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Multilinear Commutators of Multilinear Square Operators Associated with New $BMO$ Functions and New Weight Functions

Chunliang Li, Shuhui Yang, Yan Lin

Abstract

Via the new weight $A_{\vec p}^{\infty}(\varphi)$ and the new $BMO$ function, the authors introduce a new class of multilinear square operators $T$ with generalized kernels. The boundedness of multilinear commutators and multilinear iterative commutators generated by $T$ and the new $BMO$ function on weighted Lebesgue spaces and weighted Morrey spaces is obtained, respectively. The results of this article contain some known conclusions.

Multilinear Commutators of Multilinear Square Operators Associated with New $BMO$ Functions and New Weight Functions

Abstract

Via the new weight and the new function, the authors introduce a new class of multilinear square operators with generalized kernels. The boundedness of multilinear commutators and multilinear iterative commutators generated by and the new function on weighted Lebesgue spaces and weighted Morrey spaces is obtained, respectively. The results of this article contain some known conclusions.
Paper Structure (8 sections, 16 theorems, 122 equations, 1 figure)

This paper contains 8 sections, 16 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Theorems
  3. Some Preliminaries and Notations
  4. Proof of Main Results
  5. Proof of Theorem 2.1
  6. Proof of Theorem 2.2
  7. Proof of Theorem 2.3
  8. Proof of Theorem 2.4

Key Result

Theorem 2.1

Suppose that $m\geq2$, $T$ is the new multilinear square operator with generalized kernel as in Definition definition1.3. Let ${ \sum_{k=1}^{\infty }} k C_k<\infty$, $\vec{\theta}=(\theta_1,\ldots,\theta_m)$, $\theta_j\geq 0$, $j=1,\ldots,m$, $\vec{p}=(p_1,\ldots,p_m)$, $1/p=1/p_1+\ldots+1/p_m$, $\

Figures (1)

  • Figure 1: Relation among of several classes of multilinear square operators, $BMO$ functions, and weight functions

Theorems & Definitions (34)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.1
  • Remark 1.3
  • Definition 1.2
  • Remark 1.4
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • ...and 24 more