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

Let D={Z∈M(n;C);In−ZZ∗ positive definite} be the matrix ball of rank n and let HD 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:HDF(Z)=−(λ2+n2)F(Z).In,to have an L2-Poisson integral representations over the Shilov boundary S of D. Namely, F is the Poisson integral of an L2-function on the Shilov boundary S if and only if its satisfies the following growth condition of Hardy type:||F||∗,λ2=sup0⩽r<1 [(1−r2)−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 HD has an L2-Poisson representation over S if and only if its Hardy norm is finite.