New martingale inequalities and applications to Fourier analysis. (May 2019)

• Record Type: Journal Article
• Authors: Xie, Guangheng; Weisz, Ferenc; Yang, Dachun; Jiao, Yong
• Is Part Of: Nonlinear analysis. Volume 182(2019)
• Page Start: 143
• Page End: 192
• Publication Date: 2019-05
• Subjects: primary 60G42 -- secondary 60G46 -- 42B25 -- 42B35 -- 46E30; Probability space -- Musielak–Orlicz space -- Martingale Musielak–Orlicz Hardy space -- Quadratic variation -- Atom -- Doob maximal operator -- Fejér operator -- Burkholder–Davis–Gundy inequality -- Weight; 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.2018.12.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let ( Ω, F, P ) be a probability space and φ : Ω × [ 0, ∞ ) → [ 0, ∞ ) be a Musielak–Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak–Orlicz space L φ ( Ω ) . Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on L φ ( Ω ) . As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for L φ ( Ω ), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces H φ s ( Ω ), P φ ( Ω ), Q φ ( Ω ), H φ S ( Ω ) and H φ M ( Ω ) . From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak–Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on H φ S ( Ω ) and H φ M ( Ω ), the authors obtain the Burkholder–Davis–Gundy inequality associated with Musielak–Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from H φ [ 0, 1 ) to L φ [ 0, 1 ), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.