Journal of Lie Theory 28 (2018), No. 3, 805--828
Copyright Heldermann Verlag 2018
Characterization of the Lp-Range of the Poisson Transform on the Octonionic Hyperbolic Plane
Abdelhamid Boussejra
Department of Mathematics, Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco
boussejra.abdelhamid@uit.ac.ma
Nadia Ourchane
Department of Mathematics, Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco
nadia.ourchane20@gmail.com
[Abstract-pdf]
Let $ B(\mathbb{O}^2)=\{x\in \mathbb{O}^2,|x|<1\}$ be the bounded realization of the exceptional symmetric space $F_{4(-20)}/Spin(9)$. For a non-zero real number $\lambda$, we give a necessary and a sufficient condition on eigenfunctions $F$ of the Laplace-Beltrami operator on $B(\mathbb{O}^2)$ with eigenvalue $-(\lambda^2+\rho^2)$ to have an $L^p$-Poisson integral representations on the boundary $\partial B(\mathbb{O}^2)$. Namely, $F$ is the Poisson integral of an $L^p$-function on the boundary if and only if it satisfies the following growth condition of Hardy-type: \[ \sup_{0\leq r<1}(1-r^2)^{\frac{-\rho}{2}} \left(\int_{\partial B(\mathbb{O}^2)} |F(r\theta)|^p d\theta\right)^\frac{1}{p}<\infty. \] This extends previous results by the first author et al. for classical hyperbolic spaces.
Keywords: Octonionic hyperbolic plane, Poisson transform, eigenfunctions, Calderon-Zygmund estimates.
MSC: 43A85, 42B20
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