Riesz-Martin representation for positive super-polyharmonic functions in a Riemannian manifold
type: Item
instance of: scholarly article
title: Riesz-Martin representation for positive super-polyharmonic functions in a Riemannian manifold
zbMATH Open document ID: 1146.31008
DOI: 10.1155/IJMMS/2006/92176
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2007-06-14T00:00:00Z
full work available at URL: https://eudml.org/doc/54529
review text: It is known that if \(v\geq 0\) is a superharmonic function on a bounded domain \(\Omega\) in \(\mathbb{R}^n\), \(n\geq 2\), then \[ v(x)= \int_\Omega G(x,y) d\mu(y)+ \int_{\Delta_1} K(x,y) d\nu(y), \] \(x\in\Omega\), where \(G(x,y)\) is the Green kernel, \(K(x,y)\) is the Martin kernel, \(\mu\geq 0\) is a Radon measure on \(\Omega\) and \(\nu\geq 0\) is a Radon measure with support in the minimal boundary \(\Delta_1\) (which is contained in the Martin boundary \(\Delta\)). Thus, a superharmonic function \(v\geq 0\) on \(\Omega\) can be represented by two nonnegative Radon measures \(\mu\) on \(\Omega\) and \(\nu\) on Martin boundary, when the two kernels \(G(x,y)\) and \(K(x,y)\) are suitably fixed on \(\Omega\). This Riesz-Martin representation for \(v\) has been variously extended to domains \(\Omega\) in Riemannian manifolds and in locally compact spaces (in the axiomatic potential theory). This note shows that for a suitable domain \(\Omega\) in a Riemannian manifold, if \(v\) is a locally (volume-element) \(dx\)-integrable function such that \((-\Delta)^iv\geq 0\) on \(\Omega\) for \(0\leq i\leq m\), then \(v\) can be represented in the above sense (with respect to the Green kernel and suitably modified Martin kernels on \(\Omega\)) by a Radon measure \(\mu\geq 0\) on \(\Omega\) and \(m\) Radon measures \(\nu_i\geq 0\) on \(\Lambda_1\).
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31A10
Mathematics Subject Classification ID: 31A30
zbMATH DE Number: 5164260
zbMATH Keywords: Riesz-Martin representation
Wikidata QID: Q115243358
author: Victor Anandam
author: Sadoon I. Othman
reviewed by: Victor Anandam
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5164260