Discrete Reifenberg-type theorem

TL;DR

This paper establishes a link between the averaged Jones' square function of a measure and an upper bound on the measure itself, generalizing previous results related to harmonic maps.

Contribution

It introduces a generalized Reifenberg-type theorem connecting Jones' square function bounds to measure bounds under various assumptions.

Findings

Bound on Jones' square function implies measure upper bound

Generalizes Naber and Valtorta's result on harmonic map singular sets

Applicable under multiple measure assumptions

Abstract

The paper proves that a bound on the averaged Jones' square function of a measure implies an upper bound on the measure. Various types of assumptions on the measure are considered. The theorem is a generalization of a result due to A. Naber and D. Valtorta in connection with measure bounds on the singular set of harmonic maps.