  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 $\mathcal{D}=\{Z\in M(n,\mathbb{C}): I-ZZ^* > 0\}$. 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
$\widetilde{\mathcal{H}}_\nu$ which acts on $\mathcal{D}$ . Then for a given
${\mathbb{C}}$-valued function $F$ on $\mathcal{D}$ satisfying
$\widetilde{\mathcal{H}}_\nu F=-\frac{1}{4}(\lambda^2+(n-\nu)^2)
F.(\begin{smallmatrix} I&0 0&-I \end{smallmatrix})$ we prove that $F$ is the
Poisson transform by $P_{\lambda,\nu}$ of some $f\in L^p(S)$, when $1<p<\infty$
or $F=P_{\lambda,\nu}\mu$ for some Borel measure $\mu$ on the Shilov boundary
$S$, when $p=1$ if and only if \[ \sup_{0\leq r <
1}(1-r^2)^{\frac{-n(n-\nu-\Re(i\lambda))}{2}}\left( \int_S |F(rU)|^p {\rm
d}U\right) ^{\frac{1}{p}} < \infty, \] provided that the complex parameter
$\lambda$ satisfies $i\lambda \notin 2\mathbb{Z}^- +n-2\pm \nu$ and
$\Re(i\lambda)>n-1$. This generalizes the result in \cite{B1} which corresponds
to $\tau_\nu$ the trivial representation.
