Balayage, equilibrium measure, and Deny's principle of positivity of mass for $\alpha$-Green potentials

Abstract

In the theory of $g_{\alpha }$-potentials on a domain $D\subset {\mathbb{R}}^n$, $n⩾2$, $g_{\alpha }$ being the $\alpha$-Green kernel associated with the $\alpha$-Riesz kernel $|x-y{|}^{\alpha -n}$ of order $\alpha \in (0,n)$, $\alpha ⩽2$, we establish the existence and uniqueness of the $g_{\alpha }$-balayage ${\mu }^F$ of a positive Radon measure $\mu$ onto a relatively closed set $F\subset D$, we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for ${\mu }^F(D)=\mu (D)$ to hold, given in terms of the $\alpha$-harmonic measure of suitable Borel subsets of $\overline{{\mathbb{R}}^n}$, the one-point compactification of ${\mathbb{R}}^n$. As a by-product, we find necessary and/or sufficient conditions for the existence of the $g_{\alpha }$-equilibrium measure ${\gamma }_F$, ${\gamma }_F$ being understood in an extended sense where ${\gamma }_F(D)$ might be infinite. We also discover quite a surprising version of Deny's principle of positivity of mass for $g_{\alpha }$-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed.

16 pages