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 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 Fejer operator is bounded from H φ [ 0 , 1 ) to L φ [ 0 , 1 ) , which further implies some convergence results of the Fejer means; these results are new even for the weighted Hardy spaces.