A new characterization of the Muckenhoupt Ap weights through an extension of the Lorentz-Shimogaki theorem

Abstract

Given any quasi-Banach function space X over Rn it is defined an index αX that coincides with the upper Boyd index αX when the space X is rearrangement-invariant. This new index is defined by means of the local maximal operator mλf . It is shown then that the Hardy-Littlewood maximal operator M is bounded on X if and only if αX < 1 providing an extension of the classical theorem of Lorentz and Shimogaki for rearrangement-invariant X. As an application it is shown a new characterization of the Muckenhoupt Ap class of weights: u ∈ Ap if and only if for any ε > 0 there is a constant c such that for any cube Q and any measurable subset E ⊂ Q, |E| |Q| logε |Q| |E| ≤ c u(E) u(Q)!1/p. The case ε = 0 is false corresponding to the class Ap,1. Other applications are given, in particular within the context of the variable Lp spaces.