ON THE COMPACTNESS OF COMMUTATORS FOR ROUGH MARCINKIEWICZ INTEGRAL OPERATORS

Abstract

Let $b \in \operatorname{BMO}(\mathbb{R}^n)$ and $\mathscr{M}_\Omega$ be the Marcinkiewicz integral operatorwith kernel $\frac{\Omega(x)}{|x|^{n-1}}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by means of Fourier transform estimates and approximationto the operator $\mathscr{M}_\Omega$ with integral operators having smooth kernelswe show that if $b \in \operatorname{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain weak size condition, then the commutator $\mathscr{M}_{\Omega,b} = [b, \mathscr{M}_\Omega]$ generated by $b$ and $\mathscr{M}_\Omega$ is a compact operator on $L^p(\mathbb{R}^n)$ for some $1\lt p\lt \infty$.