Dual Spaces of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Abstract

Let A be a general expansive matrix on $\mathbb {R}^n$ and $H_A^{p(\cdot ),q}(\mathbb R^n)$ be the anisotropic variable Hardy–Lorentz space associated with A, where $p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty ]$ denotes a variable exponent function satisfying the globally log-Hölder continuous condition and $q\in (0,\infty )$. In this article, the authors give the appropriate dual space of $H_A^{p(\cdot ),q}(\mathbb {R}^n)$ with full range $p(\cdot )$ by introducing some anisotropic variable $\eta$-type Campanato spaces. As an application, the Carleson measure characterizations of these $\eta$-type Campanato spaces are established. To achieve these, the authors first deduce several equivalent characterizations of the anisotropic variable $\eta$-type Campanato space, and then introduce the anisotropic variable tent-Lorentz spaces and establish their atomic decomposition. All these results are new even for the isotropic Hardy–Lorentz space $H^{p,q}(\mathbb {R}^n)$ and the isotropic variable Hardy–Lorentz space $H^{p(\cdot ),q}(\mathbb {R}^n)$.