LpLp-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/KX=G/K, to have an LpLp-Poisson integral representations over the Shilov boundary of X  . More precisely, let λ∈Cλ∈C such that R(λ)>η−1 (2η being the genus of X) and let F   be a CC-valued function on X satisfying the following Hua system of second order differential equations:HqF=(λ2−η2)32η2FZ. Then F   has an LpLp-Poisson integral representation (10er(η−Rλ)t{∫K|F(kat)|pdk}1/p<+∞. In particular for λ=ηλ=η, we obtain that a Hua-harmonic function on X   has an LpLp-Poisson integral representation over the Shilov boundary of X if and only if its Hardy norm is finite.