Let A1 and A2 be expansive dilations, respectively, on R n and R m . Let A ≡ (A1, A2) and Ap( A ) be the class of product Muckenhoupt weights on Moreover, the authors prove that finite atomic norm on a dense subspace of H p w (R n ×R m ; A ) is equivalent with the standard infinite atomic decomposition norm. As an application, the authors prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space B, then T uniquely extends to a bounded sublinear operator fromThe results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting.

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Let ϕ : R n × [0, ∞) → [0, ∞) be a Musielak-Orlicz function and A an expansive dilation. Let H ϕ A (R n ) 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 ϕ A (R n ), 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 * λfunction characterization of H ϕ A (R n ) coincides with the known best conclusions in the case when H ϕ A (R n ) is the classical Hardy space H p (R n ) or the anisotropic Hardy space H p A (R n ) or their weighted variants, where p ∈ (0, 1].

Let φ : ℝn × [0, ∞)→[0, ∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H A φ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations of H A φ(ℝn) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space H A p(ℝn) with p ∈ (0,1] and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization of H A φ(ℝn), and, as an application, the authors prove that, for a given admissible triplet (φ, q, s), if T is a sublinear operator and maps all (φ, q, s)-atoms with q < ∞ (or all continuous (φ, q, s)-atoms with q = ∞) into uniformly bounded elements of some quasi-Banach spaces ℬ, then T uniquely extends to a bounded sublinear operator from H A φ(ℝn) to ℬ. These results are new even for anisotropic Orlicz-Hardy spaces on ℝn.

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