Approximation by harmonic functions in the $C^m$-Norm and harmonic $C^m$-capacity of compact sets in $\mathbb R^n$

We study the function $\Lambda^m(X)$, $0<m<1$, of compact sets $X$ in $\mathbb R^n$, $n\ge2$, defined as the distance in the space $C^m(X)\equiv\operatorname{lip}^m(X)$ from the function $|x|^2$ to the subspace $H_m(X)$ which is the closure in $C_m(X)$ of the class of functions harmonic in the neighborhood of $X$ (each function in its own neighborhood). We prove the equivalence of the conditions $\Lambda^m(X)=0$ and $C^m(X)=H^m(X)$. We derive an estimate from above that depends only on the geometrical properties of the set $X$ (on its volume).