Boundedness and compactness of commutators associated with Lipschitz functions

TL;DR

This paper investigates the boundedness and compactness of commutators associated with Lipschitz functions and singular or fractional integral operators, introducing a new CMO-type space and establishing key characterizations.

Contribution

It introduces the ${\rm CMO}_\alpha(\mathbb{R}^n)$ space, explores its relation to Lipschitz spaces, and provides new necessary and sufficient conditions for boundedness and compactness of commutators.

Findings

Established a necessary condition for boundedness of iterated commutators on weighted Lebesgue spaces.

Provided an equivalent characterization of compactness of commutators via ${\rm CMO}_\alpha(\mathbb{R}^n)$.

Presented new results even in the unweighted setting for first-order commutators.

Abstract

Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}_\alpha(\mathbb R^n)$ and the Lipchitz space $Lip_\alpha(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\Omega,\beta})^m_b$ on weighted Lebesgue spaces via functions in $Lip_\alpha(\mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{\Omega,\beta})^m_b$ via functions in ${\rm CMO}_\alpha(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators.