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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Harmonic functions on semidirect extensions of type $ H$ nilpotent groups


Author: Ewa Damek
Journal: Trans. Amer. Math. Soc. 290 (1985), 375-384
MSC: Primary 43A80; Secondary 22E27, 22E30, 31C12
MathSciNet review: 787971
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Abstract: Let $ S = NA$ be a semidirect extension of a Heisenberg type nilpotent group $ N$ by the one-parameter group of dilations, equipped with the Riemannian structure, which generalizes this of the symmetric space. Let $ {\{ {P_a}(y)\} _{a > 0}}$ be a Poisson kernel on $ N$ with respect to the Laplace-Beltrami operator. Then every bounded harmonic function $ F$ on $ S$ is a Poisson integral $ F(yb) = f \ast {P_b}(y)$ of a function $ f \in {L^\infty }(N)$. Moreover the harmonic measures $ \mu _a^b$ defined by $ {P_b} = {P_a} \ast \mu _a^b,b > a$, are radial and have smooth densities. This seems to be of interest also in the case of a symmetric space of rank $ 1$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0787971-5
PII: S 0002-9947(1985)0787971-5
Keywords: Laplace-Beltrami operator, type $ H$ nilpotent groups
Article copyright: © Copyright 1985 American Mathematical Society