Traces of vanishing Hölder spaces

Abstract

For an arbitrary subset $E\subset {\mathbb{R}}^n,$ we introduce and study the three vanishing subspaces of the Hölder space ${\dot{C}}^{0,\omega }(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/\omega (|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to \infty$ or $(3)$ $\min (|x|,|y|)\to \infty .$ 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 ${\mathbb{R}}^n.$ 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 ${\dot{C}}^{0,\omega }(E)$ by Lipschitz and boundedly supported functions.

v3: 19 pages, referee comments incorporated, to appear in The Journal of Geometric Analysis