A dual characterization of the ${\mathcal C}^{1}$ harmonic capacity and applications

Abstract

The Lipschitz and C1 harmonic capacities κ and κc in Rn can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities γ and α (resp.). In this article we provide a dual characterization of κc in the spirit of the classical one for the capacity α by means of the Garabedian function. Using this new characterization, we show that κ(E)=κ(∂oE) for any compact set E⊂Rn, where ∂oE 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.