Pseudodifferential Operators and Nonlinear PDE
For the past 25 years the theory of pseudodifferential operators has played an important role in many exciting and deep investigations into linear PDE. Over the past decade, this tool has also begun to yield interesting results in nonlinear PDE. This book
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Series Editors J. Oesterle
A. Weinstein
Michael E. TayIor
PseudodifferentialOperators and Nonlinear PDE
Springer-Science+Business Media, LLC
Michael E. Taylor Department of Mathematics University ofNorth Carolina Chapel RiU, NC 27599
Taylor, Michael Eugene, 1946Pseudodifferential operators and nonlinear POE / Michael E. Taylor. p. cm. -- (Progress in mathematics ; voI. 100) ISBN 978-0-8176-3595-4 ISBN 978-1-4612-0431-2 (eBook) DOI 10.1007/978-1-4612-0431-2 1. Pseudodifferential operators. 2. Differential equations, 1. Title. ll. Series: Progress in mathematics (Boston, Nonlinear. Mass.) : voI. 100. 91-18337 QA329.7.T39 1991 515' . 7242--dc20 CIP
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© Springer Science+Business Media New York 1991 Originaily published by Birkh!luser Boston in 1991 Ali 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, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Springer-Science+Business Media, LLC for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, D.S.A. Special requests should be addressed directly to Springer-Science+Business Media, LLC.
3556-4/91 $0.00 + .20 ISBN 978-0-8176-3595-4
Camera-ready text prepared in AMS TeX by the author.
987654321
CONTENTS Introduction. O. Pseudodifferential operators and linear PDE. §O.1 The Fourier integral representation and symbol classes §O.2 Schwartz kernels of pseudodifferential operators §O.3 Adjoints and products §O.4 Elliptic operators and parametrices §O.5 L2 estimates §O.6 Garding's inequality §O.7 The sharp Garding inequality §O.8 Hyperbolic evolution equations §O.9 Egorov's theorem §O.lO Microlocal regularity §O.l1 LP estimates §O.12 Operators on manifolds 1. Symbols with limited smoothness.
§1.1 §1.2 §1.3
Symbol classes Some simple elliptic regularity theorems Symbol smoothing 2. Operator estimates and elliptic regularity.
§2.1 §2.2 §2.3 §2.4
Bounds for operators with nonregular symbols Further elliptic regularity theorems Adjoints Sharp Garding inequality 3. Paradifferential operators.
§3.1 §3.2 §3.3 §3.4 §3.5 §3.6
Composition and paraproducts Various forms of paraproduct Nonlinear PDE and paradifferential operators Operator algebra Product estimates Commutator estimates
4. Calculus for 0 PC l S'J.
§4.1 §4.2 §4.3 §4.4
Commutator estimates Operator algebra Garding inequality C1-paradifferential calculus
5. Nonlinear hyperbolic systems. §5.1 Quasilinear symmetric hyperbolic systems §5.2 Symmetrizable hyperbolic systems
CONTENTS §5.3 §5.4
Higher order hyperbolic equations Completely nonlinear hyperbolic systems
6. Propagation of singularities. §6.1 §6.2 §6.3
Propagation of singularities Nonlinear formation of singularities Egorov's theorem
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