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

Abstract

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 Lr 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→Lr boundedness with 1p1+⋯+1pn=1r for 1<p1,…,pn<∞ and 0<r<∞. Second, our proof of Lr 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 Hörmander type multiplier theorem in the weighted Lebesgue spaces for such operators when the Fourier multiplier is only assumed with limited smoothness.