Journal of Lie Theory

no. 1
On the Generalized Poisson Transform on the Quaternionic Hyperbolic Space
Achraf Ouald Chaib1
1Dept. of Mathematics, Faculty of Sciences, University Ibn Tofail, Kénitra, Morocco
Journal of Lie Theory, Volume 35 (2025) no. 1, pp. 17-35
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 $f \in L^{1}(\partial B(\mathbb{H}^{n}))$.
DOI: 10.5802/jolt.1371
Classification: 43A85, 43A15, 33C05
Keywords: Generalized Poisson transform, hypergeometric function, quaternionic hyperbolic space

Achraf Ouald Chaib  1

1 Dept. of Mathematics, Faculty of Sciences, University Ibn Tofail, Kénitra, Morocco 
Achraf Ouald Chaib. On the Generalized  Poisson Transform on the Quaternionic Hyperbolic Space. Journal of Lie Theory, Volume 35 (2025) no. 1, pp. 17-35. doi: 10.5802/jolt.1371
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