From A 1 to A∞ : New Mixed Inequalities for Certain Maximal Operators

Abstract

In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in Berra (Proc. Am. Math. Soc. 147(10), 4259?4273, 2019). Concretely, given r ≥ 1, u ∈ A1, vr∈ A∞ and a Young function Φ with certain properties, we have that inequalityuvr({x∈ℝn:MΦ(fv)(x)v(x)>t})≤C∫ℝnΦ(|f(x)|t)u(x)vr(x)dx holds for every positive t. As an application, we furthermore exhibe and prove mixed inequalities for the generalized fractional maximal operator Mγ,Φ, where 0 < γ < n and Φ is a Young function of LlogL type.