Harmonic Measure, Equilibrium Measure, and Thinness at Infinity in the Theory of Riesz Potentials

Abstract

The paper deals with the theory of potentials with respect to the α-Riesz kernel |x − y|^α−n of order α ∈ (0,2] on $\mathbb R^{n}$, $n\geqslant 3$. Focusing first on the inner α-harmonic measure ${\varepsilon _{y}^{A}}$ (ε_y being the unit Dirac measure at $y\in \mathbb R^{n}$, and μ^A the inner α-Riesz balayage of a Radon measure μ 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 α-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 ${\varepsilon _{y}^{A}}(\mathbb R^{n})\equiv 1$ to hold. Those criteria are given in terms of newly defined concepts of inner α-thinness and inner α-ultrathinness of A at infinity that for α = 2 and A Borel coincide with the concepts of outer 2-thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to μ^A general by verifying the integral representation formula $\mu ^{A}={\int \limits \varepsilon _{y}^{A}} d\mu (y)$. We also show that for every $A\subset \mathbb R^{n}$, there exists a K_σ-set A₀ ⊂ A such that $\mu ^A=\mu ^{A_0}$ for all μ, and give various applications of this theorem. In particular, we prove the vague and strong continuity of the inner swept, resp. inner equilibrium, measure under an approximation of A arbitrary, thereby strengthening Fuglede’s result established for A Borel (Acta Math., 1960). Being new even for α = 2, the results obtained also present a further development of the theory of inner Newtonian capacities and of inner Newtonian balayage, originated by Cartan.