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Commutators of the maximal and sharp functions with weighted Lipschitz functions

Pu Zhang, Xiaomeng Zhu

Abstract

Let $M$ be the Hardy-Littlewood maximal function. Denote by $M_b$ and $[b,M]$ the maximal and the nonlinear commutators of $M$ with a function $b$. The boundedness of $M_b$ and $[b,M]$ on weighted Lebesgue spaces are characterized when the symbols $b$ belong to weighted Lipschitz (weighted Morrey-Campanato) spaces. Some new characterizations for weighted Lipschitz spaces are obtained. Similar results are also established for the nonlinear commutator of the sharp function.

Commutators of the maximal and sharp functions with weighted Lipschitz functions

Abstract

Let be the Hardy-Littlewood maximal function. Denote by and the maximal and the nonlinear commutators of with a function . The boundedness of and on weighted Lebesgue spaces are characterized when the symbols belong to weighted Lipschitz (weighted Morrey-Campanato) spaces. Some new characterizations for weighted Lipschitz spaces are obtained. Similar results are also established for the nonlinear commutator of the sharp function.
Paper Structure (4 sections, 60 equations)

This paper contains 4 sections, 60 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries and Lemmas
  3. Proof of Theorems \ref{['thm.1.1']} and \ref{['thm.1.2']}
  4. Proof of Theorem \ref{['thm.1.3']}