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

In this paper, we introduce and study two classes of multiparameter Forelli-Rudin type operators from $L^{\vec{p}}(T_B\times T_B,d{V}_{{\alpha }_1}\times d{V}_{{\alpha }_2})$ to $L^{\vec{q}}(T_B\times T_B,d{V}_{{\beta }_1}\times d{V}_{{\beta }_2})$, especially on their boundedness, where $L^{\vec{p}}(T_B\times T_B,d{V}_{{\alpha }_1}\times d{V}_{{\alpha }_2})$ and $L^{\vec{q}}(T_B\times T_B,d{V}_{{\beta }_1}\times d{V}_{{\beta }_2})$ 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 $1\le \vec{p}\le \vec{q}<\mathrm{\infty }$. Moreover, we provide the necessary and sufficient condition of the case that $\vec{q}=(\mathrm{\infty },\mathrm{\infty })$. 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.