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arXiv · 1210.0496

On the variation of the Hardy-Littlewood maximal function

Ondřej Kurka
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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)' $ is bounded from $ W^{1,1}(R) $ to $ L^{1}(R) $. This answers a question of Hajlasz and Onninen in the one-dimensional case.

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Ondřej Kurka. 2012-10-01. On the variation of the Hardy-Littlewood maximal function. https://doi.org/10.5186/aasfm.2015.4003

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