The H\"ormander Multiplier Theorem III: The complete bilinear case via interpolation

We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear H"ormander 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 $(1\mathrm{/}{p}_1,1\mathrm{/}{p}_2)$ 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$1/p=1/p_1+1/p_2$,$1<p_1,p_2<\infty$.