Littlewood-Paley Characterizations of Anisotropic Hardy Spaces of Musielak-Orlicz Type

Let φ : Rⁿ× [0,\,∞)→[0,∞) be a Musielak-Orlicz function and A an expansive dilation. Let H^φA(Rⁿ) be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. Its atomic characterization and some other maximal function characterizations of H^φA(Rⁿ), in terms of the radial, the non-tangential and the tangential maximal functions, are known. In this article, the authors further obtain their characterizations in terms of the Lusin-area function, the g-function or the g^_λ-function via first establishing an anisotropic Peetre's inequality of Musielak-Orlicz type. Moreover, the range of λ in the g^_λ-function characterization of H^φA(Rⁿ) coincides with the known best conclusions in the case when H^φA(Rⁿ) is the classical Hardy space Hᵖ(Rⁿ) or the anisotropic Hardy space HᵖA(Rⁿ) or their weighted variants, where p∈(0,\,1].