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

Long Huang, Xiaofeng Wang, Zhicheng Zeng

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.

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

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 to another space , when , equipped with possibly different weights. Using these characterizations, we establish the necessary and sufficient conditions for both boundedness of the weighted multiparameter Berezin transform and boundedness of the weighted multiparameter Bergman projection, where 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 . 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.
Paper Structure (5 sections, 33 theorems, 202 equations)

This paper contains 5 sections, 33 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Proof of necessary for boundedness of $T_{ 0,\,\vec{b},\,\vec{c}}$
  3. Proof of sufficiency for boundedness of $S_{ 0,\,\vec{b},\,\vec{c}}$
  4. Completing the proofs of Theorems \ref{['0t1']}, \ref{['0t2']}, \ref{['0t3']}, and \ref{['0t4']}
  5. Multiparameter Bergman projection and Berezin transform

Key Result

Theorem 1.3

Let $\vec{p}:=(p_1,p_2)$ and $\vec{q}:=(q_1,q_2)$ satisfy $1<p_-\le p_+\le q_-<\infty$ with $p_+:=\max\{p_1,p_2\}$, $p_-:=\min\{p_1,p_2\}$, and $q_-:=\min\{q_1,q_2\}$. Then the following statements are equivalent.

Theorems & Definitions (60)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 50 more