The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the Hardy-Littlewood maximal operator \(M\). For this, the authors consider the boundedness of \(M\) in the weighted Lorentz space \(\Lambda^p_u(w)\). Two examples are historically relevant as a motivation: If \(w=1\), this corresponds to the study of the boundedness of \(M\) on \(L^p(u)\), which was characterized by B. Muckenhoupt in 1972, and the solution is given by the so called \(A_p\) weights. The second case is when we take \(u=1\). This is a more recent theory, and was completely solved by M.A. Ariño and B. Muckenhoupt in 1991. It turns out that the boundedness of \(M\) on \(\Lambda^p(w)\) can be seen to be equivalent to the boundedness of the Hardy operator \(A\) restricted to decreasing functions of \(L^p(w)\), since the nonincreasing rearrangement of \(Mf\) is pointwise equivalent to \(Af^*\). The class of weights satisfying this boundedness is known as \(B_p\).

Even though the \(A_p\) and \(B_p\) classes enjoy some similar features, they come from very different theories, and so are the techniques used on each case: Calderón–Zygmund decompositions and covering lemmas for \(A_p\), rearrangement invariant properties and positive integral operators for \(B_p\).

This work aims to give a unified version of these two theories. Contrary to what one could expect, the solution is not given in terms of the limiting cases above considered (i.e., \(u=1\) and \(w=1\)), but in a rather more complicated condition, which reflects the difficulty of estimating the distribution function of the Hardy-Littlewood maximal operator with respect to general measures.