Molecular Characterizations of Variable Anisotropic Hardy Spaces with Applications to Boundedness of Calderón-Zygmund Operators

Abstract

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\infty ]$ be a variable exponent function satisfying the globally log-Hölder 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 Calderón-Zygmund operators on $H_A^{p(\cdot )}({\mathbb{R}}^n)$. In addition, the boundedness of anisotropic Calderón-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.