The Hörmander Multiplier Theorem III: The complete bilinear case via interpolation

Abstract

We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hörmander multiplier theorem concerning symbols that lie in the Sobolev space $L_s^r({\mathbb{R}}^{2n})$, $2\le r<\infty$, $rs>2n$, uniformly over all annuli. More precisely, given a smoothness index $s$, we find the largest open set of indices $(1/p_1,1/p_2)$ for which we have boundedness for the associated bilinear multiplier operator from $L^{p_1}({\mathbb{R}}^n)\times L^{p_2}({\mathbb{R}}^n)$ to $L^p({\mathbb{R}}^n)$ when $1/p=1/p_1+1/p_2$, $1<p_1,p_2<\infty$.