The convolution equation of Choquet and Deny on nilpotent groups
type: Item
instance of: scholarly article
title: The convolution equation of Choquet and Deny on nilpotent groups
zbMATH Open document ID: 0878.43001
DOI: 10.1007/BF01229501
author: Cho-Ho Chu
author: Titus Hilberdink
published in: Integral Equations and Operator Theory
publication date: 1997-12-15T00:00:00Z
review text: Let \(G\) be a separable, metrizable locally compact group and let \(\sigma\) be a real Borel measure on \(G\). Many authors have intensively studied the convolution equation \[ f(x)= \int_G f(y^{-1}x) d\sigma(y), \quad x\in G, \] where \(f\) is a real-valued function, and have obtained diverse applications. Various techniques have been used to treat the equation. Thus, \textit{G. Choquet} and \textit{J. Deny} [C. R. Acad. Sci. Paris 250, 799-801 (1960; Zbl 0093.12802)] proved that if \(G\) is abelian and if \(\sigma\) is a probability measure whose support generates \(G\), then the bounded solutions are constant. As a development of the above mentioned paper, \textit{J. Deny} [Sémin. Théor. Potentiel de M. Brelot (Paris 1960; Zbl 0101.31301)] has studied the positive solutions \(\mu\) of the more general equation \(\sigma* \mu =\mu\), where \(\sigma\) is a positive measure and \(\mu\) a positive Borel measure on \(G\). In the present paper, the authors adapted Choquet and Deny's methods used in the above mentioned papers to characterize the positive solutions \(\mu\) of the convolution equation \(\sigma* \mu= \mu\) on locally compact nilpotent groups, including the discrete case considered by \textit{E. B. Dynkin} and \textit{M. B. Malyutov} [Sov. Math., Doklady 2, 399-402 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1042-1045 (1961; Zbl 0214.44101)] and by \textit{G. A. Margulis} [Sov. Math., Doklady 166, 241-244 (1966); translation from Dokl. Akad. Nauk SSSR 166, 1054-1057 (1966; Zbl 0187.38902)]. The authors worked out the solutions \(\mu\) explicitly for the Heisenberg groups and some nilpotent matrix groups, by finding all the exponential functions on these groups.
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 938763
zbMATH Keywords: locally compact group
zbMATH Keywords: convolution equation
zbMATH Keywords: Heisenberg groups
zbMATH Keywords: nilpotent matrix groups
zbMATH Keywords: exponential functions
reviewed by: Anca Precupanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 938763