L2-Poisson integral representations of solutions of the Hua system on the bounded symmetric domain SU( n, n)/ S( U( n)× U( n))

Abstract

Let D={Z∈M(n; C);I n−ZZ ∗ positive definite} be the matrix ball of rank n and let H D be the associated Hua operator. For a complex number λ, such that Reiλ> n−1 we give a necessary and sufficient condition on solutions F of the following Hua system of differential equations on D: H DF(Z)=−(λ 2+n 2)F(Z).I n, to have an L 2-Poisson integral representations over the Shilov boundary S of D. Namely, F is the Poisson integral of an L 2-function on the Shilov boundary S if and only if its satisfies the following growth condition of Hardy type: ||F|| ∗,λ 2= sup 0⩽r<1 [(1−r 2) −n(n−Reiλ) ∫ S |F(rU)| 2dU]<+∞. In particular for λ=− in, this shows that every harmonic function F with respect to the Hua operator H D has an L 2-Poisson representation over S if and only if its Hardy norm is finite.