Balayage for Riesz kernels with application to potential theory for the associated Green kernels

TL;DR

This paper extends potential theory for Riesz kernels by establishing properties of the $ ext{α}$-Green kernel, including consistency, completeness of measures, and equilibrium measures, using a generalized balayage technique.

Contribution

It introduces a generalized balayage method for $ ext{α}$-Riesz kernels and proves fundamental properties of the associated Green kernels in potential theory.

Findings

Proved the consistency of the $ ext{α}$-Green kernel.

Established the completeness of the cone of finite energy measures.

Demonstrated the existence of $ ext{α}$-Green equilibrium measures.

Abstract

We study properties of the $\alpha$-Green kernel $g_D^\alpha$ of order $0<\alpha\leqslant2$ for a domain $D\subset\mathbb R^n$, $n\geqslant3$. This kernel is associated with the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $x,y\in\mathbb R^n$, in a manner particularly well known in the case $\alpha=2$. Besides the usual principles of potential theory, we establish for the $\alpha$-Green kernel the property of consistency. This allows us to prove the completeness of the cone of positive measures $\mu$ on $D$ with finite energy $g_D^\alpha(\mu,\mu):=\iint g_D^\alpha(x,y)\,d\mu(x)\,d\mu(y)$ in the topology defined by the energy norm $\|\mu\|_{g_D^\alpha}=\sqrt{g_D^\alpha(\mu,\mu)}$, as well as the existence of the $\alpha$-Green equilibrium measure for a relatively closed set in $D$ of finite $\alpha$-Green capacity. The main tool is a generalization of Cartan's theory of balayage (sweeping) for the Newtonian kernel to the $\alpha$-Riesz kernels with $0<\alpha<2$.