Weighted $\alpha$-subharmonic measure

TL;DR

This paper introduces a weighted version of the $\alpha$-subharmonic measure, explores its properties, and characterizes regularity and continuity conditions related to it.

Contribution

It extends the classical $\alpha$-subharmonic measure by incorporating a weight function and provides new characterizations of regularity and continuity.

Findings

Weighted $\alpha$-subharmonic measure generalizes the classical measure.

Characterization of $(\alpha,\psi)$-regularity via measure continuity.

H"older continuity of the measure on a compact set implies global H"older continuity.

Abstract

In this paper, we introduce and study the weighted $\alpha$-subharmonic measure associated with a weight function $\psi$, extending the usual $\alpha$-subharmonic measure and reducing to it when $\psi \equiv -1$. Furthermore, we study the relationship between the weighted $\alpha$-subharmonic measure and $\alpha$-regular compact sets. We also obtain a characterization of $(\alpha,\psi)$-regularity in terms of the continuity of the corresponding weighted $\alpha$-subharmonic measure. Finally, we prove that if the weighted $\alpha$-subharmonic measure of the compact set $K$ is H\"older continuous with respect to $K$, then it is H\"older continuous everywhere.