Generalization of Hardy–Littlewood maximal inequality with variable exponent

Abstract

Let 𝑝(⋅) be a measurable function defined on ℝ 𝑑 and 𝑝 − ∶= inf 𝑥∈ℝ 𝑑 𝑝(𝑥). 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 𝐿 𝑞(⋅) -means. Let 𝑝(⋅) and 𝑞(⋅) satisfy the log-Hülder condition and 𝑝(⋅) = 𝑞(⋅)𝑟(⋅). Then, we prove that the maximal operator is bounded on 𝐿 𝑝(⋅) if 1 < 𝑟 − ≤ ∞ and is bounded from 𝐿 𝑝(⋅) to the weak 𝐿 𝑝(⋅) if 1 ≤ 𝑟 − ≤ ∞. We generalize also the theorem about the Lebesgue points.