Littlewood-Paley Characterizations of Anisotropic Hardy-Lorentz Spaces

Abstract

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