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

We show that $m$-linear homogeneous rough singular integral operators $\LL_Ω$ associated with $L^\infty$ functions $Ω$ 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