The multilinear Hormander multiplier theorem with a Lorentz-Sobolev condition

Abstract

In this article, we provide a multilinear version of the Hörmander multiplier theorem with a Lorentz-Sobolev space condition. The work is motivated by the recent result of the first author and Slavíková where an analogous version of classical Hörmander 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 $$\Vert T_{\sigma }(f_1,\dots ,f_m){\Vert }_{L^p((\mathbb{R}{)}^n)}\lesssim \underset{k\in \mathbb{Z}}{\sup }\Vert \sigma (2^k\vec{\cdot })\widehat{{\Psi }^{(m)}}{\Vert }_{L_s^{mn/s,1}({\mathbb{R}}^{mn})}\Vert f_1{\Vert }_{L^{p_1}((\mathbb{R}{)}^n)}\cdots \Vert f_m{\Vert }_{L^{p_m}((\mathbb{R}{)}^n)}$$ for certain $p,p_1,\dots ,p_m$ with $1/p=1/p_1+\cdots +1/p_m$. 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 $r<mn/s$, $0<q\le \infty$, or by $L_s^{mn/s,q}$ for $q>1$.