Muckenhoupt has given a
necessary and sufficient condition to be satisfied by the weight functions U and V in
order that the Hardy-Littlewood maximal function Mf should satisfy a weighted
weak type (1,1) inequality. In this note, conditions on the weight functions U and V
are given in order that the sense of this inequality may be reversed. This is then
applied to give conditions which ensure that the integrability of Mf with respect to a
weight implies that f belongs to a weighted Zygmund class LlogL, thus extending a
result of Stein. Analogous results related to the strong maximal function and the
classes L(logL)k are also given. These extend certain results of Favo, Gatto and
Gutiérrez.
