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

Abstract

Let $I_{\alpha }$ be the bilinear fractional integral operator, $B_{\alpha }$ be a more singular family of bilinear fractional integral operators and $\vec{b}=(b,b)$. Bényi et al. in \cite{B1} showed that if $b\in \mathrm{CMO}$, the {\rm BMO}-closure of $C_c^{\infty }({\mathbb{R}}^n)$, the commutator $[b,B_{\alpha }{]}_i(i=1,2)$ is a separately compact operator. In this paper, it is proved that $b\in \mathrm{CMO}$ 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 $b\in \mathrm{CMO}$.

19 pages

Cited by in corpus (1)

• Sharp estimates for commutators of bilinear operators on Morrey type spaces