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

Abstract

Let Mg+f be the one-sided Hardy–Littlewood maximal function, φ1 be a nonnegative and nondecreasing function on [0,∞), γ be a positive and nondecreasing function defined on [0,∞); let φ2 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 φ1(λ)∫{Mg+f>λ}u(x)g(x)dx≤C∫−∞+∞φ2(C|f(x)|v(x)γ(λ))w(x)g(x)dx holds. Then, we unify the weak and extra-weak-type one-sided Hardy–Littlewood maximal inequalities in the above inequality.