Journal of Lie Theory 35 (2025), No. 1, 017--035
Copyright Heldermann Verlag 2025
On the Generalized Poisson Transform on the Quaternionic Hyperbolic Space
Achraf Ouald Chaib
Dept. of Mathematics, Faculty of Sciences, University Ibn Tofail, Kénitra, Morocco
achraf.oualdchaib@uit.ac.ma
[Abstract-pdf]
Let $B(\mathbb{H}^{n})=Sp(n,1)/Sp(n)\times Sp(1)$ be the quaternionic hyperbolic space. We consider a generalized Poisson transform $\mathcal{P}_{\lambda,l}$ associated with a character of a class of irreducible representations of $Sp(n)\times Sp(1)$. In this paper, we show that if $f$ is a hyperfunction on the boundary of $B(\mathbb{H}^{n})$, then $f$ belongs to the space $L^{p}(\partial B(\mathbb{H}^{n}))$ if and only if either its generalized Poisson transform $\mathcal{P}_{\lambda,l}f$ satisfies a Hardy-type condition, or the modified admissible maximal function of $\mathcal{P}_{\lambda,l}f$ belongs to $L^{p}(\partial B(\mathbb{H}^{n}))$. In addition, we study the admissible convergence of the generalized Poisson transform $\mathcal{P}_{\lambda,l}f$ for $f \in L^{1}(\partial B(\mathbb{H}^{n}))$.
Keywords: Generalized Poisson transform, hypergeometric function, quaternionic hyperbolic space.
MSC: 43A85, 43A15, 33C05.
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