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

摘要

We prove that if f : I subset of R -> R is of bounded variation, then the uncentered maximal function Mf is absolutely continuous, and its derivative satisfies the sharp inequality vertical bar vertical bar DMf vertical bar vertical bar (L1(I)) <= vertical bar Df vertical bar(I). This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.