Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$L^p-L^q$ boundedness of Forelli-Rudin type operators on the unit ball of $\mathbb{C}^n$

Ruhan Zhao, Lifang Zhou

Abstract

We completely characterize $L^p-L^q$ boundedness of two classes of Forelli-Rudin type operators on the unit ball of $\mathbb{C}^n$ for all $(p, q)\in [1, \infty]\times [1, \infty]$. The results are not only a complement to some previous results on Forelli-Rudin type operators by Kures and Zhu in 2006 and the first author in 2015, but also a high dimension extension of some results by Cheng, Fang, Wang and Yu in 2017.

$L^p-L^q$ boundedness of Forelli-Rudin type operators on the unit ball of $\mathbb{C}^n$

Abstract

We completely characterize boundedness of two classes of Forelli-Rudin type operators on the unit ball of for all . The results are not only a complement to some previous results on Forelli-Rudin type operators by Kures and Zhu in 2006 and the first author in 2015, but also a high dimension extension of some results by Cheng, Fang, Wang and Yu in 2017.
Paper Structure (14 sections, 33 theorems, 107 equations)

This paper contains 14 sections, 33 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some integral formulas
  4. Schur's tests
  5. Proof of sufficiency for Boundedness of $S_{a, b,c}$
  6. Proof of (iii)$\Rightarrow$(i) for Theorem \ref{['thm:pq3']}
  7. Proof of (iii)$\Rightarrow$ (i) for Theorem \ref{['thm:thm4']} and \ref{['thm:thm5']}
  8. Proof of necessity for Boundedness of $T_{a, b,c}$
  9. Proof of (ii)$\Rightarrow$(iii) for Theorem \ref{['thm:pq3']}
  10. Case I: $1\leq q<p< \infty$
  11. Case II: $p=\infty$ and $1\leq q<\infty$
  12. Proof of (ii)$\Rightarrow$(iii) for Theorem \ref{['thm:thm4']}
  13. Proof of (ii)$\Rightarrow$(iii) for Theorem \ref{['thm:thm5']}
  14. Applications

Key Result

Theorem 1.1

Suppose $1\leq q<p\leq\infty$. Then the following conditions are equivalent.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 45 more