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 Calderón–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.