Weighted $α$-subharmonic measure

Abstract

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