Molecular Characterizations of Variable Anisotropic Hardy Spaces with Applications to Boundedness of Calder\'on-Zygmund Operators

Abstract

Let $p(\cdot):\ \mathbb{R}^n\to(0,\infty]$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the non-tangential grand maximal function. In this article, via the known atomic characterization of $H_A^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its molecular characterization with the known best possible decay of molecules. As an application, the author obtains a criterion on the boundedness of linear operators on $H_A^{p(\cdot)}(\mathbb{R}^n)$, which is used to prove the boundedness of anisotropic Calder\'on-Zygmund operators on $H_A^{p(\cdot)}(\mathbb{R}^n)$. In addition, the boundedness of anisotropic Calder\'on-Zygmund operators from $H_A^{p(\cdot)}(\mathbb{R}^n)$ to the variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^n)$ is also presented. All these results are new even in the classical isotropic setting.