On Riesz decomposition for super-polyharmonic functions in \({\mathbb R}^{n}\)
type: Item
instance of: scholarly article
title: On Riesz decomposition for super-polyharmonic functions in \({\mathbb R}^{n}\)
zbMATH Open document ID: 1335.31004
published in: Potential Analysis
publication date: 2016-01-27T00:00:00Z
review text: The Riesz decomposition theorem tells us that a superharmonic function \(u\) on \(\mathbb{R}^{n}\), \(n\geq 3\), can be written as the sum of a Newtonian potential and a harmonic function if and only if the mean values of \(u\) over the spheres \(\{|x| =r\}\), \(r>0\), are bounded below. The author proves an analogous decomposition theorem for functions \(u\) that are \(m\)-superharmonic, which means roughly that the distribution \((-\Delta )^{m}u\) can be identified with a positive measure. The particular case where \(m=2\) had previously been investigated by \textit{K. Kitaura} and \textit{Y. Mizuta} [J. Math. Soc. Japan 58, No. 2, 521--533 (2006; Zbl 1103.31004)].
Mathematics Subject Classification ID: 31B30
zbMATH DE Number: 6536134
zbMATH Keywords: polyharmonic function
zbMATH Keywords: Riesz decomposition
author: Alexander Tovstolis
reviewed by: Stephen J. Gardiner
MaRDI profile type: Publication
full work available at URL: s11118-015-9474-5
OpenAlex ID: W1000904761
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DOI: 10.1007/S11118-015-9474-5
description: scientific article; zbMATH DE number 6536134