A sparse domination for the Marcinkiewicz integral with rough kernel and applications

TL;DR

This paper proves a bilinear sparse domination for the Marcinkiewicz integral with rough kernel, leading to new weighted bounds, advancing understanding of its behavior in harmonic analysis.

Contribution

The authors establish a bilinear sparse domination for the Marcinkiewicz integral with rough kernel, enabling new quantitative weighted bounds.

Findings

Established bilinear sparse domination for the Marcinkiewicz integral

Derived new weighted bounds for the operator

Extended analysis to rough kernels with minimal regularity

Abstract

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mu_{\Omega}$ be the higher-dimensional Marcinkiewicz integral defined by $$\mu_\Omega(f)(x)= \Big(\int_0^\infty\Big|\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-1}}f(y)dy\Big|^2\frac{dt}{t^3}\Big)^{1/2}. $$ In this paper, the authors establish a bilinear sparse domination for $\mu_{\Omega}$ under the assumption $\Omega\in L^{\infty}(S^{n-1})$. As applications, some quantitative weighted bounds for $\mu_{\Omega}$ are obtained.