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We study the problem of whether the centered Hardy--Littlewood maximalfunction 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 $d$-Ahlforsregular measure associated with $E$. We prove that for the cumulativedistribution function $f(x)=\mu([0,x])$ its maximal function $Mf$ is absolutelycontinuous. We then adapt our method to the multiparameter case and show thatthe same is true in the positive cone defined by these functions, i.e., forfunctions 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 thefirst improvement of regularity for the classical centered maximal operator,and can be seen as a partial analogue of the result of Aldaz and P\'erezL\'azaro about the uncentered maximal operator.

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