Harmonic Functions on Homogeneous Spaces |
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Authors: | Cho-Ho Chu and Chi-Wai Leung |
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Institution: | (1) Goldsmiths College, University of London, London SE14 6NW, United Kingdom;(2) Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1 Edmonton, Alberta, Canada |
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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. |
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