Square functions and uniform rectifiability

TL;DR

This paper characterizes uniform rectifiability of Ahlfors-David measures in Euclidean space using square function estimates, providing both necessary and sufficient conditions and alternative characterizations involving smoother square functions.

Contribution

It establishes a new equivalence between uniform rectifiability and square function bounds, extending previous characterizations with smoother variants.

Findings

Characterization of uniform rectifiability via square functions.

Equivalence between rectifiability and integral square function bounds.

Alternative characterizations using smoother square functions.

Abstract

In this paper it is shown that an Ahlfors-David $n$-dimensional measure $\mu$ on $\mathbb{R}^d$ is uniformly $n$-rectifiable if and only if for any ball $B(x_0,R)$ centered at $\operatorname{supp}(\mu)$, $$ \int_0^R \int_{x\in B(x_0,R)} \left|\frac{\mu(B(x,r))}{r^n} - \frac{\mu(B(x,2r))}{(2r)^n} \right|^2\,d\mu(x)\,\frac{dr}r \leq c\, R^n.$$ Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.