Let τν (ν∈Z) be a character of K=S(U(n)×U(n)), and SU(n,n)×KC the associated homogeneous line bundle over D={Z∈M(n,C):I−ZZ⁎>0}. Let Hν be the Hua operator on the sections of SU(n,n)×KC. Identifying ...sections of SU(n,n)×KC with functions on D we transfer the operator Hν to an equivalent matrix-valued operator H˜ν which acts on D. Then for a given C-valued function F on D satisfying H˜νF=−14(λ2+(n−ν)2)F.(I00−I) we prove that F is the Poisson transform by Pλ,ν of some f∈Lp(S), when 1<p<∞ or F=Pλ,νμ for some Borel measure μ on the Shilov boundary S, when p=1 if and only ifsup0≤r<1(1−r2)−n(n−ν−ℜ(iλ))2(∫S|F(rU)|pdU)1p<∞, provided that the complex parameter λ satisfies iλ∉2Z−+n−2±ν and ℜ(iλ)>n−1.
This generalizes the result in 1 which corresponds to τν the trivial representation.
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