# Variation of the uncentered maximal characteristic function

### Julian Weigt

Aalto University, Finland

## Abstract

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

## Cite this article

Julian Weigt, Variation of the uncentered maximal characteristic function. Rev. Mat. Iberoam. 38 (2022), no. 3, pp. 823–849

DOI 10.4171/RMI/1312