A Poisson formula for solvable Lie groups
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 \(\alpha\) on \(X\) and a probability measure \(\nu\) on \(X\) such that the Poisson representation (*) holds with \(G/H\) replaced by \(X\). In previous papers [Pac. J. Math. 165, 115-129 (1994); 170, 517-533 (1995; Zbl 0849.60007)] the author has shown that \(\mu\)-boundaries of a spread out probability measure satisfy a certain deeper property called Strong Approximate Transitivity (SAT). In this paper it is shown that the SAT property of a \(G\)-space for a connected solvable Lie group \(G\) implies transitivity of the action. The class of subgroups \(H\subset G\) such that the homogeneous space \(G/H\) is SAT is characterized what leads to an identification of \(\mu\)-boundaries for the case of such groups with a class of contractive homogeneous spaces of \(G\).
scientific article; zbMATH DE number 927200
scientific article