Summary: Given any quasi-Banach function space \(X\) over \(\mathbb R^n\), an index \(\alpha_X\) is defined that coincides with the upper Boyd index \(\overline{\alpha}_X\) when the space \(X\) is rearrangement-invariant. This new index is defined by means of the local maximal operator \(m_\lambda f\). Then, it is shown 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, a new characterization of the Muckenhoupt \(A_p\) class of weights is shown: \(u\in A_p\) if and only if, for any \(\varepsilon>0\), there is a constant \(c\) such that, for any cube \(Q\) and any measurable subset \(E\subset Q\),
\[ \frac{|E|}{|Q|} \log^\varepsilon \bigg(\frac{|Q|}{|E|}\bigg)\leq c\bigg(\frac{u(E)}{u(Q)} \bigg)^{1/p}. \]
The case \(\varepsilon=0\) corresponding to the class \(A_{p,1}\) is false.
Other applications are given, in particular within the context of variable \(L^p\) spaces.