Harmonic functions on homogeneous spaces
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AbstractGiven a locally compact group G acting on a locally compact space X and a probability measure sigma on G, a real Borel function f on X is called sigma-harmonic if it satisfies the convolution equation f = sigma*f. We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of sigma-admissible neighbourhoods of the identity, relative to X, then every bounded sigma-harmonic function on X is constant. Consequently, for spread out sigma, the bounded sigma-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 sigma-harmonic functions on X are constant which extends Furstenberg's result for connected semisimple Lie groups.
All Author(s) ListChu CH, Leung CW
Journal nameMonatshefte für Mathematik
Volume Number128
Issue Number3
Pages227 - 235
LanguagesEnglish-United Kingdom
Keywords[SIN]-group; harmonic function; homogeneous space; Liouville property
Web of Science Subject CategoriesMathematics; MATHEMATICS

Last updated on 2021-13-04 at 23:37