Fine Topology and Estimates for Potentials and Subharmonic Functions

Abstract

The paper concerns classical potential theory and the theory of subharmonic and meromorphic functions. Some estimates for Riesz and Green potentials and for δ-subharmonic functions are obtained. The estimates hold outside some small exceptional sets described in terms of capacity. The formulations involve some weight functions which reflect the growth of Borel measure defining a given potential; for the subharmonic case the weight functions concern Riesz measure and Nevanlinna characteristic of a given δ-subharmonic function. The exceptional sets satisfy some relations which are similar to the Wiener criterion in the Dirichlet problem. Some results on the decrease of meromorphic functions on small sets — even some of those which were called exceptional in the above-mentioned estimates — are presented. Our results improve or complement some classical theorems related to notions of thinness and fine topology.