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

TL;DR

This paper establishes Poisson integral representations for generalized Hua operators on line bundles over certain symmetric spaces, extending previous results to more general settings involving matrix-valued functions and complex parameters.

Contribution

It generalizes existing Poisson transform characterizations for Hua operators to line bundles over SU(n,n)/S(U(n)×U(n)), incorporating matrix-valued functions and complex parameters.

Findings

Characterization of solutions to Hua operator equations as Poisson transforms.

Extension of previous results to non-trivial line bundles and matrix-valued functions.

Conditions on parameters for the Poisson integral representation to hold.

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 $\mathcal{D}=\{Z\in M(n,\mathbb{C}): I-ZZ^* > 0\}$. 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.