Compactness for Commutators of Marcinkiewicz Integrals in Morrey Spaces

Abstract

In this paper the authors give a characterization of the compactness for the commutator $[b,\mu_\Omega]$ in the Morrey spaces $L^{p,\,\lambda}(\mathbb R^n)$, where $\mu_\Omega$ denotes the Marcinkiewicz integral. More precisely, the authors prove that if $b\in \mathrm{VMO}(\mathbb{R}^n)$, the $\mathrm{BMO}(\mathbb{R}^n)$-closure of $C_c^\infty(\mathbb{R}^n)$, then the commutators $[b,\mu_\Omega]$ is a compact operator in the Morrey spaces $L^{p,\,\lambda}(\mathbb{R}^n)$ for $1 \lt p \lt \infty$ and $0 \lt \lambda \lt n$. Conversely, if $b \in \mathrm{BMO}(\mathbb{R}^n)$ and $[b, \mu_{\Omega}]$ is a compact operator in $L^{p,\,\lambda}(\mathbb{R}^n)$ for some $p \in (1, \infty)$ and $\lambda \in (0,n)$, then $b \in \mathrm{VMO}(\mathbb{R}^n)$. In the above results, the kernel function $\Omega$ of the operator $\mu_{\Omega}$ is assumed to satisfy a very weak condition on $S^{n-1}$.