Sparse domination for rough multilinear singular integrals

Let $\mathrm{\Omega }$ be a function on ${\mathbb{R}}^{mn}$, homogeneous of degree zero, and satisfy a cancellation condition on the unit sphere ${\mathbb{S}}^{mn-1}$. In this paper, we show that the multilinear singular integral operator $${\mathcal{T}}_{\mathrm{\Omega }}(f_1,\dots ,f_m)(x):=\mathrm{p}\mathrm{.}\mathrm{v}\mathrm{.}{\int }_{{\mathbb{R}}^{mn}}\frac{\mathrm{\Omega }(x-y_1,\dots ,x-y_m)}{\mathrm{\mid }x-\vec{y}{\mathrm{\mid }}^{mn}}\prod _{i=1}^m{f}_i(y_i)d\vec{y},$$ associated with a rough kernel $\mathrm{\Omega }\in L^r({\mathbb{S}}^{mn-1})$, $r>1$, admits a sparse domination, where $\vec{y}=(y_1,\dots ,y_m)$ and $d\vec{y}=d{y}_1\cdots d{y}_m$. As a consequence, we derive some {quantitative weighted norm inequalities} for ${\mathcal{T}}_{\mathrm{\Omega }}$.