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A theory of inner Riesz balayage and its applications

Zorii Natalia

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 1

41--67

We establish a theory of balayage for the Riesz kernel │x - y│α-n, α∈(0; 2], on Rn, 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 γA for A ⊂ Rnμ arbitrary, in particular given in terms of the total mass of the inner swept measure μA with μ suitably chosen. For example, ϒA exists if and only if εA* ≠ε, where ε 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 α-irregularity. The results obtained are illustrated by examples.