On weighted inequalities for B -maximal operators

Let 1 < p < ∞ and w, v be two non-negative weight functions. In this paper, we provide a sufficient condition on w, v for the new maximal operator associated with the generalized translate operator Ty for each to be bounded from Lp,γ (v) to Lp,γ (w). Our condition is stronger, but closely related, to Muckenhoupt’s condition for two weights.