L^p-subharmonic functions in R^n

Abstract

We prove that if u is an $L^p$-subharmonic function defined outside a compact set in $\mathbb{R}^n$, it is bounded above near infinity, in particular, if the subharmonic function u is in $L^p(\mathbb{R}^n)$, $1\leq p<\infty $, then u is non-positive. Some of the consequences of this property are obtained. We discuss the properties of subharmonic functions defined outside a compact set in $\mathbb{R}^n$ if they are also $L^p$ functions.

Keywords

• $L^p$-subharmonic functions
• harmonic Hardy class