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 R-n, an index alpha(X) is defined that coincides with the upper Boyd index (alpha) over bar (X) when the space X is rearrangement-invariant. This new index is defined by means of the local maximal operator m(lambda)f. It is shown then that the Hardy-Littlewood maximal operator M is bounded on X if and only if alpha(X) < 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 is an element of A(p) if and only if for any epsilon > 0 there is a constant c such that for any cube Q and any measurable subset E subset of Q, [GRAPHICS] The case epsilon = 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.