Weighted $L^p$ bounds for the 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 authors proved that if $\mathrm{\Omega }\in L^q(S^{n-1})$ for some $q\in (1,,\infty]$,thenfor$p\in (q’,,\infty)$and$w\in A_{p}($\mathbb{R}$^n)$, the bound of ${\mathcal{M}}_{\mathrm{\Omega }}$ on $L^p({\mathbb{R}}^n,w)$ is less than $C[w{]}_{A_{p\mathrm{/}{q}^{\mathrm{\prime }}}}^{2\max \{1,\frac{1}{p-q^{\mathrm{\prime }}}\}}$.