We show that a function $ f $ of bounded variation satisfies $$ \Var Mf \leq C \Var f $$ where $ Mf $ is the centered Hardy-Littlewood maximal function of $ f $. Consequently, the operator $ f \mapsto (Mf)' $ is bounded from $ W^{ 1,1}(R) $ to $ L^{ 1}(R) $. This answers a question of Hajlasz and Onninen in the one-dimensional case.
Categories: math.CA