Electron. J. Differential Equations, Vol. 2016 (2016), No. 322, pp. 1-4.

$L^p$-subharmonic functions in $\mathbb{R}^n$

Moustafa Damlakhi

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.

Submitted August 8, 2016. Published December 20, 2016.
Math Subject Classifications: 31B05, 30D55.
Key Words: L^p-subharmonic functions; harmonic Hardy class

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Moustafa Damlakhi
Department of Mathematics
College of Science, King Saud University
P.O. Box 2455, Riyadh 11451, Saudi Arabia
email: damlakhi@ksu.edu.sa

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