Singular Operators with Antisymmetric Kernels, Related Capacities, and Wolff Potentials

Abstract

We consider a generalization of the Riesz operator in $\mathbb R^d$ and obtain estimates for its norm and for related capacities via the modified Wolff potential. These estimates are based on the certain version of T1 theorem for Calderon–Zygmund operators in metric spaces. We extend two versions of Calderon–Zygmund capacities in $\mathbb R^d$ to metric spaces and establish their equivalence (under certain conditions). As an application, we extend the known relations between s-Riesz capacities, 0<s<d, and the capacities in Nonlinear Potential Theory, to the case s=0.