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 W₏ () (Rⁿ) and A₏ () (Rⁿ). Then, by regarding the weighted variable Lebesgue space L^p () _ (Rⁿ) with ₏ () (Rⁿ) as a special case of X and applying known results of the Hardy-type space Hₗ (Rⁿ) associated with X, we further obtain several equivalent characterizations of the weighted variable Hardy space H^p () _ () and the boundedness of some sublinear operators on H^p () _ (). All of these results coincide with or improve existing ones, or are completely new.