The Variable Muckenhoupt Weight Revisited

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.