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 $\Omega(\frac{y}{|y|})|y|^{-mn}$, where $\Omega$ is a function in $L^{q}(\mathbb S^{mn-1})$ with vanishing integral and $q>1$.