The convolution equation of Choquet and Deny on nilpotent groups
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.
scientific article; zbMATH DE number 938763
scientific article