A theory of inner Riesz balayage and its applications

Natalia Zorii¹

¹ Institute of Mathematics Academy of Sciences of Ukraine Tereshchenkivska 3 01601 Kyiv, Ukraine <a href="https://orcid.org/0000-0003-3228-8654">ORCID: 0000-0003-3228-8654</a>

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 68 (2020) no. 1, pp. 41-67.

Résumé

We establish a theory of balayage for the Riesz kernel $|x-y|^{\alpha -n}$, $\alpha \in (0,2]$, on $\mathbb R^n$, $n\ge 3$, 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 $\varepsilon ^{A^*}\ne \varepsilon $, where $\varepsilon $ is the Dirac measure at $x=0$ and $A^*$ 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.

Détail

DOI : 10.4064/ba191104-31-1

Keywords: establish theory balayage riesz kernel x y alpha n alpha mathbb alternative suggested book landkof caused the balayage book defined means integral representation which however far completely justified alternative approach mainly based cartan ideas concerning inner balayage formulated him newtonian kernel applying theory inner riesz balayage thereby developed obtain number criteria existence inner equilibrium measure gamma subset mathbb arbitrary particular given terms total mass inner swept measure suitably chosen example gamma exists only varepsilon * varepsilon where varepsilon dirac measure * inverse relative sphere which leads wiener type criterion inner alpha irregularity results obtained illustrated examples