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 \(L^{\vec {p}}\left( {\mathcal {D}}\right) \) to \(L^{\vec {q}}\left( {\mathcal {D}}\right) \), especially on their boundedness, where \(L^{\vec {p}}\left( {\mathcal {D}}\right) \) and \(L^{\vec {q}}\left( {\mathcal {D}}\right) \) are both weighted Lebesgue spaces over the Cartesian product of two tubular domains \(T_B\), 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.