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$L^{\vec{p}}-L^{\vec{q}}$ Boundedness of Multiparameter Forelli-Rudin Type Operators on the Product of Unit Balls of $\mathbb{C}^n$ (2304.04942v2)

Published 11 Apr 2023 in math.CV

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.

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