论文标题
在高复合条件中达尔伯格定理的逆向障碍
On an obstacle to the converse of Dahlberg's theorem in high codimensions
论文作者
论文摘要
最近已经了解到，边界上的谐波度量$ e = \ partialω$ in $ \ mathbb {r}^n $与hausdorff量$ \ mathcal {h}^h}^{h}^{n -1} $ on $ e $ in $ e $ IF和边界$ e $ e $ e $ e $ e $ e $ e $ ectiffiffiffiffiffiffiffiffiffiffififift的$ \ \ mathbb {r}^n $相对于Hausdorff量值$ \ Mathcal {h}^n -1} $是绝对连续的。然后，由G. David，M。Engelstein，J。Feneuil，S。Mayboroda和其他合着者，对Ahlfors -groumar -e $ e $ E $的谐波度量的概念借助运营商$ $l_α= - $d_α$是集合$ e $的某些正规距离函数。启动了一项计划，以建立类似于经典案例的等效性，而较高的量子集合$ e $与（新的）谐波和Hausdorff措施的良好关系之间的良好关系。仅获得了对定量绝对连续性的可重新讨论能力的充分性。对于另一个方向，主要障碍是证明，只有在集合$ e $是超平面时，方程式$l_αd_α= 0 $才是正确的。在本文中，我们证明了一些第一个结果，这些结果表明后者的猜想可能是正确的。我们还解释说，解决该问题的某种自然策略直到最后才起作用。
It has been recently understood that the harmonic measure on the boundary $E = \partial Ω$ of a domain $Ω$ in $\mathbb{R}^n$ is absolutely continuous with respect to the Hausdorff measure $\mathcal{H}^{n - 1}$ on $E$ if and only if the boundary $E$ is rectifiable. Then, by G. David, M. Engelstein, J. Feneuil, S. Mayboroda and other coauthors, a notion of harmonic measure for Ahlfors-regular sets $E$ of higher codimension $n - d$ was developed with the aid of the operator $L_α= -\mbox{div} D_α^{-n + d + α} \nabla$, where $α> 0$ and $D_α$ is a certain regularized distance function to the set $E$. A program was launched to establish analogous to the classical case equivalence between rectifiability of the higher-codimensional set $E$ and good relations of the (new) harmonic and Hausdorff measures. The sufficiency of rectifiability for quantitative absolute continuity was only just obtained. For the other direction the main obstacle is to prove that, roughly, the equation $L_αD_α= 0$ is true only when the set $E$ is a hyperplane. In this paper we prove some first results which indicate that the latter conjecture may be true. We also explain that a certain natural strategy to tackle the problem does not work till the end.
