The centered maximal operator removes the non-concave Cantor part from the gradient

We study regularity of the centered Hardy–Littlewood maximal function $Mf$ of a function $f$ of bounded variation in ${\mathbb{R}}^d$, $d\in \mathbb{N}$. In particular, we show that at $\mathrm{\mid }{D}^{c}f\mathrm{\mid }$-a.e. point $x$ where $f$ has a non-concave blow-up, it holds that $Mf(x)>f^{\ast }(x)$. We further deduce from this that if the variation measure of $f$ has no jump part and its Cantor part has non-concave blow-ups, then BV regularity of $Mf$ can be upgraded to Sobolev regularity.