  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 $1/p=1/p_1+1/p_2$, $1<p_1,p_2<\infty$.
