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

Abstract

Let $\vec{p}\in (0,\infty {)}^n$, $A$ be an expansive dilation on ${\mathbb{R}}^n$,and $H_A^{\vec{p}}({\mathbb{R}}^n)$ 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 $H_A^{\vec{p}}({\mathbb{R}}^n)$. As an application, the authors obtain a boundedness criterion for a class of linear operators from $H_A^{\vec{p}}({\mathbb{R}}^n)$ to $H_A^{\vec{p}}({\mathbb{R}}^n)$. 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).

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