  We provide characterizations for boundedness of multilinear Fourier operators
on Hardy-Lebesgue spaces with symbols locally in Sobolev spaces. Let
$H^q(\mathbb R^n)$ denote the Hardy space when $0<q\le 1$ and the Lebesgue
space $L^q(\mathbb R^n)$ when $1<q\le \infty$. We find optimal conditions on
$m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times
\cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of
local $L^2$-Sobolev space estimates for the symbol of the operator. Our
conditions provide multilinear analogues of the linear results of Calder\'on
and Torchinsky
[http://www.sciencedirect.com/science/article/pii/S0001870877800169] and of the
bilinear results of Miyachi and Tomita
[http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=29&iss=2&rank=4].
The extension to general $m$ is significantly more complicated both technically
and combinatorially, the optimal Sobolev space smoothness required of the
symbol depends on the Hardy-Lebesgue exponents and is constant on various
convex simplices formed by configurations of $m2^{m-1} +1$ points in
$[0,\infty)^m$.
