Dyadic Maximal Operators on Martingale Musielak–Orlicz Hardy Type Spaces and Applications

Let φ : [ 0 , 1 ) × [ 0 , ∞ ) → [ 0 , ∞ ) be a Musielak–Orlicz function and q ∈ ( 0 , ∞ ] . In this article, the authors characterize the martingale Musielak–Orlicz Hardy space H φ [ 0 , 1 ) and the martingale Musielak–Orlicz–Lorentz Hardy space H φ , q [ 0 , 1 ) via dyadic maximal operators. As applications, the authors prove that the maximal Fejér operator is bounded from the space H φ [ 0 , 1 ) to the Musielak–Orlicz space L φ [ 0 , 1 ) and from H φ , q [ 0 , 1 ) to the Musielak–Orlicz–Lorentz space L φ , q [ 0 , 1 ) , which further implies some convergence results of the Fejér means.