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

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.