Characterization of CMO via compactness of the commutators of bilinear fractional integral operators

TL;DR

This paper characterizes the conditions under which commutators of bilinear fractional integral operators are compact, establishing that membership in CMO is both necessary and sufficient for compactness.

Contribution

It proves the necessity of CMO for the compactness of commutators and characterizes the compactness of iterated commutators of bilinear fractional integral operators.

Findings

CMO membership is necessary for commutator compactness.

The compactness of the iterated commutator is characterized by CMO.

The paper extends previous results by providing a full characterization.

Abstract

Let $I_{\alpha}$ be the bilinear fractional integral operator, $B_{\alpha}$ be a more singular family of bilinear fractional integral operators and $\vec{b}=(b,b)$. B\'{e}nyi et al. in \cite{B1} showed that if $b\in {\rm CMO}$, the {\rm BMO}-closure of $C^{\infty}_{c}(\mathbb{R}^n)$, the commutator $[b,B_{\alpha}]_{i}(i=1,2)$ is a separately compact operator. In this paper, it is proved that $b\in {\rm CMO}$ is necessary for $[b,B_{\alpha}]_{i}(i=1,2)$ is a compact operator. Also, the authors characterize the compactness of the {\bf iterated} commutator $[\Pi\vec{b},I_{\alpha}]$ of bilinear fractional integral operator. More precisely, the commutator $[\Pi\vec{b},I_{\alpha}]$ is a compact operator if and only if $b\in {\rm CMO}$.