Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces

Let p∈ (0,1], q ∈(0,∞] and A be a general expansive matrix on ℝn. Let HAp,q(ℝn) be the anisotropic Hardy-Lorentz spaces associated with A defined via the non-tangential grand maximal function. In this article, the authors characterize HAp,q(ℝn) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley gλ*-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space Lp,qℝn. All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on ℝn. Moreover, the range of λ in the gλ*-function characterization of HAp,q(ℝn) coincides with the best known one in the classical Hardy space Hp(ℝn) or in the anisotropic Hardy space HAp(ℝn).