Note about square function estimates and uniformly rectifiable measures

TL;DR

This paper generalizes and provides a new proof for a recent $L^2$ square function estimate on uniformly rectifiable sets, utilizing Tolsa's $eta$-numbers for a more concise argument.

Contribution

It offers a simplified proof of an existing square function estimate on UR sets, expanding the understanding of geometric measure theory techniques.

Findings

New proof of $L^2$ square function estimate on UR sets

Utilizes $eta$-numbers for a concise argument

Enhances understanding of geometric measure theory methods

Abstract

We generalise and offer a different proof of a recent $L^2$ square function estimate on UR sets by Hofmann, Mitrea, Mitrea and Morris. The proof is a short argument using the $\alpha$-numbers of Tolsa.