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

Given any quasi-Banach function space X over ℝⁿ, an index αX is defined that coincides with the upper Boyd index ᾱX 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 A_p class of weights: u ∈ A_p 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|log^ε (|Q|/|E|) ≤ c (u(E)/u(Q))^1/p The case ε = 0 is false corresponding to the class A_P,1. Other applications are given, in particular within the context of the variable L^p spaces.