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

Lancien, Gilles; Randrianantoanina, Beata

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

Authors

Description

We study the isometric extension problem for Hölder 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 $α$-Hölder map, with $0<α\leq 1$, 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 $α$'s for which all $α$-Hölder 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 $K$.
16 pages