不可达素端的素端容量、可分辨性和Kellogg性质
The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property
摘要：
在支持$p$ -Poincaré不等式的完备加倍度量测度空间中，研究整环$Ω$的素端边界$\partial_PΩ$.引入了可矫正（不）可接近和（不）可矫正远距离素端的概念，并将其用于素端的分类。我们证明了，对于一个给定的域，素数端容量的集合的所有rectifiably不可访问的素数端在一起将所有非singleton素数端是零。本文证明了$\partial_PΩ$上的连续函数的分解性，这些连续函数是关于Mazurkiewicz度量的Lipschitz连续函数，当它们被限制在所有可达素数端点的集合$\partial_{SP}Ω$上时.此外，在$\partial_PΩ\setminus\partial_{SP}Ω$中，这些函数的有界扰动产生相同的Perron解。在文章的最后一部分，我们证明了度量空间中有界域的素端边界的（分解）Kellogg性质。文中给出的一些概念都用实例加以说明。
英文摘要：
Prime end boundaries $\partial_PΩ$ of domains $Ω$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincaré inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_PΩ$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}Ω$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_PΩ\setminus\partial_{SP}Ω$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples.
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