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 $p\in (q^{\prime },\infty )$ and $w\in A_p({\mathbb{R}}^n)$, the bound of ${\mathcal{M}}_{\Omega }$ on $L^p({\mathbb{R}}^n,w)$ is less than $C[w{]}_{A_{p/q^{\prime }}}^{2\max \{1,\frac{1}{p-q^{\prime }}\}}$.

10 pages

References in corpus (1)

• Weak and strong $A_p$-$A_{\infty }$ estimates for square functions and related operators