Multilinear rough singular integral operators

Abstract

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