Abstract Hardy-Littlewood Maximal Inequality

TL;DR

This paper introduces an abstract version of the Hardy-Littlewood Maximal Inequality applicable to any outer measure and family of sets, providing conditions for its finiteness and applications in various geometric contexts.

Contribution

It generalizes the classical maximal inequality to an abstract setting and offers new criteria and bounds for the associated maximal constant across different spaces.

Findings

Finiteness of the Hardy-Littlewood maximal constant is characterized.

Upper bounds are established for various geometric families.

Application to mass density estimates in Classical Mechanics.

Abstract

In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1) maximal function inequality, associated to any outer measure and a family of subsets on a set is introduced. The inequality is (effectively) satisfied if and only if a special numerical constant called Hardy-Littelwood maximal constant is finite. Two general sufficient conditions for the finiteness of this constant are given and upper bounds for this constant associated to the family of (centered) balls in homogeneous spaces, family of dyadic cubes in Euclidean spaces, family of admissible trapezoids in homogeneous trees, and family of Calder\'{o}n-Zygmund sets in (ax+b)-group, are considered. Also a very simple application to find some nontrivial estimates about mass density in Classical Mechanics is given.