Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions
Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions
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来自卡尔森测量估计的均匀可整流性和有界调和函数的 ε 逼近性
作者：
J. Garnett;Mihalis Mourgoglou;X. Tolsa
设$\Omega\subset\mathbb R^{n+1}$，$n\geq1$，是一个具有Ahlfors-David正则边界的螺旋域.在本文中，我们证明了$\partial\Omega$是一致n$-可求长的，如果$\Omega$上的每一个有界调和函数是$\vareps $-可逼近的，或者如果$\Omega$上的每一个有界调和函数满足一个合适的平方函数Carleson测度估计。特别地，这适用于$\Omega=\mathbb R^{n+1}\setminus E$且$E$是Ahlfors-David正则的情况。我们的结果解决了霍夫曼，马特尔和梅博罗达在最近的工作中提出的一个猜想，他们证明了匡威的陈述。在这里，我们还获得了两个额外的标准一致可求直。一个是在所谓的“$S<N$”的估计方面，另一个是在一个合适的电晕分解涉及谐波措施。
Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every bounded harmonic function on $\Omega$ is $\varepsilon$-approximable or if every bounded harmonic function on $\Omega$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $\Omega=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "$S<N$" estimates, and another in terms of a suitable corona decomposition involving harmonic measure.