New real-variable characterizations of Musielak–Orlicz Hardy spaces

AbstractLet φ:Rn×[0,∞)→[0,∞) be such that φ(x,⋅) is an Orlicz function and φ(⋅,t) is a Muckenhoupt A∞(Rn) weight. The Musielak–Orlicz Hardy space Hφ(Rn) is defined to be the space of all f∈S′(Rn) such that the grand maximal function f∗ belongs to the Musielak–Orlicz space Lφ(Rn). Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of Hφ(Rn) in terms of the vertical or the non-tangential maximal functions, or the Littlewood–Paley g-function or gλ∗-function, via first establishing a Musielak–Orlicz Fefferman–Stein vector-valued inequality. Moreover, the range of λ in the gλ∗-function characterization of Hφ(Rn) coincides with the known best results, when Hφ(Rn) is the classical Hardy space Hp(Rn), with p∈(0,1], or its weighted variant