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On the extension of Hölder maps with values in spaces of continuous functions

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Abstract

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc 0 or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc 0 can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030, Besançon, France

    Gilles Lancien

  2. Department of Mathematics and Statistics, Miami University, 45056, Oxford, OH, USA

    Beata Randrianantoanina

Authors
  1. Gilles Lancien
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  2. Beata Randrianantoanina
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Correspondence to Gilles Lancien.

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Partially funded by a CFR Grant from Miami University.

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Lancien, G., Randrianantoanina, B. On the extension of Hölder maps with values in spaces of continuous functions. Isr. J. Math. 147, 75–92 (2005). https://doi.org/10.1007/BF02785360

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  • DOI: https://doi.org/10.1007/BF02785360

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