Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities

Abstract

We prove that if $f:I\subset \mathbb{R}\to \mathbb{R}$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\Vert DMf{\Vert }_1\le |Df|(I)$. This allows us obtain, under less regularity, versions of classical inequalities involving derivatives.

To appear, TAMS, 21 pages

References in corpus (1)

• Boundedness and unboundedness results for some maximal operators on functions of bounded variation