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 $$H^{p(\cdot ),q}(\mathbb {R}^n)$$ .