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A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between Lp spaces

Part of:Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

Assaf Naor
Assaf Naor
Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel.Current address: Theory Group, Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, U.S.A. E-mail: anaor@microsoft.com
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Abstract

It is shown that it is possible to extend α Hölder maps from subsets of Lp to Lq (1 < p, q ≤ 2) isometrically if and only if α≤p/q*, and isomorphically if and only if α≤p/2. It is also proved that the set of αs which allow an isomorphic extension for α Hölder maps from subsets of X to Y is monotone when Y is a dual Banach space. Finally, the isometric and isomorphic extension problems for Hölder functions between Lp and Lq is studied for general p, q ≥ 1, and a question posed by K. Ball is solved by showing that it is not true that all Lipschitz maps from subsets of Hilbert space into normed spaces extend to the whole of Hilbert space.

MSC classification

Secondary:46B20: Geometry and structure of normed linear spaces

Information

Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001, pp. 253 - 271
Copyright
Copyright © University College London 2001

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