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, Tome 35 (2025) no. 1, pp. 17-35

Résumé

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 $f \in L^{1}(\partial B(\mathbb{H}^{n}))$.

Détail

Classification : 43A85, 43A15, 33C05
Mots-clés : Generalized Poisson transform, hypergeometric function, quaternionic hyperbolic space