The Variable Muckenhoupt Weight Revisited

TL;DR

This paper explores the relationship between variable weights and function spaces, providing new characterizations of weighted variable Hardy spaces and establishing boundedness of certain operators, thus advancing the theory of variable exponent analysis.

Contribution

It introduces new connections between variable weights and function spaces, and offers novel characterizations and boundedness results for weighted variable Hardy spaces.

Findings

Established equivalences for weighted variable Hardy spaces.

Proved boundedness of sublinear operators on these spaces.

Improved or extended existing theoretical results.

Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function and $X$ a ball quasi-Banach function space. In this paper, we first study the relationship between two kinds of variable weights $\mathcal{W}_{p(\cdot)}(\mathbb{R}^n)$ and $A_{p(\cdot)}(\mathbb{R}^n)$. Then, by regarding the weighted variable Lebesgue space $L^{p(\cdot)}_{\omega}(\mathbb{R}^n)$ with $\omega\in\mathcal{W}_{p(\cdot)}(\mathbb{R}^n)$ as a special case of $X$ and applying known results of the Hardy-type space $H_{X}(\mathbb{R}^n)$ associated with $X$, we further obtain several equivalent characterizations of the weighted variable Hardy space $H^{p(\cdot)}_{\omega}(\rn)$ and the boundedness of some sublinear operators on $H^{p(\cdot)}_{\omega}(\rn)$. All of these results coincide with or improve existing ones, or are completely new.