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Littlewood-Paley Characterizations of Anisotropic Hardy-Lorentz Spaces (1601.05242v2)

Published 20 Jan 2016 in math.CA and math.FA

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 $Hp(\mathbb{R}n)$ or in the anisotropic Hardy space $Hp_A(\mathbb{R}n)$.

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