A theory of inner Riesz balayage and its applications

Abstract

We establish the theory of balayage for the Riesz kernel $|x-y{|}^{\alpha -n}$, $\alpha \in (0,2]$, on ${\mathbb{R}}^n$, $n⩾3$, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan's ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner equilibrium measure ${\gamma }_A$ for $A\subset {\mathbb{R}}^n$ arbitrary, in particular given in terms of the total mass of the inner swept measure ${\mu }^A$ with $\mu$ suitably chosen. For example, ${\gamma }_A$ exists if and only if ${\epsilon }^{A^{\ast }}\ne \epsilon$, where $\epsilon$ is a Dirac measure at $x=0$ and $A^{\ast }$ the inverse of $A$ relative to the sphere $|x|=1$, which leads to a Wiener type criterion of inner $\alpha$-irregularity. The results obtained are illustrated by examples.

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