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

In the theory of \(g_\alpha \)-potentials on a domain \(D\subset \mathbb R^n\), \(n\geqslant 2\), \(g_\alpha \) being the \(\alpha \)-Green kernel associated with the \(\alpha \)-Riesz kernel \(|x-y|^{\alpha -n}\) of order \(\alpha \in (0,n)\), \(\alpha \leqslant 2\), 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 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.