  In this paper, we study multilinear Fourier multiplier operators on Hardy
spaces. In particular, we prove that the multilinear Fourier multiplier
operator of H\"ormander type is bounded from $H^{p_1} \times \cdots \times
H^{p_m}$ to $H^p$ for $0<p_1,\dots,p_m\le 1$ with $1/p_1 + \cdots 1/p_m = 1/p$,
under suitable cancellation conditions. As a result, we extend the trilinear
estimates of the authors(arXiv:2107.00225) to general multilinear ones and
improve the boundedness result of the authors, Lee, Heo, Hong, Park, and
Yang(Math. Ann. 381 : 499-555, 2021) in limiting situations.
