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

TL;DR

This paper introduces anisotropic mixed-norm Hardy spaces using maximal functions, characterizes them via various methods, explores their duality with Campanato spaces, and proves boundedness of certain operators on these spaces.

Contribution

It provides the first comprehensive characterization of anisotropic mixed-norm Hardy spaces and establishes their duality and operator boundedness properties.

Findings

Characterization of $H_A^{oldsymbol{p}}(oldsymbol{R}^n)$ via maximal functions, atoms, and Littlewood-Paley functions.

Duality between $H_A^{oldsymbol{p}}(oldsymbol{R}^n)$ and anisotropic mixed-norm Campanato spaces.

Boundedness of Calderón-Zygmund operators on $H_A^{oldsymbol{p}}(oldsymbol{R}^n)$ and $L^{oldsymbol{p}}(oldsymbol{R}^n)$.

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 Calder\'{o}n-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 Calder\'on-Zygmund operators on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$ with $\vec{p}\in(1,\infty)^n$ is also presented.