Multilinear rough singular integral operators may not be of the weak type at the endpoint

Abstract

We show that $m$-linear homogeneous rough singular integral operators $\LL_Ω$ associated with $L^{\infty }$ functions $\Omega$ on ${\mathbb{S}}^{mn-1}$ with mean value zero may not be bounded from $L^1({\mathbb{R}}^n)\times \cdots \times L^1({\mathbb{R}}^n)$ to $L^{\frac 1 m,\nf}(\mathbb R^n)$ except possibly for the case $n=1$ and $m=2$.