Square functions and uniform rectifiability

Chousionis, Vasileios; Garnett, John; Le, Triet; Tolsa, Xavier

doi:10.1090/tran/6557

Square functions and uniform rectifiability
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by Vasileios Chousionis, John Garnett, Triet Le and Xavier Tolsa PDF

Trans. Amer. Math. Soc. 368 (2016), 6063-6102 Request permission

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.

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