New real-variable characterizations of Musielak–Orlicz Hardy spaces

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