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Square functions and uniform rectifiability (1401.3382v1)
Published 14 Jan 2014 in math.CA
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 $\operatorname{supp}(\mu)$, $$ \int_0R \int_{x\in B(x_0,R)} \left|\frac{\mu(B(x,r))}{rn} - \frac{\mu(B(x,2r))}{(2r)n} \right|2\,d\mu(x)\,\frac{dr}r \leq c\, Rn.$$ Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.