On those weights satisfying a weak-type inequality for the maximal operator and fractional maximal operator

Muckenhoupt and Wheeden [Indiana Univ. Math. J. 26 (1977), pp. 801–816] formulated a weighted weak ( p , p ) (p,p) inequality where the weight for the weak L p L^p space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood maximal operator and the Hilbert transform for weights in the class A p A_p , while also deriving necessary conditions to characterize the weights for which these estimates hold. In this paper, we establish the sufficiency of these conditions for the maximal operator when p > 1 p > 1 and present corresponding results for the fractional maximal operators. This completes the characterization and resolves the open problem posed by Muckenhoupt and Wheeden for p > 1 p > 1 .