The capacity associated to signed Riesz kernels, and Wolff potentials

Abstract

There is a natural capacity associated to any vector valued Riesz kernel of a given homogeneity. If we are in the plane and the kernel is the Cauchy kernel, then this capacity is analytic capacity. Our main result states that if the homogeneity of the kernel is negative and larger than minus one, then the capacity is comparable to one of the well studied capacities of non-linear potential theory.