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

Abstract

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 $q>1$.