The Maximal Function of the Devil’s Staircase is Absolutely Continuous

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 Pérez Lázaro about the uncentered maximal operator.