On the theory of capacities on locally compact spaces and its interaction with the theory of balayage

TL;DR

This paper advances the mathematical understanding of capacities on locally compact spaces by providing new characterizations and linking them with balayage, thereby extending classical theories to more general kernels.

Contribution

It introduces new characterizations of inner and outer capacities and measures, and rigorously justifies Fuglede's theories, extending results to various kernels.

Findings

New characterizations of capacities and measures

Rigorous justification of Fuglede's theories

Extensions to classical kernels like logarithmic and Newtonian

Abstract

The paper deals with the theory of inner (outer) capacities on locally compact spaces with respect to general function kernels, the main emphasis being placed on the establishment of alternative characterizations of inner (outer) capacities and inner (outer) capacitary measures for arbitrary sets. The analysis is substantially based on the close interaction between the theory of capacities and that of balayage. As a by-product, we provide a rigorous justification of Fuglede's theories of inner and outer capacitary measures and capacitability (Acta Math., 1960). The results obtained are largely new even for the logarithmic, Newtonian, Green, $\alpha$-Riesz, and $\alpha$-Green kernels.