Atomic and Littlewood-Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Abstract

Let $\vec{a}:=(a_1,\dots ,a_n)\in [1,\infty {)}^n$, $\vec{p}:=(p_1,\dots ,p_n)\in (0,\infty {)}^n$ and $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$ be the anisotropic mixed-norm Hardy space associated with $\vec{a}$ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderón-Zygmund decomposition and a discrete Calderón reproducing formula, the authors then characterize $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$, respectively, by means of atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\hat{I}}{»}^{\ast }$-function. The obtained Littlewood-Paley $g$-function characterization of $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$ coincidentally confirms a conjecture proposed by Hart et al. [Trans. Amer. Math. Soc. (2017), DOI: 10.1090/tran/7312]. Applying the aforementioned Calderón-Zygmund decomposition as well as the atomic characterization of $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$, the authors establish a finite atomic characterization of $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$, which further induces a criterion on the boundedness of sublinear operators from $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$ to itself [or to $L^{\vec{p}}({\mathbb{R}}^n)$]. The obtained atomic characterizations of $H_{\vec{a}}^{\vec{p}}({\mathbb{R}}^n)$ and boundedness of anisotropic Calderón-Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. in [J. Geom. Anal. 27 (2017), 2758-2787]. All these results are new even for the isotropic mixed-norm Hardy spaces on ${\mathbb{R}}^n$.

64 pages; Submitted