Square functions and uniform rectifiability

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.