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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 anisotropicmixed-norm Hardy space defined via the non-tangential grand maximal functionstudied by \cite{hlyy20}. In this paper, the authors establish new atomic andmolecular decompositions of $H^{\vec{p}}_A({\mathbb {R}}^n)$. As anapplication, the authors obtain a boundedness criterion for a class of linearoperators from $H^{\vec{p}}_{A}(\mathbb{R}^n)$ to$H^{\vec{p}}_{A}(\mathbb{R}^n)$. Part of results are still new even in theclassical isotropic setting (in the case $A:=2\mathrm I_{n\times n}$,${\mathrm{I}}_{n\times n}$ denotes the $n\times n$ unit matrix).

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