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.