New martingale inequalities and applications to Fourier analysis

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 Lφ(Ω). Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on Lφ(Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for Lφ(Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces Hφs(Ω), Pφ(Ω), Qφ(Ω), HφS(Ω) and HφM(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 HφS(Ω) and HφM(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 Hφ[0,1) to Lφ[0,1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.