Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness

The main purpose of this paper is three-fold. First of all, we are concerned with the limited smoothness conditions in the spirit of Hörmander on the multi-linear and multi-parameter Coifman–Meyer type Fourier multipliers studied by C. Muscalu, J. Pipher, T. Tao, C. Thiele (2004, 2006) where they established the LrLr estimates for the multiplier operators under the assumption that the multiplier has smoothness of sufficiently large order. Under our limited smoothness assumption, we will prove the Lp1×⋯×Lpn→LrLp1×⋯×Lpn→Lr boundedness with 1p1+⋯+1pn=1r for 1