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 $\LL_{\Omega}$ 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 $m=2$.