A dual characterization of the C1 harmonic capacity and applications

Abstract

The Lipschitz and $C^1$ harmonic capacities $\mathrm{κ}$ and ${\mathrm{κ}}_c$ in ${\mathbb{R}}^n$ can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities $\mathrm{γ}$ and $\mathrm{Î\pm }$ (resp.). In this article we provide a dual characterization of ${\mathrm{κ}}_c$ in the spirit of the classical one for the capacity $\mathrm{Î\pm }$ by means of the Garabedian function. Using this new characterization, we show that $\mathrm{κ}(E)=\mathrm{κ}({∂}_{o}E)$ for any compact set $E⊂{\mathbb{R}}^n$, where ${∂}_{o}E$ is the outer boundary of $E$, and we solve an open problem posed by A. Volberg, which consists in estimating from below the Lipschitz harmonic capacity of a graph of a continuous function.