Real-Variable Characterizations of New Anisotropic Mixed-Norm Hardy Spaces

Abstract

Let $\vec{p}\in (0,\infty {)}^n$ and $A$ be a general expansive matrix on ${\mathbb{R}}^n$. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $H_A^{\vec{p}}({\mathbb{R}}^n)$ associated with $A$ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $H_A^{\vec{p}}({\mathbb{R}}^n)$, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley $g$-functions or $g_{\lambda }^{\ast }$-functions via first establishing an anisotropic Fefferman-Stein vector-valued inequality on the mixed-norm Lebesgue space $L^{\vec{p}}({\mathbb{R}}^n)$. In addition, the authors also obtain the duality between $H_A^{\vec{p}}({\mathbb{R}}^n)$ and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from $H_A^{\vec{p}}({\mathbb{R}}^n)$ into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional $\delta$-type and non-convolutional $\beta$-order Calderón-Zygmund operators from $H_A^{\vec{p}}({\mathbb{R}}^n)$ to itself [or to $L^{\vec{p}}({\mathbb{R}}^n)$]. As a corollary, the boundedness of anisotropic convolutional $\delta$-type Calderón-Zygmund operators on the mixed-norm Lebesgue space $L^{\vec{p}}({\mathbb{R}}^n)$ with $\vec{p}\in (1,\infty {)}^n$ is also presented.

52 pages, Submitted. arXiv admin note: text overlap with arXiv:1801.06251, arXiv:1908.03291