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

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