Weighted norm inequalities for maximal functions from the Muckenhoupt conditions (Q1344458)

A generalized maximal operator \[ (M_\phi f) (x) = \sup\{\phi(Q) |Q|^{-1} \int_Q |f(y)|dy\} \] is considered. Here the supremum is taken with respect to all cubes \(Q \ni x\) and \(\phi\) is a map defined on the set of cubes, taking its values in \((0,\infty)\) and satisfying certain conditions. Necessary conditions and sufficient conditions, which are close to each other, for the validity of the weighted inequality \[ |M_\phi f|_{L^q_u} \leq C|f|_{L^p_v} \] are established for the case in which \(1 < q < p < \infty\).