  We provide necessary and sufficient conditions for multilinear multiplier
operators with symbols in $L^r$-based product-type Sobolev spaces uniformly
over all annuli to be bounded from products of Hardy spaces to a Lebesgue
space. We consider the case $1<r\leq 2$ and we characterize boundedness in
terms of inequalities relating the Lebesgue indices (or Hardy indices), the
dimension, and the regularity and integrability indices of the Sobolev space.
The case $r>2$ cannot be handled by known techniques and remains open. Our
result not only extends but also establishes the sharpness of previous results
of Miyachi, Nguyen, Tomita, and the first author, who only considered the case
$r=2$.
