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A poisson formula for solvable Lie groups

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Abstract

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computing Science, Dalhousie University, B3H 3J5, Nova Scotia, Halifax, Canada

    Wojciech Jaworski

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  1. Wojciech Jaworski
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Jaworski, W. A poisson formula for solvable Lie groups. J. Anal. Math. 68, 183–208 (1996). https://doi.org/10.1007/BF02790209

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  • DOI: https://doi.org/10.1007/BF02790209

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