Square functions and uniform rectifiability (Q2790723)

A Radon measure \(\mu\) in \(\mathbb{R}^d\) is said to be \(n\)-dimensional Ahlfors-David regular with constant \(c_0\) or, for short, \(n\)-AD-regular,if NEWLINE\[NEWLINEc_0^{-1}r^n\leq\mu(B(x,r))\leq c_0r^nNEWLINE\]NEWLINE for all \(x\in\mathrm{supp}(\mu)\) and \(0<r\leq \mathrm{diam}(\mathrm{supp}(\mu))\). An \(n\)-AD-regular measure \(\mu\) is said to be uniformly \(n\)-rectifiable if there exist \(\theta\), \(M>0\) such that, for all \(x\in\mathrm{supp}(\mu)\) and all \(r>0\), there exists a Lipschitz mapping \(\rho\) from the ball \(B_n(0,r)\) in \(\mathbb{R}^n\) to \(\mathbb{R}^d\) with \(\mathrm{Lip}(\rho)\leq M\) such that NEWLINE\[NEWLINE\mu(B(x,r)\cap\rho(B_n(0,r)))\geq\theta r^n.NEWLINE\]NEWLINE In this paper, the authors first show that, if \(\mu\) is an \(n\)-AD-regular measure, then \(\mu\) is uniformly \(n\)-rectifiable if and only if there exists a positive constant \(c\) such that, for any ball \(B(x_0,R)\) centered at \(x_0\in \mathrm{supp}(\mu)\), NEWLINE\[NEWLINE\int_0^R\int_{x\in B(x_0,R)}\left|\frac{\mu(B(x,r))}{r^n}-\frac{\mu(B(x,2r))} {(2r)^n}\right|^2\,d\mu(x)\frac{dr}{r}\leq cR^n.NEWLINE\]NEWLINE Also, the authors obtain characterizations of uniform \(n\)-rectifiable \(n\)-AD-regular measures in terms of smoother square functions NEWLINE\[NEWLINE\Delta_{\mu,\varphi}(x,t):=\int [\varphi_t(y-x)-\varphi_{2t}(y-x)]\,d\mu(y)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\widetilde{\Delta}_{\mu,\varphi}(x,t):=\int \partial_\varphi(y-x,t)\,d\mu(y),NEWLINE\]NEWLINE where \(\varphi\) is a Borel function from \(\mathbb{R}^d\) to \(\mathbb{R}\), \(\varphi_t(x):=\frac1{t^n}\varphi(\frac x t)\) and \(\partial_\varphi(x,t):=t\partial_t\varphi_t(x)\) for all \(x\in \mathbb{R}^d\) and \(t>0\).NEWLINENEWLINEMoreover, by arguments analogous to the above, another equivalent square function condition for uniform rectifiability is also obtained.