem in an setting and Garding’s inequality. We then give a proof of the sharp Garding inequal ity. This result leads to a theorem on propagation of singularities of solutions of pseudo-differential equations. This result together with the results of Chapter 4 then leads, for example, to a local existence theorem for operators with real principal symbol and simple real characteristics. Each chapter begins with an introduction which gives a detailed descrip tion of the contents of the chapter and, in some cases, some historical remarks. Additional historical remarks and some exercises are scattered throughout the text. IX

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