Capacities associated with scalar signed Riesz kernels, and analytic capacity

TL;DR

This paper demonstrates that capacities linked to scalar Riesz kernels are comparable to classical analytic capacity, highlighting the real-variable nature of analytic capacity and extending the results to higher dimensions.

Contribution

It establishes the equivalence between capacities associated with scalar Riesz kernels and classical analytic capacity, including higher-dimensional generalizations.

Findings

Capacities associated with scalar Riesz kernels are comparable to analytic capacity.

The results emphasize the real variables aspect of analytic capacity.

Higher dimensional analogues of the main result are also provided.

Abstract

The real and imaginari parts of the Cauchy kernel in the plane are scalar Riesz kernels of homogeneity -1. One can associate with each of them a natural notion of capacity related to bounded potentials. The main result of the paper asserts that these capacities are comparable to classical analytic capacity, thus stressing the real variables nature of analytic capacity. Higher dimensional versions of this result are also considered.