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

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