Mean values of positive superharmonic functions on half-spaces
type: Item
instance of: scholarly article
title: Mean values of positive superharmonic functions on half-spaces
zbMATH Open document ID: 0645.31002
DOI: 10.1016/0022-247X(86)90178-2
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: Let w denote a non-negative superharmonic function on the open half-space \(D=R^ n\times]0,\infty[\), \(n\geq 1\), and let \(\Phi\) be a non-decreasing function \(\Phi: [0,\infty[\to [0,\infty[,\) not necessarily concave. The present paper is devoted to the study of mean values, \[ M(\Phi(w),t)=\int_{R^ n}\Phi (w(x,t))dx, \] and extends results of the first author [J. Lond. Math. Soc., II. Ser. 23, 129-136 (1981; Zbl 0427.31002)] and of the second author [Mathematika 29, 55-57 (1982; Zbl 0508.31004)]. For instance, under the stated conditions, there exists \(C>0\) (possibly depending on w) such that \[ \liminf_{t\to \infty}M(\Phi(w),t)/(t^ n\Phi(Ct^{-n}))>0. \] Stronger results are obtained under further conditions on the function \(\Phi\). Conditions on \(M(\Phi(w),t)\) are given that imply \(w\equiv 0\) on D.
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B25
zbMATH DE Number: 4051982
zbMATH Keywords: estimates of mean values
zbMATH Keywords: uniqueness
zbMATH Keywords: non-negative superharmonic
zbMATH Keywords: open half-space
zbMATH Keywords: mean values
author: D. H. Armitage
author: Neil A. Watson
reviewed by: Edwin J. Akutowicz
MaRDI profile type: Publication
cites work: Hyperplane Mean Values of Positive Superharmonic Functions
cites work: Mean Values of Strongly Subharmonic Functions on Half-Spaces
cites work: Mean Values of Subharmonic Functions on Half-Spaces
cites work: Harmonic Majorizations in Half-Balls and Half-Spaces
cites work: A Limit Function Associated with Harmonic Majorization on Half-Spaces
cites work: Mean values of superharmonic functions on half‐spaces
description: scientific article; zbMATH DE number 4051982