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

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc0 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 intoc0 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.