2025

Mohanta, Kaushik; Mudarra, Carlos; Oikari, Tuomas

For an arbitrary subset E⊂Rn, we introduce and study the three vanishing subspaces of the Hölder space C˙0,ω(E) consisting of those functions for which the ratio |f(x)−f(y)|/ω(|x−y|) vanishes, when (1) |x−y|→0 , (2) |x−y|→∞ or (3) min(|x|,|y|)→∞. We prove that the Whitney extension operator maps each of these vanishing subspaces from E to the corresponding vanishing spaces defined on the whole ambient space Rn. In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions C˙0,ω(E) by Lipschitz and boundedly supported functions.