Singular sets for harmonic measure on locally flat domains with locally finite surface measure
Singular sets for harmonic measure on locally flat domains with locally finite surface measure
复制标题
具有局部有限表面测量的局部平坦域上谐波测量的奇异集
DOI：
10.1093/imrn/rnw121
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
X. Tolsa
中科院分区：
文献类型：
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作者：
Jonas Azzam;Mihalis Mourgoglou;X. Tolsa
David 和 Jerison 定理断言，在具有 Ahlfors 正则边界的 NTA 域中，调和测度相对于表面测度是绝对连续的。我们证明，如果我们放宽 Ahlfors 正则假设，通过证明对于每个 $d>1$，存在一个 Reifenberg 平坦域 $\Omega\subset \mathbb{R}^{d+1}$ 和 $\mathcal{H}^{d}(\partial\Omega)<\infty$ 以及一个子集 $E\subset \partial \Omega$ 且正谐波测量值为零，则在高维上会失败。 $\mathcal{H}^{d}$-测量。特别是，这意味着 F. 和 M. Riesz 定理的经典定理在此类域的更高维度上失败。
A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $\Omega\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partial\Omega)<\infty$ and a subset $E\subset \partial \Omega$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimensions for this type of domains.