NONLINEAR EMBEDDINGS OF SPACES OF CONTINUOUS FUNCTIONS

We provide a nonlinear version of an extension of the classical Holsztynski theorem due to Jarosz (1984) concerning the into isomorphisms of spaces of continuous functions. More precisely, supposing that K and S are locally compact Hausdorff spaces, we prove that if there exists a map T from an extremely regular subspace A of C-0(K) to C-0(S) satisfying 1/M parallel to f - g parallel to - L <= parallel to T(f) - T(g)parallel to <= M parallel to f - g parallel to + L for all f, g is an element of A, with 1 <= M-2 < 2 and L >= 0, then there exist a subset S-0 of S and a proper mapping phi of S-0 onto K. We show that phi is not only continuous in the obvious case when K is compact, but also in the case when M-2 < 4/3, where S-0 can be taken locally compact. In the Lipschitz case, that is, when L = 0, and K and S are intervals of ordinal numbers, our result improves some others by Prochazka and Sanchez-Gonzalez (2017) concerning the Lipschitz embeddings between C(K) spaces and also solves a problem raised by them.