A remark on the Hardy-Littlewood maximal functions

TL;DR

This paper compares centered and non-centered Hardy-Littlewood maximal functions, establishing conditions under which they coincide or differ, especially in ultrametric spaces and Riemannian manifolds, using discretization techniques.

Contribution

It provides new characterizations of metric spaces based on the equality of maximal functions and explores their differences in Riemannian geometry contexts.

Findings

Space is ultrametric iff the two maximal operators are identical for all discrete measures.

The uncentered maximal operator exceeds the centered one on Riemannian manifolds.

Discretization technique is used to analyze maximal functions in different spaces.

Abstract

We investigate the magnitude relation of the non-centered Hardy-Littlewood maximal operators and centered one. By using a discretization technique, we prove two facts: the first one is that the space is ultrametric if and only if the two maximal operators are identical for all discrete measure; the second is, the uncentred maximal operator is strictly greater than the centered one if $(M,d_g)$ is a Riemannian manifold and $\mu$ is the Riemannian volume measure.