Hormander 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 Hormander 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 L-r 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 L-p1 x ... x L-pn -> L-r boundedness with 1/p(1) + ... + 1/p(n) = 1/r for 1 < p(1), ..., p(n) < infinity and 0 < r < infinity. Second, our proof of L-r estimates also offers a different and more direct approach than the one given in Muscalu et al. (2004, 2006) where they use the deep analysis of multi-linear and multi-parameter paraproducts. Third, we also prove a Hormander type multiplier theorem in the weighted Lebesgue spaces for such operators when the Fourier multiplier is only assumed with limited smoothness. (C) 2014 Elsevier Ltd. All rights reserved.