Separately harmonic and subharmonic functions
type: Item
instance of: scholarly article
title: Separately harmonic and subharmonic functions
zbMATH Open document ID: 0859.31005
DOI: 10.1007/BF00275514
author: Slawomir Kołodziej
author: Johan Thorbiörnson
published in: Potential Analysis
publication date: 1996-12-04T00:00:00Z
review text: Let \(u(x,y)\) be defined on \(B_1 \times B_2\), where \(B_1\) is the unit ball of \(\mathbb{R}^m\) and \(B_2\) is the unit ball of \(\mathbb{R}^n\). It is known that, if \(u(x, \cdot)\) is harmonic on \(B_2\) for each \(x\) and \(u (\cdot,y)\) is harmonic on \(B_1\) for each \(y\), then \(u\) is harmonic on \(B_1 \times B_2\). This statement fails if ``harmonic'' is replaced by ``subharmonic'' throughout. A hybrid problem, which has remained unsolved for some time, concerns the case where \(u(x, \cdot)\) is harmonic on \(B_2\) for each \(x\) and \(u(\cdot,y)\) is subharmonic on \(B\) for each \(y\). Is \(u\) then subharmonic on \(B_1 \times B_2\)? \textit{S. A. Imomkulov} [Dokl. Akad. Nauk UzSSR 1990, No. 2, 8-10 (1990; Zbl 0708.31005)], has pointed out that the answer is ``yes'' under the rather strong additional assumption that \(u(\cdot,y)\) is real analytic for each \(y\). The present paper shows, by a different argument, that it is enough to assume that \(u(\cdot,y)\) is of class \(C^2\) for each \(y\). The general problem of whether one needs any assumption on \(u(\cdot,y)\), apart from subharmonicity, remains open.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 936755
zbMATH Keywords: separately harmonic functions
zbMATH Keywords: separately subharmonic functions
MaRDI profile type: Publication
cites work: Conditions for separately subharmonic functions to be subharmonic
cites work: On Subharmonicity of Doubly Subharmonic Functions
cites work: Q3253275
cites work: Fonctions plurisousharmoniques et fonctions doublement sousharmoniques
cites work: Q3490263
cites work: Les fonctions plurisousharmoniques
cites work: Fonctions plurisousharmoniques et fonctions analytiques de variables réelles
cites work: On a theorem of Avanissian-Arsove
cites work: Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of $C^n$
cites work: Separately Subharmonic Functions Need not be Subharmonic
cites work: Q3976260
full work available at URL: https://doi.org/10.1007/bf00275514
OpenAlex ID: W1988241564
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description: scientific article; zbMATH DE number 936755