Inequalities for Maximal Operators Associated with a Family of General Sets

Article Type

Research Article

Publication Title

Results in Mathematics

Abstract

Let E={Er(x):r>0,x∈X} be a family of open subsets of a topological space X equipped with a nonnegative Borel measure μ satisfying some basic properties. We establish sharp quantitative weighted norm inequalities for the Hardy–Littlewood maximal operator ME associated with E in terms of mixed Ap–A∞ constants. The main ingredient to prove this result is a sharp form of a weak reverse Hölder inequality for the A∞,E weights. As an application of this inequality, we also provide a quantitative version of the open property for Ap,E weights. Next, we prove a covering lemma in this setting and using this lemma establish the endpoint Fefferman–Stein weighted inequality for the maximal operator ME. Moreover, vector-valued extensions for maximal inequalities are also obtained in this context.

DOI

10.1007/s00025-024-02224-1

Publication Date

8-1-2024

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