Weighted complete continuity for the commutator of Marcinkiewicz integral

TL;DR

This paper investigates the weighted complete continuity of the commutator of the Marcinkiewicz integral with CMO functions on weighted Lebesgue and Morrey spaces, extending understanding of operator behavior in harmonic analysis.

Contribution

It establishes weighted complete continuity results for the commutator of the Marcinkiewicz integral with CMO functions on various weighted function spaces, under specific size conditions on mega.

Findings

Proves weighted complete continuity on weighted L^p spaces.

Establishes results on weighted Morrey spaces.

Extends previous work to higher-dimensional settings.

Abstract

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mathcal{M}_{\Omega}$ be the higher-dimensional Marcinkiewicz integral associated with $\Omega$. In this paper, the author considers the complete continuity on weighted $L^p(\mathbb{R}^n)$ spaces with $A_p(\mathbb{R}^n)$ weights, weighted Morrey spaces with $A_p(\mathbb{R}^n)$ weights, for the commutator generated by ${\rm CMO}(\mathbb{R}^n)$ functions and $\mathcal{M}_{\Omega}$ when $\Omega$ satisfies certain size conditions.