On those Weights Satisfying a Weak-Type Inequality for the Maximal Operator and Fractional Maximal Operator

Abstract

Muckenhoupt and Wheeden previously formulated a weighted weak (p,p) inequality where the weight for the weak Lp 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 Ap, 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 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.

Brandon Sweeting

NSF MPS-Ascend Postdoctoral Research Fellow, Mathematics

My current research interests are in harmonic analysis, specifically weighted norm inequalities for singular integral operators and Riesz potentials in both the scalar and matrix setting.