Abstract Hardy-Littlewood Maximal Inequality

The main aim of this note is to unify some concepts and technics in various generalizations of maximal function theory. We consider two abstract versions of the Vitali covering lemma and introduce and study an abstract Hardy-Littlewood maximal inequality that generalizes the weak type (1, 1) maximal function inequality. Our abstract inequality can be stated for any outer measure on an arbitrary set with a class of subsets. It turns out that 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 the upper bounds associated with 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 Calderon-Zygmund sets in (ax+b)-group are considered. Also, as a very simple application, we find some nontrivial estimates for mass density of classical mechanical systems in Euclidean space.