On the variation of the Hardy-Littlewood maximal function

TL;DR

This paper proves that the centered Hardy-Littlewood maximal function preserves bounded variation in one dimension, establishing a boundedness result for its derivative operator from Sobolev space to Lebesgue space.

Contribution

It demonstrates that for functions of bounded variation, the maximal function's variation is controlled, answering a specific open question in the one-dimensional setting.

Findings

Bounded variation of functions is preserved under the maximal operator.

The derivative operator associated with the maximal function is bounded from W^{1,1} to L^1.

Provides a positive answer to a previously open question in the field.

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.