A Poisson formula for solvable Lie groups
type: Item
instance of: scholarly article
title: A Poisson formula for solvable Lie groups
zbMATH Open document ID: 0865.43008
DOI: 10.1007/BF02790209
author: Wojciech Jaworski
published in: Journal d'Analyse Mathématique
publication date: 1997-07-06T00:00:00Z
review text: Given a probability measure \(\mu\) on a locally compact group \(G\) a bounded Borel function \(h:G \to\mathbb{C}\) is called \(\mu\)-harmonic if it satisfies the \(\mu\)-mean value property, viz. \(h(g)= \int_Gh(gg') \mu(dg')\), \(g\in G\). It is known that certain conditions on \(G\) and \(\mu\) lead to the Poisson representation for such functions in terms of \(L^\infty\) functions on an appropriate homogeneous space \(G/H\) of \(G\). In fact, if \(\mu\) is a spread out probability measure (meaning that some convolution power of \(\mu\) is nonsingular with respect to the Haar measure on \(G)\) on a connected semisimple Lie group with finite center, then a classical result of \textit{H. Furstenberg} [Ann. Math., II. Ser. 77, 335-385 (1963; Zbl 0192.12704)] asserts the validity of the Poisson representation (*) \(h(g)= \int_{G/H} f(gx) \nu (dx)\), where \(f\in L^\infty (G/H)\), \(\nu\) is a probability measure on \(G/H\) and \(H\) a suitable subgroup such that \((MAN)_0 \subset H \subset MAN\), \(MAN\) denoting the minimal parabolic subgroup of \(G\) (and subscript 0 meaning as usual the connected component of identity). With an additional restriction of finiteness of the first order moment of \(\mu\) \textit{A. Raugi} [Bull. Soc. Math. France, Suppl., Mem. 54, 5-18 (1977; Zbl 0389.60003)] has extended this result to all almost connected locally compact second countable groups. The author investigates the case when \(G\) is a connected solvable Lie group and shows that the Poisson representation holds also without the first moment restriction. However, the class of homogeneous spaces needed to represent bounded \(\mu\)-harmonic functions without this restriction is strictly larger than that in the presence of this restriction. The author approaches the solution of the problem by the study of \(\mu\)-boundaries of \(G\), by which a triple \((X,\alpha, \nu)\) is meant consisting of a Borel \(G\)-space \(X\) with a \(\sigma\)-finite quasiinvariant measure \(\alpha\) o…
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 28C10
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 927200
zbMATH Keywords: harmonic functions
zbMATH Keywords: Poisson formula
zbMATH Keywords: transitive action
zbMATH Keywords: probability measure
zbMATH Keywords: locally compact group
zbMATH Keywords: Poisson representation
zbMATH Keywords: homogeneous space
zbMATH Keywords: semisimple Lie groups
zbMATH Keywords: solvable Lie group
Wikidata QID: Q115391594
reviewed by: A. Strasburger
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 927200