A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators

In this paper, we investigate the Hörmander type theorems for the multi-linear and multi-parameter Fourier multipliers. When the multipliers are characterized by \(L^u\)-based Sobolev norms for \(1<u\le 2\), our results on the smoothness assumptions are sharp in the multi-parameter and bilinear case. In the multi-parameter and multi-linear case, our results are almost sharp. Moreover, even in the one-parameter and multi-linear case, our results improve earlier ones in the literature.