Local and global majorization of subharmonic functions
type: Item
instance of: scholarly article
title: Local and global majorization of subharmonic functions
zbMATH Open document ID: 0536.31004
DOI: 10.1007/BF02786877
published in: Journal d'Analyse Mathématique
publication date: 1983-01-01T00:00:00Z
review text: Let s be subharmonic in a bounded domain in Euclidean space \(R^ n\) (\(n\geq 2)\). The question of whether the existence of a global harmonic majorant for s is equivalent to the existence of a harmonic majorant on a neighbourhood of each boundary point has received some attention in recent years. In particular, it follows from the work of \textit{P. M. Gauthier} and \textit{M. Goldstein} [J. Lond. Math. Soc., II. Ser. 16, 458- 466 (1977; Zbl 0379.31006)] that equivalence holds for Lipschitz domains. The present paper considers the same question for two special types of harmonic majorization, namely (i) majorization by a quasi-bounded harmonic function, as defined by \textit{M. Parreau} [Ann. Inst. Fourier 3, 103-197 (1952; Zbl 0047.320)], and (ii) majorization by the Perron- Wiener-Brelot solution of the generalized Dirichlet problem for the corresponding open set. Equivalence of local and global majorization is again established for Lipschitz domains, and an n-dimensional counter- example is provided to show that this result does not hold for arbitrary bounded domains.
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 3851483
zbMATH Keywords: majorization
zbMATH Keywords: subharmonic functions
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Lipschitz domain
zbMATH Keywords: Perron-Wiener-Brelot solution
author: Stephen J. Gardiner
MaRDI profile type: Publication
cites work: A Strong Type of Regularity for the PWB Solution of the Dirichlet Problem
cites work: Harmonic majorization of subharmonic functions in unbounded domains
cites work: From Local to Global Properties of Subharmonic Functions on Green Spaces
cites work: Local harmonic majorants of functions subharmonic in the unit disc
cites work: Q5581926
cites work: Über die Existenz harmonischer Minoranten von superharmonischen Funktionen
cites work: On the Boundary Values of Harmonic Functions
cites work: Positive Harmonic Functions on Lipschitz Domains
cites work: On local harmonic majorants of subharmonic functions
cites work: Q4112197
cites work: On Half-Spherical Means of Subharmonic Functions in Half-Spaces
cites work: Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann
description: scientific article; zbMATH DE number 3851483