Methods of Geometric Analysis in Extension and Trace Problems Volume
This is the first 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 (Whi
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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 1
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 978-3-0348-0208-6 e-ISBN 978-3-0348-0209-3 DOI 10.1007/978-3-0348-0209-3 Library of Congress Control Number: 2011939996 © 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
xvii
I
Classical Extension-Trace Theorems and Related Results
1
Continuous and Lipschitz Functions Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation and definitions . . . . . . . . . . . . . . . . . . . . . 1.2 Extension and trace problems: formulations and examples . . 1.2.1 Example: Continuous functions . . . . . . . . . . . . . 1.2.2 Example: Uniformly continuous functions . . . . . . . 1.2.3 Example: Continuously differentiable functions on Rn 1.2.4 Example: BMO and Sobolev spaces . . . . . . . . . . 1.3 Continuous selections . . . . . . . . . . . . . . . . . . . . . . . 1.4 Simultaneous continuous extensions . . . . . . . . . . . . . . . 1.5 Extensions of continuous maps acting between metric spaces . 1.6 Absolute metric retracts . . . . . . . . . . . . . . . . . . . . . Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Notation and definitions . . . . . . . . . . . . . . . . . . . . . 1.8 Trace and extension problems for Lipschitz functions . . . . . 1.9 Lipschitz selection problem . . . . . . . . . . . . . . . . . . . 1.9.1 Counterexample . . . . . . . . . . . . . . . . . . . . . 1.9.2 Combinatorial–geometric selection results . . . . . . . 1.10 Extensions preserving Lipschitz constants . . . . . . . . . . . 1.10.1 Banach-valued Lipschitz functions . . . . . . . . . . . 1.10.2 Extension and the intersection property of bal
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