Balayage, equilibrium measure, and Deny’s principle of positivity of mass for \(\alpha \)-Green potentials

In the theory of \(g_\alpha \) $g_{\alpha }$ -potentials on a domain \(D\subset \mathbb R^n\) $D\subset {\mathbb{R}}^n$ , \(n\geqslant 2\) $n⩾2$ , \(g_\alpha \) $g_{\alpha }$ being the \(\alpha \) $\alpha$ -Green kernel associated with the \(\alpha \) $\alpha$ -Riesz kernel \(|x-y|^{\alpha -n}\) ${|x-y|}^{\alpha -n}$ of order \(\alpha \in (0,n)\) $\alpha \in (0,n)$ , \(\alpha \leqslant 2\) $\alpha ⩽2$ , we establish the existence and uniqueness of the \(g_\alpha \) $g_{\alpha }$ -balayage \(\mu ^F\) ${\mu }^F$ of a positive Radon measure \(\mu \) $\mu$ onto a relatively closed set \(F\subset D\) $F\subset D$ , we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for \(\mu ^F(D)=\mu (D)\) ${\mu }^F(D)=\mu (D)$ to hold, given in terms of the \(\alpha \) $\alpha$ -harmonic measure of suitable Borel subsets of \(\overline{\mathbb R^n}\) $\overline{{\mathbb{R}}^n}$ , the one-point compactification of \(\mathbb R^n\) ${\mathbb{R}}^n$ . As a by-product, we find necessary and/or sufficient conditions for the existence of the \(g_\alpha \) $g_{\alpha }$ -equilibrium measure \(\gamma _F\) ${\gamma }_F$ , \(\gamma _F\) ${\gamma }_F$ being understood in an extended sense where \(\gamma _F(D)\) ${\gamma }_F(D)$ might be infinite. We also discover quite a surprising version of Deny’s principle of positivity of mass for \(g_\alpha \) $g_{\alpha }$ -potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann Acad Sci Fenn Math 43:121–145, 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed.