The multilinear Hormander multiplier theorem with a Lorentz-Sobolev condition

In this article, we provide a multilinear version of the H\"ormander multiplier theorem with a Lorentz-Sobolev space condition. The work is motivated by the recent result of the first author and Slav\'ikov\'a where an analogous version of classical H\"ormander multiplier theorem was obtained; this version is sharp in many ways and reduces the number of indices that appear in the statement of the theorem. As a natural extension of the linear case, in this work, we prove that if $mn/2<s<mn$, then $$\big\Vert T_{\sigma}(f_1,\dots,f_m)\big\Vert_{L^p((\mathbb{R})^n)}\lesssim \sup_{k\in\mathbb{Z}}\big\Vert \sigma(2^k\;\vec{\cdot}\;)\widehat{\Psi^{(m)}}\big\Vert_{L_{s}^{mn/s,1}(\mathbb{R}^{mn})}\Vert f_1\Vert_{L^{p_1}((\mathbb{R})^n)}\cdots \Vert f_m\Vert_{L^{p_m}((\mathbb{R})^n)}$$ for certain $p,p_1,\dots,p_m$ with $1/p=1/p_1+\dots+1/p_m$. We also show that the above estimate is sharp, in the sense that the Lorentz-Sobolev space $L_s^{mn/s,1}$ cannot be replaced by $L_{s}^{r,q}$ for $r<mn/s$, $0<q\leq \infty$, or by $L_s^{mn/s,q}$ for $q>1$.