A Vitali-type lemma for the boundary

TL;DR

This paper introduces a boundary version of the Vitali covering lemma, establishing a relationship between disjoint balls' perimeters and the perimeter of their union, with applications to regularity of maximal functions.

Contribution

It presents a Vitali-type lemma for boundaries, along with volume-perimeter results, counterexamples, and applications to maximal function regularity.

Findings

Disjoint balls' perimeters control the union's perimeter.

Counterexamples illustrate limitations of the results.

Applications include simplified proofs of regularity for maximal functions.

Abstract

Take a set of balls in $\mathbb R^d$. We find a subset of pairwise disjoint balls whose combined perimeter controls the perimeter of the union of the original balls. This can be seen as a boundary version of the Vitali covering lemma. We further prove combined volume-perimeter results and counterexamples and apply them to find short proofs of some regularity statements for maximal functions.