The Characterizations of Anisotropic Mixed-Norm Hardy Spaces on $\mathbb{R}^n$ by Atoms and Molecules

TL;DR

This paper develops atomic and molecular decompositions for anisotropic mixed-norm Hardy spaces on ^n, providing new tools for analyzing linear operators and extending classical results to more general settings.

Contribution

It introduces novel atomic and molecular characterizations of anisotropic mixed-norm Hardy spaces, including in the classical isotropic case, and establishes a boundedness criterion for linear operators.

Findings

New atomic and molecular decompositions for the spaces.

Boundedness criteria for linear operators on these spaces.

Results extend classical isotropic Hardy space theory.

Abstract

Let $\vec{p}\in(0,\,\infty)^n$, $A$ be an expansive dilation on $\mathbb{R}^n$,and $H^{\vec{p}}_A({\mathbb {R}}^n)$ be the anisotropic mixed-norm Hardy space defined via the non-tangential grand maximal function studied by \cite{hlyy20}. In this paper, the authors establish new atomic and molecular decompositions of $H^{\vec{p}}_A({\mathbb {R}}^n)$. As an application, the authors obtain a boundedness criterion for a class of linear operators from $H^{\vec{p}}_{A}(\mathbb{R}^n)$ to $H^{\vec{p}}_{A}(\mathbb{R}^n)$. Part of results are still new even in the classical isotropic setting (in the case $A:=2\mathrm I_{n\times n}$, ${\mathrm{I}}_{n\times n}$ denotes the $n\times n$ unit matrix).