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In this paper, we investigate the H\"ormander type theorems for themulti-linear and multi-parameter Fourier multipliers. When the multipliers arecharacterized by $L^u$-based Sobolev norms for $1<u\le 2$ , our results on thesmoothness assumptions are sharp in the multi-parameter and bilinear case. Inthe multi-parameter and multi-linear case, our results are almost sharp.Moreover, even in the one-parameter and multi-linear case, our results improveearlier ones in the literature.

A sharp H\"{o}rmander estimate for multi-parameter and multi-linear Fourier multiplier operators