Estimates of $C^m$-Capacity of Compact Sets in $\mathbb{R}^N$

For a given homogeneous elliptic partial differential operator $L$ with constant complex coefficients, the Banach space $V$ of distributions in $\mathbb{R}^N$ and a compact set $X$ in $\mathbb{R}^N$, we study the quantity $\lambda_{V,L}(X)$ equal to the distance in $V$ from the class of functions $f_0$ satisfying the equation $Lf_0 = 1$ in a neighborhood of $X$ (depending on $f_0$) to the solution space of the equation $Lf= 0$ in the neighborhoods of $X$. For $V=BC^m$, we obtain upper and lower bounds for $\lambda_{V,L}(X)$ in terms of the metric properties of the set $X$, which allows us to obtain estimates for $\lambda_{V,L}(X)$ for a wide class of spaces $V$.