The Variable Muckenhoupt Weight Revisited

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\mathrm{\infty })$ 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 $L_{\omega }^{p(\cdot )}({\mathbb{R}}^n)$ 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 $H_{\omega }^{p(\cdot )}(\text{rn})$ and the boundedness of some sublinear operators on $H_{\omega }^{p(\cdot )}(\text{rn})$. All of these results coincide with or improve existing ones, or are completely new.