Formula omitted]-Poisson integral representations of solutions of the Hua system on Hermitian symmetric spaces of tube type

Abstract

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 L p -Poisson integral representations over the Shilov boundary of X. More precisely, let λ ∈ C such that R ( λ ) > η − 1 (2 η 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: H q F = ( λ 2 − η 2 ) 32 η 2 F Z . Then F has an L p -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 e r ( η − R λ ) t { ∫ K | F ( k a t ) | p d k } 1 / p < + ∞ . In particular for λ = η , we obtain that a Hua-harmonic function on X has an L p -Poisson integral representation over the Shilov boundary of X if and only if its Hardy norm is finite.