The compactness of commutators of Calder\'on-Zgymund operators with Dini condition

TL;DR

This paper proves that certain commutators of Calderón-Zygmund operators with Dini conditions are compact on weighted Lebesgue spaces, extending understanding of their operator properties.

Contribution

It establishes the compactness of commutators generated by Calderón-Zygmund operators with Dini conditions on weighted spaces, a novel result in harmonic analysis.

Findings

Commutators are compact on weighted L^p spaces.

Maximal commutators share the compactness property.

Results hold for Muckenhoupt A_p weights.

Abstract

Let $T$ be the $\theta$-type Calder\'on-Zgymund operator with Dini condition. In this paper, we prove that for $b\in {\rm CMO}(\mathbb R^n)$, the commutator generated by $T$ with $b$ and the corresponding maximal commutator, are both compact operators on $L^{p}(\omega)$ spaces, where $\omega$ be the Muchenhoupt $A_p$ weight function and $1<p<\infty$.