Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 28 (2018), No. 3, 805--828Copyright 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 [ Fulltext-pdf  (196  KB)] for subscribers only.