On harmonic approximation of Lipschitz functions on compacts in ${\mathbb{R}}^d$

Abstract

Given a porous compact $K\subset {\mathbb{R}}^d$ and a continuity modulus $\ddot{I}‰$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class ${\mathrm{Lip}}_{\ddot{I}}‰(K)$ if and only if $f$ can be approximated uniformly on $K$ with a rate of $\ddot{I}‰(\hat{I}´)$ by a function that is harmonic in the $\hat{I}´$-neighborhood of $K$, provided the uniform estimate $\ddot{I}‰(\hat{I}´)/\hat{I}´$ on the gradient holds.