Lp-Poisson integral representations for solutions of the generalized Hua system on line bundles over SU(n, n)/S(U(n) x U(n))

Let tau(nu) (nu is an element of Z) be a character of K = S (U(n) x U(n)), and SU (n, n) x (K) C the associated homogeneous line bundle over D = {Z is an element of M(n, C) : I - ZZ* > 0}. Let H-nu be the Hua operator on the sections of SU(n, n) x (K) C. Identifying sections of SU(n, n) x (K) C with functions on D we transfer the operator H-nu to an equivalent matrix-valued operator (H) over tilde (nu) which acts on D. Then for a given C-valued function F on D satisfying (H) over tilde (nu) F = - 1/4(lambda(2) + (n - nu)F-2. [GRAPHICS] we prove that F is the Poisson transform by P-lambda,P-nu of some f is an element of L-p(S), when 1 < p < infinity or F = P-lambda,P-nu mu for some Borel measure mu on the Shilov boundary S, when p = 1 if and only if sup(0 <= r<1 )(1 - r(2)) (-n(n - nu - R(i lambda))/2 )(integral(S)vertical bar F(rU)vertical bar(p )dU)(1/p) < infinity, provided that the complex parameter lambda satisfies i lambda is not an element of 2Z(- )+ n - 2 +/- nu and R(i lambda) > n - 1. This generalizes the result in [1] which corresponds to tau(nu) the trivial representation. (C) 2020 Elsevier Inc. All rights reserved.