# Anisotropic Variable Hardy-Lorentz Spaces and Their Real Interpolation

## TL;DR

This paper introduces anisotropic variable Hardy-Lorentz spaces, characterizes them via maximal functions and atoms, and explores their interpolation properties, linking them to variable Lorentz spaces under certain conditions.

## Contribution

The authors define and characterize anisotropic variable Hardy-Lorentz spaces and establish their interpolation relations with other function spaces, extending existing theory.

## Key findings

- Characterization of $H_A^{p( ext{·}),q}( ext{ℝ}^n)$ via maximal functions and atoms.
- Identification of $H_A^{p( ext{·}),q}( ext{ℝ}^n)$ as an intermediate space through real interpolation.
- Equivalence of $H_A^{p( ext{·}),q}( ext{ℝ}^n)$ and variable Lorentz spaces when $ ext{essinf}_{x} p(x) ext{ in } (1, ext{∞})$.

## Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot),q}(\mathbb R^n)$ associated with $A$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of $H_A^{p(\cdot),q}(\mathbb R^n)$, respectively, in terms of the atom and the Lusin area function. As an application, the authors prove that the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot),q}(\mathbb R^n)$ severs as the intermediate space between the anisotropic variable Hardy space $H_A^{p(\cdot)}(\mathbb R^n)$ and the space $L^\infty(\mathbb R^n)$ via the real interpolation. This, together with a special case of the real interpolation theorem of H. Kempka and J. Vyb\'iral on the variable Lorentz space, further implies the coincidence between $H_A^{p(\cdot),q}(\mathbb R^n)$ and the variable Lorentz space $L^{p(\cdot),q}(\mathbb R^n)$ when $\mathop\mathrm{essinf}_{x\in\mathbb{R}^n}p(x)\in (1,\infty)$.