Singular operators with antisymmetric kernels, related capacities, and Wolff potentials

TL;DR

This paper generalizes the Riesz operator with antisymmetric kernels, establishing new estimates for its norm and capacities via Wolff potentials, and extends capacity theories from Euclidean spaces to metric spaces.

Contribution

It introduces a generalized Riesz operator with antisymmetric kernels, develops capacity estimates using Wolff potentials, and extends Calderón-Zygmund capacity concepts to metric spaces.

Findings

Established estimates for the generalized Riesz operator's norm.

Extended Calderón-Zygmund capacities to metric spaces.

Linked capacities for s=0 with nonlinear potential theory.

Abstract

We consider a generalization of the Riesz operator in $R^d$ 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 Calder\'on-Zygmund operators in metric spaces. We extend two versions of Calder\'on-Zygmund capacities in $R^d$ 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 $s=0$.