Weighted complete continuity for the commutator of Marcinkiewicz integral

Let $\mathrm{\Omega }$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and ${\mathcal{M}}_{\mathrm{\Omega }}$ be the higher-dimensional Marcinkiewicz integral associated with $\mathrm{\Omega }$. In this paper, the author considers the complete continuity on weighted $L^p({\mathbb{R}}^n)$ spaces with $A_p({\mathbb{R}}^n)$ weights, weighted Morrey spaces with $A_p({\mathbb{R}}^n)$ weights, for the commutator generated by ${\rm CMO}($\mathbb{R}$^n)$functionsand$$\mathcal{M}$_{\Omega}$when$\Omega$ satisfies certain size conditions.