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In this work, we provide a complete characterization of the boundedness oftwo classes of multiparameter Forelli-Rudin type operators from one mixed-normLebesgue space $L^{\vec p}$ to another space $L^{\vec q}$, when $1\leq\vec{p}\leq \vec q<\infty$, equipped with possibly different weights. Usingthese characterizations, we establish the necessary and sufficient conditionsfor both $L^{\vec p}-L^{\vec q}$ boundedness of the weighted multiparameterBerezin transform and $L^{\vec p}-A^{\vec q}$ boundedness of the weightedmultiparameter Bergman projection, where $A^{\vec q}$ denotes the mixed-normBergman space. Our approach presents several novelties. Firstly, we conductrefined integral estimates of holomorphic functions on the unit ball in$\mathbb{C}^n$. Secondly, we adapt the classical Schur's test to differentweighted mixed-norm Lebesgue spaces. These improvements are crucial in ourproofs and allow us to establish the desired characterization and sharpconditions.

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