The Variable Muckenhoupt Weight Revisited

Let p(·):\ Rⁿ→(0,∞) 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 Wp(·)(Rⁿ) and Ap(·)(Rⁿ). Then, by regarding the weighted variable Lebesgue space Lp(·)ω(Rⁿ) with ωp(·)(Rⁿ) as a special case of X and applying known results of the Hardy-type space HX(Rⁿ) associated with X, we further obtain several equivalent characterizations of the weighted variable Hardy space Hp(·)ω() and the boundedness of some sublinear operators on Hp(·)ω(). All of these results coincide with or improve existing ones, or are completely new.