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