Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces

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