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

We study regularity of the centered Hardy--Littlewood maximal function $M f$ of a function f of bounded variation in Rᵈ, d∈ N. In particular, we show that at |Dᶜ f|-a.e. point x where f has a non-concave blow-up, it holds that M f(x)>f^*(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 $M f$ can be upgraded to Sobolev regularity.