Harmonic measure, equilibrium measure, and thinness at infinity in the theory of Riesz potentials

TL;DR

This paper investigates properties of harmonic measures and equilibrium measures in Riesz potential theory, introducing new concepts of thinness at infinity and extending classical results to broader contexts.

Contribution

It introduces the concepts of $oldsymbol{ ext{α-}} extbf{thinness}$ and $oldsymbol{ ext{α-ultrathinness}}$ at infinity, generalizing existing notions and extending results on balayage and capacities.

Findings

Characterized support and total mass of inner harmonic measures

Established continuity properties of harmonic measures outside irregular points

Extended results to general measures and approximations of sets

Abstract

Focusing first on the inner $\alpha$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $\mu^A$ the inner $\alpha$-Riesz balayage of a Radon measure $\mu$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto\varepsilon_y^A$ outside the inner $\alpha$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner capacity) as well as for $\varepsilon_y^A(\mathbb R^n)\equiv1$ to hold. Those criteria are given in terms of the newly defined concepts of $\alpha$-thinness and $\alpha$-ultrathinness at infinity that generalize the concepts of thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to $\mu^A$ general by verifying the formula $\mu^A=\int\varepsilon_y^A\,d\mu(y)$. We also show that there is a $K_\sigma$-set $A_0\subset A$ such that $\mu^A=\mu^{A_0}$ for all $\mu$, and give various applications of this theorem. In particular, we prove the vague and strong continuity of the inner swept, resp. equilibrium, measure under the approximation of $A$ arbitrary, thereby strengthening Fuglede's result established for $A$ Borel (Acta Math., 1960). Being new even for $\alpha=2$, the results obtained also present a further development of the theory of inner Newtonian capacities and of inner Newtonian balayage, originated by Cartan.