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

Key Points

Maximal operator performance is characterized based on two specific weight conditions, ensuring efficacy.
Establishment of necessary and sufficient conditions for weights in the context of weak-type inequalities occurs.
Characterization relates to the Hardy-Littlewood maximal operator, focusing on weights within specific limits.
Implications solidify understanding of weight necessities, indicating foundational aspects for further exploration.

Abstract

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 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 A p 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 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.

Mark Helpful

Bookmark

Relay