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

Let ${\tau }_{\nu }$ ($\nu \in \mathbb{Z}$) be a character of $K=S(U(n)\times U(n))$,and$SU(n,n)\times_K$\mathbb{C}$$ the associated homogeneous line bundle over $\mathcal{D}=\{Z\in M(n,\mathbb{C}):I-Z{Z}^{\ast }>0\}$. Let ${\mathcal{H}}_{\nu }$ be the Hua operator on the sections of $SU(n,n){\times }_K\mathbb{C}$. Identifying sections of $SU(n,n){\times }_K\mathbb{C}$ with functions on $\mathcal{D}$ we transfer the operator ${\mathcal{H}}_{\nu }$ to an equivalent matrix-valued operator ${\tilde{\mathcal{H}}}_{\nu }$ which acts on $\mathcal{D}$ . Then for a given $\mathbb{C}$-valued function $F$ on $\mathcal{D}$ satisfying $\widetilde{$\mathcal{H}$}_\nu F=-\frac{1}{4}(\lambda^2+(n-\nu)^2) F.(\begin{smallmatrix} I&0 0&-I \end{smallmatrix})$weprovethat$F$ is the Poisson transform by $P_{\lambda ,\nu }$ of some $f\in L^p(S)$, when $1<p<\mathrm{\infty }$ or $F=P_{\lambda ,\nu }\mu$ for some Borel measure $\mu$ on the Shilov boundary $S$, when $p=1$ if and only if $$\underset{0\le r<1}{\sup }(1-r^2{)}^{\frac{-n(n-\nu -\mathrm{\Re }(i\lambda ))}{2}}{({\int }_S\mathrm{\mid }F(rU){\mathrm{\mid }}^p\mathrm{d}U)}^{\frac{1}{p}}<\mathrm{\infty },$$ provided that the complex parameter $\lambda$ satisfies $i\lambda \mathrm{\notin }2{\mathbb{Z}}^{-}+n-2\pm \nu$ and $\mathrm{\Re }(i\lambda )>n-1$. This generalizes the result in \cite{B1} which corresponds to ${\tau }_{\nu }$ the trivial representation.