Variation of the uncentered maximal characteristic Function

Abstract

Let $\mathcal{M}$ be the uncentered Hardy-Littlewood maximal operator or the dyadic maximal operator and $d\ge 1$. We prove that for a set $E\subset {\mathbb{R}}^d$ of finite perimeter the bound $\mathrm{var}\mathcal{M}{1}_E\le C_d\mathrm{var}{1}_E$ holds. We also prove this for the local maximal operator.

arXiv commentsignificantly shortened the proof for the optimal rate in lambda thanks to the anonymous referee