Harmonic functions on homogeneous spaces
type: Item
instance of: scholarly article
title: Harmonic functions on homogeneous spaces
zbMATH Open document ID: 0957.43005
author: Cho-Ho Chu
author: C. W. Leung
published in: Monatshefte für Mathematik
publication date: 2001-04-01T00:00:00Z
review text: Let \(G\) be a locally compact group acting on a locally compact space \(X\) and \(\sigma\) a probability measure on \(G\). A real Borel function on \(X\) is said to be \(\sigma\)-harmonic if it satisfies the convolution equation \(f=\sigma *f\). In 1963, \textit{H. Furstenberg} [Ann. Math. (2) 77, 335-386 (1963; Zbl 0192.12704)] showed that for a connected semisimple Lie group \(G\) with finite centre and maximal compact subgroup \(K\), the homogeneous space \(G/K\) has the Liouville property, that is, every bounded harmonic function on it is necessarily constant. In this paper, the authors show that Furstenberg's proof of the Liouville property for \(G/K\) is valid in a wider setting. More precisely, introducing first the concept of \(\sigma\)-admissible neighbourhood of the identity relative to \(X\), where \(X\) is a homogeneous space of \(G\), and adapting Furstenberg's arguments, they show that if \(G\) is a union of \(\sigma\)-admissible neighborhoods, then the bounded \(\sigma\)-harmonic functions on \(X\) are constant. Then they present some interesting consequences, pointing out several examples of such spaces \(X\) having the Liouville property.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 1382945
zbMATH Keywords: \(\sigma\)-harmonic functions
zbMATH Keywords: locally compact group
zbMATH Keywords: real Borel function
zbMATH Keywords: homogeneous space
zbMATH Keywords: harmonic function
zbMATH Keywords: Liouville property
reviewed by: Anca Precupanu
MaRDI profile type: Publication
full work available at URL: s006050050060
OpenAlex ID: W2056090822
DOI: 10.1007/S006050050060
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description: scientific article; zbMATH DE number 1382945