Integral representation of balayage on locally compact spaces and its application

TL;DR

This paper establishes integral representations for balayage of measures on locally compact spaces with general kernels, providing new insights and potential applications in classical and modern potential theory.

Contribution

It introduces conditions for integral representations of balayage, extending the theory to new kernels and potential applications in fractional Green kernels.

Findings

Integral representations hold under new conditions.

Results are largely new for classical and modern kernels.

Application to measure mass variation under fractional Green kernels.

Abstract

In the theory of inner and outer balayage of positive Radon measures on a locally compact space $X$ to arbitrary $A\subset X$ with respect to suitable, quite general function kernels, developed in a series of the author's recent papers, we find conditions ensuring the validity of the integral representations. The results thereby obtained do hold and seem to be largely new even for several interesting kernels in classical and modern potential theory, which looks promising for possible applications. As an example of such applications, we analyze how the total mass of a measure varies under its balayage with respect to fractional Green kernels.