Weighted Inequalities For Maximal Operators

In this paper, we study and get Ap type conditions which are sufficient for the two-weight strong inequalities and several results for a maximal operator N on (0,∞). We also get in this paper some Sharp weighted inequalities for the vector-valued maximal function Nq.For a maximal operator N on (0, ∞) we get Ap type conditions which are sufficient for the two-weight strong inequalitiesFor the vector-valued maximal function Nq we also get in this paper some Sharp weighted inequalities. Let 1<q<p<∞.(1) There exists a constant C>0 such that for all locally integrable functions f, ω≥ 0.(2) Part (1) is Sharp since there exists no constant C> 0 such that for all locally integrable functions f, ω≥ 0.