Integral Equations and Operator Theory

, Volume 40, Issue 4, pp 454–464 | Cite as

Harmonic functions on nilpotent groups

  • B E Johnson
Article

Abstract

For a probability measure σ on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL(G) is called σ-harmonic if ∫F(st)dσ(t)=F(s), for almost alls inG. Constant functions are σ-harmonic and it is known that for abelianG 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.

Mathematics Subject Classification 2000

Primary 43A05 Secondary 22D99 

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Copyright information

© Birkhäuser Verlag 2001

Authors and Affiliations

  • B E Johnson
    • 1
  1. 1.Department of MathematicsUniversity of NewcastleNewcastle upon TyneEngland

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