The maximal function of the Devil's staircase is absolutely continuous

TL;DR

This paper proves that the maximal function of the Devil's staircase, associated with certain fractal measures, is absolutely continuous, extending the regularity results to multiparameter cases and improving understanding of the classical maximal operator.

Contribution

The authors establish absolute continuity of the maximal function for functions derived from $d$-Ahlfors regular measures, including multiparameter extensions, advancing regularity theory for maximal operators.

Findings

Maximal functions of the Devil's staircase are absolutely continuous.

Extension of absolute continuity results to multiparameter functions.

First improvement of regularity for the classical centered maximal operator.

Abstract

We study the problem of whether the centered Hardy--Littlewood maximal function of a singular function is absolutely continuous. For a parameter $d \in (0,1)$ and a closed set $E\subset [0,1]$, let $\mu$ be a $d$-Ahlfors regular measure associated with $E$. We prove that for the cumulative distribution function $f(x)=\mu([0,x])$ its maximal function $Mf$ is absolutely continuous. We then adapt our method to the multiparameter case and show that the same is true in the positive cone defined by these functions, i.e., for functions of the form $f(x)=\sum_{i=1}^{n}\mu_i([0,x])$ where $\{\mu_i\}_{i=1}^{n}$ is any collection of $d_i$-Ahlfors regular measures, $d_i \in (0,1)$, associated with closed sets $E_i\subset [0,1]$. This provides the first improvement of regularity for the classical centered maximal operator, and can be seen as a partial analogue of the result of Aldaz and P\'erez L\'azaro about the uncentered maximal operator.