Family of Subharmonic Functions and Separately Subharmonic Functions

TL;DR

This paper investigates the properties of families of subharmonic functions and shows that separately subharmonic functions are subharmonic outside a small, negligible set, extending previous results in potential theory.

Contribution

It generalizes a known result by proving that the upper envelope of a family of subharmonic functions is locally bounded outside a negligible set, and applies this to separately subharmonic functions.

Findings

Upper envelope of subharmonic functions is locally bounded outside a closed nowhere dense set.

Separately subharmonic functions are subharmonic outside a closed nowhere dense set.

Generalizes a result by Cegrell and Sadullaev.

Abstract

We prove that the upper envelope of a family of subharmonic functions defined on an open subset of $\mathbb{R}^{N}$, $(N\geq2)$, that is finite every where, is locally bounded above outside a closed nowhere dense set with no bounded components. Then we conclude as a consequence that a separately subharmonic function is subharmonic outside a closed nowhere dense set with no bounded components. It generalizes a result due to Cegrell and Sadullaev.