  We find optimal conditions on $m$-linear Fourier multipliers to give rise to
bounded operators from a product of Hardy spaces $H^{p_j}$, $0<p_j\le 1$, to
Lebesgue spaces $L^p$. The conditions we obtain are necessary and sufficient
for boundedness and are expressed in terms of $L^2$-based Sobolev spaces. Our
results extend those obtained in the linear case ($m=1 $) by Calder\'on and
Torchinsky [http://www.sciencedirect.com/science/article/pii/S0001870877800169]
and in the bilinear case ($m=2$) by Miyachi and Tomita
[http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=29&iss=2&rank=4].
We also prove a coordinate-type H\"ormander integral condition which we use to
obtain certain extreme cases.

Analysis of PDEs