Journal of Lie Theory

no. 1
On the Generalized Poisson Transform on the Quaternionic Hyperbolic Space
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 
@article{JOLT_2025_35_1_a1,
     author = {A. Ouald Chaib},
     title = {On the {Generalized}  {Poisson} {Transform} on the {Quaternionic} {Hyperbolic} {Space}},
     journal = {Journal of Lie Theory},
     pages = {17--35},
     year = {2025},
     volume = {35},
     number = {1},
     doi = {10.5802/jolt.1371},
     zbl = {1573.43010},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1371/}
}
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A. 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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