Multilinear rough singular integral operators

We study $m$-linear homogeneous rough singular integral operators ${\mathcal{L}}_{\mathrm{\Omega }}$ associated with integrable functions $\mathrm{\Omega }$ on ${\mathbb{S}}^{mn-1}$ with mean value zero. We prove boundedness for ${\mathcal{L}}_{\mathrm{\Omega }}$ from $L^{p_1}\times \cdots \times L^{p_m}$ to $L^p$ when $1<p_1,\dots ,p_m<\mathrm{\infty }$ and $1\mathrm{/}p=1\mathrm{/}{p}_1+\cdots +1\mathrm{/}{p}_m$ in the largest possible open set of exponents when $\mathrm{\Omega }\in L^q({\mathbb{S}}^{mn-1})$ and $q\ge 2$. This set can be described by a convex polyhedron in ${\mathbb{R}}^m$.