$L^{\vec{p}}-L^{\vec{q}}$ Boundedness of Multiparameter Forelli-Rudin Type Operators on the Product of Unit Balls of ${\mathbb{C}}^n$

In this work, we provide a complete characterization of the boundedness of two classes of multiparameter Forelli-Rudin type operators from one mixed-norm Lebesgue space $L^{\vec{p}}$ to another space $L^{\vec{q}}$, when $1\le \vec{p}\le \vec{q}<\mathrm{\infty }$, equipped with possibly different weights. Using these characterizations, we establish the necessary and sufficient conditions for both $L^{\vec{p}}-L^{\vec{q}}$ boundedness of the weighted multiparameter Berezin transform and $L^{\vec{p}}-A^{\vec{q}}$ boundedness of the weighted multiparameter Bergman projection, where $A^{\vec{q}}$ denotes the mixed-norm Bergman space. Our approach presents several novelties. Firstly, we conduct refined integral estimates of holomorphic functions on the unit ball in ${\mathbb{C}}^n$. Secondly, we adapt the classical Schur’s test to different weighted mixed-norm Lebesgue spaces. These improvements are crucial in our proofs and allow us to establish the desired characterization and sharp conditions.