Subharmonicity without upper semicontinuity
type: Item
instance of: scholarly article
title: Subharmonicity without upper semicontinuity
zbMATH Open document ID: 0873.31007
published in: Journal of Functional Analysis
publication date: 1997-11-04T00:00:00Z
review text: Let \(\Omega\subseteq \mathbb{R}^d\) be open and let \(x\in\Omega\). There are many probability measures \(\mu\) with compact support in \(\Omega\) which have the following property: \(u(x)\leq\int u d\mu\) for every subharmonic function \(u\) on \(\Omega\). (Such a measure is called a Jensen measure for \(x\).) Familiar examples are normalized surface measure on the sphere \(\partial B(x,r)\) and normalized volume measure on the ball \(B(x,r)\) whenever \(\overline {B(x,r)}\subset \Omega\). Further, if \(\omega\) is any bounded open set such that \(x\in\omega\) and \(\overline {\omega}\subset\Omega\), then harmonic measure for \(\omega\) and \(x\) is also a Jensen measure for \(x\). The idea of this paper is to study functions \(u\) (satisfying mild measurability conditions) on \(\Omega\) with the property that \(u(x)\leq\int u d\mu\) for each pair \((x,\mu)\) such that \(\mu\) is a Jensen measure for \(x\). Such functions are called quasi-subharmonic. The first main result of the paper is that, if \(u\) is quasi-subharmonic on \(\Omega\), then the u.s.c. regularization \(u^*\) is subharmonic on \(\Omega\) and \(u^*=u\) outside a polar set. (The converse is obvious.) The potential usefulness of this result is shown by the ease with which one can deduce the classical Cartan theorem concerning the supremum of a family of subharmonic functions which is locally uniformly bounded above. Another result is a converse to the Cartan theorem: under very mild assumptions, any quasi-subharmonic function \(u\) which is locally bounded above can be expressed as the supremum of all subharmonic minorants of \(u\). This leads to an interesting duality theorem. The final section discusses corresponding results for plurisubharmonic functions. The paper is very well written.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 31C10
zbMATH DE Number: 1046932
zbMATH Keywords: subharmonic function
zbMATH Keywords: Jensen measure
zbMATH Keywords: plurisubharmonic function
author: Brian J. Cole
author: Thomas J. Ransford
MaRDI profile type: Publication
full work available at URL: jfan.1996.3070
OpenAlex ID: W2050717044
cites work: Q4112980
cites work: Théorie du potentiel newtonien : énergie, capacité, suites de potentiels
cites work: A Counterexample to the "Fine" Problem in Pluripotential Theory
cites work: Q5522606
cites work: Q3851127
cites work: Subharmonicity for uniform algebras
cites work: Q4378605
cites work: Q3852455
cites work: Q4002495
cites work: Q3711155
DOI: 10.1006/JFAN.1996.3070
description: scientific article; zbMATH DE number 1046932