Journal of Lie Theory

no. 3
Characterization of the Lp-Range of the Poisson Transform on the Octonionic Hyperbolic Plane
Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 805-828
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: \[ \sup_{0\leq r1}(1-r^2)^{\frac{-\rho}{2}} \left(\int_{\partial B(\mathbb{O}^2)} |F(r\theta)|^p d\theta\right)^\frac{1}{p}\infty. \] This extends previous results by the first author et al. for classical hyperbolic spaces.
DOI: 10.5802/jolt.1027
Classification: 43A85, 42B20
Keywords: Octonionic hyperbolic plane, Poisson transform, eigenfunctions, Calderon-Zygmund estimates 
@article{JOLT_2018_28_3_a11,
     author = {A. Boussejra and N. Ourchane},
     title = {Characterization of the {L\protect\textsuperscript{p}-Range} of the {Poisson} {Transform} on the {Octonionic} {Hyperbolic} {Plane}},
     journal = {Journal of Lie Theory},
     pages = {805--828},
     year = {2018},
     volume = {28},
     number = {3},
     doi = {10.5802/jolt.1027},
     zbl = {1395.43006},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1027/}
}
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%A N. Ourchane
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%J Journal of Lie Theory
%D 2018
%P 805-828
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%F JOLT_2018_28_3_a11
A. Boussejra; N. Ourchane. Characterization of the Lp-Range of the Poisson Transform on the Octonionic Hyperbolic Plane. Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 805-828. doi: 10.5802/jolt.1027

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