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Harmonic functions on nilpotent groups

Lookup NU author(s): Professor Barry Johnson

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Abstract

For a probability measure σ on a locally compact group G which is not supported on any proper closed subgroup, an element F of L∞(G) is called σ-harmonic if (latin small letter esh) F(st)dσ(t) = F(s) for almost all s in G. Constant functions are σ-harmonic and it is known that for abelian G all σ -harmonic functions are constant. For other groups it is known that non constant σ-harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant σ-harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.

Publication metadata

Author(s): Johnson BE

Publication type: Article

Publication status: Published

Journal: Integral Equations and Operator Theory

Year: 2001

Volume: 40

Issue: 4

Pages: 454-464

Print publication date: 01/01/2001

ISSN (print): 0378-620X

ISSN (electronic): 1420-8989

Publisher: Birkhaeuser Verlag AG

URL: http://dx.doi.org/10.1007/BF01198140

DOI: 10.1007/BF01198140

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