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

Abstract

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 ${\tilde{\mathcal{H}}}_{\nu }F=-\frac{1}{4}({\lambda }^2+(n-\nu {)}^2)F.(\begin{array}{ccc}I& 00& -I\end{array})$ 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 \[ \sup_{0\leq r < 1}(1-r^2)^{\frac{-n(n-ν-\Re(iλ))}{2}}\left( \int_S |F(rU)|^p {\rm d}U\right) ^{\frac{1}{p}} < \infty, \] 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.