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 $H_A^{p,q}({\mathbb{R}}^n)$ be the anisotropic Hardy-Lorentz spaces associated with $A$ defined via the non-tangential grand maximal function. In this article, the authors characterize $H_A^{p,q}({\mathbb{R}}^n)$ in terms of the Lusin-area function, the Littlewood-Paley $g$-function or the Littlewood-Paley $g_{\lambda }^{\ast }$-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 $g_{\lambda }^{\ast }$-function characterization of $H_A^{p,q}({\mathbb{R}}^n)$ coincides with the best known one in the classical Hardy space $H^p({\mathbb{R}}^n)$ or in the anisotropic Hardy space $H_A^p({\mathbb{R}}^n)$.

40 pages; Submitted. arXiv admin note: text overlap with arXiv:1512.05081

Cited by in corpus (1)

• Anisotropic Hardy-Lorentz Spaces and Their Applications