Boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces

The main purpose of this paper is three-fold. First, we prove that under the limited smoothness conditions, multi-parameter Fourier multiplier operators are bounded on multi-parameter Triebel-Lizorkin and Besov-Lipschitz spaces by the Littlewood-Paley decomposition and the strong maximal operator. Second, we offer a different and more direct method to deal with the boundedness instead of transforming Fourier multiplier operators into multi-parameter Calderon-Zygmund operators. Third, we also prove the boundedness of multi-parameter Fourier multiplier operators on weighted multi-parameter Triebel-Lizorkin and Besov-Lipschitz spaces when the Fourier multiplier is only assumed with limited smoothness. (C) 2016 Elsevier Ltd. All rights reserved.