Multilinear singular integrals with homogeneous kernels near $L^1$

We obtain the optimal open range of $L^{p_1}({\mathbb{R}}^n)\times \cdots \times L^{p_m}({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)$ bounds for multilinear singular integral operators with homogeneous kernels of the form $\mathrm{\Omega }(\frac{y}{\mathrm{\mid }y\mathrm{\mid }})\mathrm{\mid }y{\mathrm{\mid }}^{-mn}$, where $\mathrm{\Omega }$ is a function in $L^q({\mathbb{S}}^{mn-1})$ with vanishing integral and $q>1$.