Nonelliptic Partial Differential Equations Analytic Hypoellipticity
This book fills a real gap in the analytical literature. After many years and many results of analytic regularity for partial differential equations, the only access to the technique known as $(T^p)_\phi$ has remained embedded in the research papers thems
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Developments in Mathematics VOLUME 22 Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California
For further volumes: http://www.springer.com/series/5834
David S. Tartakoff
Nonelliptic Partial Differential Equations Analytic Hypoellipticity and the Courage to Localize High Powers of T
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David S. Tartakoff S Morgan St 851 60607-7042 Chicago Illinois USA [email protected]
ISSN 1389-2177 ISBN 978-1-4419-9812-5 e-ISBN 978-1-4419-9813-2 DOI 10.1007/978-1-4419-9813-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011931713 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents
1
What This Book Is and Is Not . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1
2
Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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3
Overview of Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 A Few Preliminary Definitions .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Elliptic Equations and Boundary Value Problems.. . . . . . . . . . . . . . . . 3.3 The Simplest Subelliptic Case . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Subelliptic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Local C 1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Proving C 1 Regularity .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Gevrey Regularity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8 Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.1 Symmetrization of the Estimates . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.2 Proof viNa the Norm Estimate . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9 Nonelliptic Operators .. . . . . . . . . . .
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