# Dichotomy property for maximal operators in non-doubling setting

## TL;DR

This paper explores the dichotomy property of Hardy--Littlewood maximal operators in non-doubling metric measure spaces, providing a full spectrum analysis and characterizations in Euclidean spaces with arbitrary norms.

## Contribution

It offers a comprehensive analysis of the dichotomy property for maximal operators in non-doubling spaces and characterizes when the centered operator has this property in Euclidean spaces.

## Key findings

- Full spectrum of cases for the dichotomy property in non-doubling spaces
- Characterization of when $M^c$ has the property in Euclidean spaces with arbitrary norms
- Conditions on measure $mbda$ for the dichotomy property in $b R^d$

## Abstract

We investigate a dichotomy property for Hardy--Littlewood maximal operators, non-centered $M$ and centered $M^c$, that was noticed by Bennett, DeVore and Sharpley. We illustrate the full spectrum of possible cases related to the occurrence or not of this property for $M$ and $M^c$ in the context of non-doubling metric measure spaces $(X, \rho, \mu)$. In addition, if $X = \mathbb{R}^d$, $d \geq 1$, and $\rho$ is the metric induced by an arbitrary norm on $\mathbb{R}^d$, then we give the exact characterization (in terms of $\mu$) of situations in which $M^c$ possesses the dichotomy property provided that $\mu$ satisfies some very mild assumptions.