Harmonic functions on semidirect extensions of type 𝐻 nilpotent groups

Damek, Ewa

doi:10.1090/S0002-9947-1985-0787971-5

Harmonic functions on semidirect extensions of type $H$ nilpotent groups
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Trans. Amer. Math. Soc. 290 (1985), 375-384 Request permission

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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