A criterion for approximability by harmonic functions in Lipschitz spaces

Let $X$ be a compact subset of $\mathbb R^3$, $f$ be a function harmonic inside $X$, from Lipschitz space $C^\gamma(X)$, $0<\gamma<1$. A criterion for approximability of $f$ on $X$ in $C^\gamma(X)$ by functions harmonic on neighborhoods of $X$ is obtained in terms of Hausdorff content of order $1+\gamma$. The proof is completely constructive, and Vitushkin's scheme of singularities separation and approximation by parts is applied.