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 $${\mu }_{\Omega }(f)(x)=\left({\int }_0^{\infty }\right|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{|x-y{|}^{n-1}}f(y)dy{|}^2\frac{dt}{t^3}{)}^{1/2}.$$ 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.

18 pages