Weighted $L^p$ bounds for the Marcinkiewicz integral

TL;DR

This paper establishes weighted $L^p$ bounds for the higher-dimensional Marcinkiewicz integral with kernels in $L^q$, providing explicit bounds depending on the Muckenhoupt weight class, extending previous results in harmonic analysis.

Contribution

The paper proves new weighted $L^p$ bounds for the Marcinkiewicz integral with kernels in $L^q$, including explicit dependence on the $A_p$ weight characteristic.

Findings

Bound of $ ext{Marcinkiewicz}$ integral on $L^p(w)$ is controlled by $[w]_{A_{p/q'}}^{2 ext{max}igrace{1, rac{1}{p-q'}}}$.

Results extend weighted bounds to kernels in $L^q$ with $q eq 2$, broadening applicability.

Provides explicit quantitative bounds depending on weight characteristics.

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 $C[w]_{A_{p/q'}}^{2\max\{1,\,\frac{1}{p-q'}\}}$.