A Vitali-type lemma for the boundary

2025/07/21 by Weigt, Julian
#28A75 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.15333

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.

Citations

• The Variation of the Uncentered Maximal Operator with respect to Cubes
• Variation of the dyadic maximal function
• Variation of the uncentered maximal characteristic Function
• The variation of the maximal function of a radial function

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