Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces. (March 2016)

• Record Type: Journal Article
• Authors: Chen, Lu; Lu, Guozhen; Luo, Xiang
• Is Part Of: Nonlinear analysis. Volume 134(2016)
• Page Start: 55
• Page End: 69
• Publication Date: 2016-03
• Subjects: 42B15 -- 42B25; Multi-parameter Fourier multiplier -- Multi-parameter Triebel–Lizorkin spaces -- Multi-parameter Besov–Lipschitz spaces -- Strong maximal functions -- Littlewood–Paley decomposition; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.12.016
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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.