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 $\mathrm{supp}(\mu )$, $${\int }_0^R{\int }_{x\in B(x_0,R)}{\left|\frac{\mu (B(x,r))}{r^n}-\frac{\mu (B(x,2r))}{(2r{)}^n}\right|}^{2}d\mu (x)\frac{dr}{r}\le c{R}^n.$$ Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.