Multilinear rough singular integral operators

We study m$m$-linear homogeneous rough singular integral operators L omega$\mathcal {L}_{\Omega }$ associated with integrable functions omega$\Omega$ on Smn-1$\mathbb {S}<^>{mn-1}$ with mean value zero. We prove boundedness for L omega$\mathcal {L}_{\Omega }$ from Lp1xMIDLINE HORIZONTAL ELLIPSISxLpm$L<^>{p_1}\times \cdots \times L<^>{p_m}$ to Lp$L<^>p$ when 1q(\mathbb {S}<^>{mn-1})$ and q > 2$q\geqslant 2$. This set can be described by a convex polyhedron in Rm$\mathbb {R}<^>m$.