Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T07:11:54.728Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[B]Ball, K.. Markov chains, Riesz transforms and Lipschitz maps. Geometric and Functional Analysis, 2 (1992), 137–172.Google Scholar
[B-L]Benyamini, Y. and Lindenstrauss, J.. Geometric Nonlinear Functional Analysis, Volume 1. Colloquium Publications, 48 (2000).Google Scholar
[Be-Lo]Bergh, J. and Lofstrom, J.. Interpolation Spaces, An Introduction. Grundlehren der math. Wissenschaften, 223 (Springer-Verlag, 1976).Google Scholar
[K]Kirszbraun, M. D.. Über die zussammenziehenden und Lipschitzchen Transformationen. Fund. Math., 22 (1934), 77–108.Google Scholar
[L]Lindenstrauss, J.. On nonlinear projections in Banach spaces. Michigan Math. J., 11 (1964), 263–287.Google Scholar
[G-K-L]Godefroy, G.Kalton, N. J.Lancien, G.. L'espace de Banach c 0 est déterminée par sa metrique. C.R. Acad. Sci. Paris, 327 (1998), 817–822.CrossRefGoogle Scholar
[T]Tsarkov, I. G.. Smoothing of uniformly continuous mappings in L p spaces (in Russian). Mat. Zametki, 54 (1993), 123–140. English translation: Math Notes, 54 (1993), 957–967.Google Scholar
[W-W]Wells, J. H. and Williams, L. R.. Embeddings and Extensions in Analysis. Ergebnisse, 84 (Springer-Verlag, 1975).Google Scholar