Monatshefte für Mathematik

, Volume 128, Issue 3, pp 227–235 | Cite as

Harmonic Functions on Homogeneous Spaces

  • Cho-Ho Chu
  • Chi-Wai Leung

Abstract.

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation \(\). We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.

1991 Mathematics Subject Classification: 43A05 31C05 45E10 
Key words: Harmonic function homogeneous space Liouville property [SIN]-group 

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Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • Cho-Ho Chu
    • 1
  • Chi-Wai Leung
    • 2
  1. 1. University of London, UKGB
  2. 2. Chinese University of Hong KongHK

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