A Unified Version of Weighted Weak-Type Inequalities for the One-Sided Hardy–Littlewood Maximal Function in Orlicz Classes

Erxin Zhang ()
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Erxin Zhang: School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, China

Mathematics, 2024, vol. 12, issue 18, 1-8

Abstract: Let M g + f be the one-sided Hardy–Littlewood maximal function, φ 1 be a nonnegative and nondecreasing function on [ 0 , ∞ ) , γ be a positive and nondecreasing function defined on [ 0 , ∞ ) ; let φ 2 be a quasi-convex function and u , v , w be three weight functions. In this paper, we present necessary and sufficient conditions on weight functions ( u , v , w ) such that the inequality φ 1 ( λ ) ∫ { M g + f > λ } u ( x ) g ( x ) d x ≤ C ∫ − ∞ + ∞ φ 2 ( C | f ( x ) | v ( x ) γ ( λ ) ) w ( x ) g ( x ) d x holds. Then, we unify the weak and extra-weak-type one-sided Hardy–Littlewood maximal inequalities in the above inequality.

Keywords: weight; weak-type inequality; one-sided maximal function; Orlicz classes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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