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

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

2004-05-28

arxiv.org/abs/math/0405565v1

Mathematics

Functional Analysis

16 pages

Scientific paper

We study the isometric extension problem for H\"{o}lder maps from subsets of any Banach space into $c_0$ or into a space of continuous functions. For a Banach space $X$, we prove that any $\alpha$-H\"{o}lder map, with $0<\alpha\leq 1$, from a subset of $X$ into $c_0$ can be isometrically extended to $X$ if and only if $X$ is finite dimensional. For a finite dimensional normed space $X$ and for a compact metric space $K$, we prove that the set of $\alpha$'s for which all $\alpha$-H\"{o}lder maps from a subset of $X$ into $C(K)$ 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 space $X$, the described above set of $\al$'s does not depend on $K$, but only on finiteness of $K$.

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