Functions of bounded variation, the derivative of the one dimensional   maximal function, and applications to inequalities
  We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain, under less regularity, versions of classical inequalities involving derivatives. 
J. M. Aldaz and J. P\'erez L\'azaro