Square functions and uniform rectifiability

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

No reviews yet.

No comments yet.