Traces of Vanishing Hölder Spaces

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