Capacities commensurable with harmonic ones

Abstract

Let $\mathcal L$ be a second-order homogeneous elliptic differential operator in $\mathbb R^N$, $N\ge3$, with constant complex coefficients. Removable singularities of $\mathrm L^{\infty}$-bounded solutions of the equation $\mathcal Lf=0$ are described in terms of the capacities $\gamma_{\mathcal L}$, where $\gamma_{\Delta}$ is the classical harmonic capacity from potential theory. It is shown for the corresponding values of $N$ that $\gamma_{\mathcal L}$ and $\gamma_{\Delta}$ are commensurable for all $\mathcal L$. Some ideas due to Tolsa are used in the proof. Various consequences of this commensurability are presented; in particular, criteria for the uniform approximation of functions by solutions of the equation $\mathcal Lf=0$ are stated in terms of harmonic capacities. Bibliography: 19 titles.