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

TL;DR

This paper characterizes the boundedness of multiparameter Forelli-Rudin type operators between weighted mixed-norm Lebesgue spaces on product unit balls in complex space, using refined estimates and adapted Schur's test.

Contribution

It provides the first complete characterization of boundedness for these operators on product spaces with weights, introducing new integral estimates and a modified Schur's test.

Findings

Established necessary and sufficient conditions for operator boundedness.

Derived refined integral estimates for holomorphic functions.

Adapted Schur's test for weighted mixed-norm spaces.

Abstract

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 $1\leq \vec{p}\leq \vec q<\infty$, 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.