Boundedness of Multiparameter Forelli-Rudin Type Operators on Product $L^p$ Spaces over Tubular Domains

Abstract

In this paper, we introduce and study two classes of multiparameter Forelli-Rudin type operators from $L^{\vec{p}}\left(T_B\times T_B, dV_{\alpha_1}\times dV_{\alpha_2}\right)$ to $L^{\vec{q}}\left(T_B\times T_B, dV_{\beta_1}\times dV_{\beta_2}\right)$, especially on their boundedness, where $L^{\vec{p}}\left(T_B\times T_B, dV_{\alpha_1}\times dV_{\alpha_2}\right)$ and $L^{\vec{q}}\left(T_B\times T_B, dV_{\beta_1}\times dV_{\beta_2}\right)$ are both weighted Lebesgue spaces over the Cartesian product of two tubular domains $T_B\times T_B$, with mixed-norm and appropriate weights. We completely characterize the boundedness of these two operators when $1\le \vec{p}\le \vec{q}<\infty$. Moreover, we provide the necessary and sufficient condition of the case that $\vec{q}=(\infty,\infty)$. As an application, we obtain the boundedness of three common classes of integral operators, including the weighted multiparameter Bergman-type projection and the weighted multiparameter Berezin-type transform.