Characterization of the Lp-range of the Poisson transform on the Octonionic Hyperbolic Plane

Abstract

Let B(O 2 ) = {x ∈ O 2 , | x |< 1} be the bounded realization of the exceptional symmetric space F 4(−20) /Spin(9). For a nonzero real number λ, we give a necessary and a sufficient condition on eigenfunctions F of the Laplace-Beltrami operator on B(O 2 ) with eigenvalue −(λ 2 + ρ 2 ) to have an L p -Poisson integral representations on the boundary ∂B(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:This extends previous results by the first author et al. for classical hyperbolic spaces.