Theory of Sobolev Multipliers With Applications to Differential and
The purpose of this book is to give a comprehensive exposition of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The theory was essentially developed by the authors during the last thirty years and the present v
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maz’ya
1
vladimir g. maz’ya tatyana o. shaposhnikova
Volume 337 Grundlehren der mathematischen Wissenschaften
theory of
A Series of Comprehensive Studies in Mathematics
theory of sobolev multipliers with applications to differential and integral operators
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Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics
Series editors M. Berger B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane A. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates
S.R.S. Varadhan
337
, Vladimir G. Maz ya Tatyana O. Shaposhnikova
Theory of Sobolev Multipliers With Applications to Differential and Integral Operators
ABC
, Vladimir Maz ya
Tatyana Shaposhnikova
Department of Mathematical Sciences M&O Building University of Liverpool Liverpool L69 3BX UK
Department of Mathematics Linkö ping University SE-581 83 Linkö ping Sweden [email protected]
and
Department of Mathematics Linkö ping University SE-581 83 Linkö ping Sweden [email protected]
ISBN: 978-3-540-69490-8
e-ISBN: 978-3-540-69492-2
Library of Congress Control Number: 2008 932182 Mathematics Subject Classification Numbers (2000): 26D10, 46E25, 42B25 c 2009 Springer-Verlag Berlin Heidelberg 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. 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. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper springer.com
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part I Description and Properties of Multipliers 1
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Trace Inequalities for Functions in Sobolev Spaces . . . . . . . . . 1.1 Trace Inequalities for Functions in w1m and W1m . . . . . . . . . . . . . 1.1.1 The Case m = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Case m ≥ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Trace Inequalities for Functions in wpm and Wpm , p > 1 . . . . . . . 1.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The (p, m)-Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Estimate for the Integral of Capacity of a Set Bounded by a L
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