Lp-Poisson integral representations of solutions of the Hua system on Hermitian symmetric spaces of tube type

In this paper, we give a necessary and sufficient condition on eigenfunctions of the Hua operator on a Hermitian symmetric space of tube type X = G/K, to have an Lp-Poisson integral representations over the Shilov boundary of X. More precisely, let λ ∈ C such that (λ) > η − 1 (2η being the genus of X) and let F be a C-valued function on X satisfying the following Hua system of second order differential equations: HqF = (λ2 −η2) 32η2 FZ. Then F has an Lp-Poisson integral representation (1 < p <+∞) over the Shilov boundary of X if and only if it satisfies the following growth condition of Hardy type: sup t>0 er(η− λ)t K F(kat ) p dk 1/p <+∞. In particular for λ = η, we obtain that a Hua-harmonic function on X has an Lp-Poisson integral representation over the Shilov boundary of X if and only if its Hardy norm is finite. © 2005 Elsevier Inc. All rights reserved.