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

Let $\mathrm{\Omega }$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and ${\mu }_{\mathrm{\Omega }}$ be the higher-dimensional Marcinkiewicz integral defined by $${\mu }_{\mathrm{\Omega }}(f)(x)=({\int }_0^{\mathrm{\infty }}\mid {\int }_{\mathrm{\mid }x-y\mathrm{\mid }\le t}\frac{\mathrm{\Omega }(x-y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n-1}}f(y)dy{\mid }^2\frac{dt}{t^3}{)}^{1\mathrm{/}2}\mathrm{.}$$ In this paper, the authors establish a bilinear sparse domination for ${\mu }_{\mathrm{\Omega }}$ under the assumption $\mathrm{\Omega }\in L^{\mathrm{\infty }}(S^{n-1})$. As applications, some quantitative weighted bounds for ${\mu }_{\mathrm{\Omega }}$ are obtained.