Generalization of Hardy–Littlewood maximal inequality with variable exponent

Abstract

AbstractLet be a measurable function defined on and . In this paper, we generalize the Hardy–Littlewood maximal operator. In the definition, instead of cubes or balls, we take the supremum over all rectangles the side lengths of which are in a cone‐like set defined by a given function ψ. Moreover, instead of the integral means, we consider the ‐means. Let and satisfy the log‐Hülder condition and . Then, we prove that the maximal operator is bounded on if and is bounded from to the weak if . We generalize also the theorem about the Lebesgue points.