Weighted complete continuity for the commutator of Marcinkiewicz integral

Abstract

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and ${\mathcal{M}}_{\Omega }$ be the higher-dimensional Marcinkiewicz integral associated with $\Omega$. 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 $\mathrm{CMO}({\mathbb{R}}^n)$ functions and ${\mathcal{M}}_{\Omega }$ when $\Omega$ satisfies certain size conditions.

24 pages