Molecular characterization of anisotropic variable Hardy-Lorentz spaces

Abstract

Let $A$ be an expansive dilation on $\mathbb{R}^n$, $q\in(0, \infty]$ and $p(\cdot):\mathbb{R}^n\rightarrow(0, \infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. Let $H^{p(\cdot), q}_A({\mathbb{R}}^n)$ be the anisotropic variable Hardy-Lorentz space defined via the radial grand maximal function. In this paper, the authors first establish its molecular characterization via the atomic characterization of $H^{p(\cdot), q}_A(\mathbb{R}^n)$. Then, as applications, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from $H^{p(\cdot), q}_{A}(\mathbb{R}^n)$ to $L^{p(\cdot), q}(\mathbb{R}^n)$ or from $H^{p(\cdot), q}_{A}(\mathbb{R}^n)$ to itself. All these results are still new even in the classical isotropic setting.