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 $L^r_s(\mathbb R^{2n})$, $2\le r<\infty$, $rs>2n$, 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 $1/p=1/p_1+1/p_2$, $1<p_1,p_2<\infty$.