L^p-Poisson integral representations of the generalized Hua operators on line bundles over SU(n,n)/S(U(n)xU(n))

Abstract

Let τ ν (ν ∈ Z) be a character of K = S(U (n) × U (n)), and SU (n, n) × K C the associated homogeneous line bundle over D = {Z ∈ M (n, C) : I − ZZ * > 0}. Let H ν be the Hua operator on the sections of SU (n, n) × K C. Identifying sections of SU (n, n) × K C with functions on D we transfer the operator H ν to an equivalent matrix-valued operator H ν which acts on D . Then for a given C-valued function F on D satisfying H ν F = − 1 4 (λ 2 + (n − ν) 2 )F.( I 0 0 −I ) we prove that F is the Poisson transform by P λ,ν of some f ∈ L p (S), when 1 < p < ∞ or F = P λ,ν µ for some Borel measure µ on the Shilov boundary S, when p = 1 if and only if sup 0≤r<1