Green kernels associated with Riesz kernels

Bent Fuglede, Natalia Zorii

Published 2016-10-02Version 1

We study properties of the $\alpha$-Green kernel of order $0<\alpha\leqslant2$ for a domain $D\subset\mathbb R^n$, $n\geqslant3$. This kernel is associated with the M. Riesz kernel $|x-y|^{\alpha-n}$, $x,y\in\mathbb R^n$, in a manner well known in the case $\alpha=2$. The usual principles in potential theory are established for the $\alpha$-Green kernel, as well as the property of consistency which allows us to prove 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 H. Cartan's theory of balayage (sweeping) for the Newtonian kernel to the $\alpha$-Riesz kernels with $0<\alpha<2$.