Fixed points of the uncentered Hardy-Littlewood maximal operator

TL;DR

This paper surveys known and new results on fixed points of the uncentered Hardy-Littlewood maximal operator in metric measure spaces, proving that fixed points are constant under certain conditions.

Contribution

It provides a comprehensive overview and introduces new results on fixed points of the maximal operator in general metric measure spaces, including conditions for constancy.

Findings

Fixed points are constant functions if measure support is connected and satisfies mild continuity.

The paper extends known results to more general metric measure space settings.

Provides new insights into the structure of fixed points of the maximal operator.

Abstract

We give a survey, known and new results on the beingness of fixed points of the maximal operator in the more general settings of metric measure space. In particular, we prove that the fixed points of the uncentered one must be the constant function if the measure satisfies a mild continuity assumption and its support is connected.