New real-variable characterizations of Musielak-Orlicz Hardy spaces

Let phi : R-n x [0, infinity) -> [0, infinity) be such that phi(x,.) is an Orlicz function and (phi(., t) is a Muckenhoupt A(infinity)(R-n) weight. The Musielak-Orlicz Hardy space H-phi(R-n) is defined to be the space of all f is an element of s'(R-n) such that the grand maximal function f* belongs to the Musielak-Orlicz space L phi(R-n). Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of H-phi(R-n) in terms of the vertical or the non-tangential maximal functions, or the Littlewood-Paley g-function or g(lambda)*-function, via first establishing a Musielak-Orlicz Fefferman-Stein vector-valued inequality. Moreover, the range of lambda in the g(lambda)*-function characterization of H-phi(R-n) coincides with the known best results, when H-phi(R-n) is the classical Hardy space H-p(R-n), with p is an element of (0, 1], or its weighted variant. (C) 2012 Elsevier Inc. All rights reserved.