Methods of Geometric Analysis in Extension and Trace Problems Volume
This is the second of a two-volume work presenting a comprehensive exposition of extension results for maps between different geometric objects and of extension-trace results for smooth functions on subsets with no a priori differential structure (Wh
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Managing Editors: H. Amann Universität Zürich, Switzerland J.-P. Bourguignon IHES, Bures-sur-Yvette, France K. Grove University of Maryland, College Park, USA P.-L. Lions Université de Paris-Dauphine, France Associate Editors: H. Araki, Kyoto University F. Brezzi, Università di Pavia K.C. Chang, Peking University N. Hitchin, University of Warwick H. Hofer, Courant Institute, New York H. Knörrer, ETH Zürich K. Masuda, University of Tokyo D. Zagier, Max-Planck-Institut Bonn
Alexander Brudnyi • Yuri Brudnyi
Methods of Geometric Analysis in Extension and Trace Problems Volume 2
Alexander Brudnyi Department of Mathematics & Statistics University of Calgary 2500 University Dr. NW Calgary, Alberta, Canada, T2N 1N4 [email protected]
Yuri Brudnyi Mathematics Department Technion - Israel Institute of Technology Haifa 32000 Israel [email protected]
2010 Mathematics Subject Classification: 26A16, 26B35 , 46B85, 46B70, 51H25, 52A07, 53C23, 54E35, 54E40 ISBN 978-3-0348-0211-6 e-ISBN 978-3-0348-0212-3 DOI 10.1007/978-3-0348-0212-3 Library of Congress Control Number: 2011939775 © Springer Basel AG 2012 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
Contents Preface
xi
Basic Terms and Notation
III 6
7
8
xiii
Lipschitz Extensions from Subsets of Metric Spaces Extensions of Lipschitz Maps 6.1 Lipschitz n-connectedness . . . . . . . . . . . . . . . . . 6.2 Whitney covers . . . . . . . . . . . . . . . . . . . . . . . 6.3 Main extension theorem . . . . . . . . . . . . . . . . . . 6.4 Corollaries of the main extension theorem . . . . . . . . 6.5 Nonlinear Lipschitz extension constants . . . . . . . . . 6.5.1 The classical spaceforms of nonpositive curvature 6.5.2 Lipschitz maps between Banach spaces . . . . . . 6.5.3 Extensions preserving Lipschitz constants . . . . Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 4 13 20 28 33 33 42 44 47
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49 50 50 61 69 89 105 115
Linearity and Nonlinearity 8.1 Snowflake stability of Lipschitz extension properties . . . . . . . 8.2 Relation between linear and nonlinear extension constants . . . . 8.3 Metric spaces without simultaneous Lipschitz extension property 8.3.1 Two dimensional space of bounded geometry . . . . . . .
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117 117 123 137 137
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Simultaneous Lipschitz Extensions 7.1 Characterization of simultaneous Lipschitz extension spaces . 7.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Finiteness property . . . . . . . . . . . . . . . . . . . . 7.2 Main extension result . . . . . . . .
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