Variation of the uncentered maximal characteristic function

Abstract

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