On the variation of the Hardy-Littlewood maximal function

Abstract

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 $f\mapsto (Mf{)}^{\prime }$ is bounded from $W^{1,1}(R)$ to $L^1(R)$. This answers a question of Hajlasz and Onninen in the one-dimensional case.

25 pages

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