Prime ends for domains in metric spaces

Abstract

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carath\'eodory and N\"akki. Modulus ends and prime ends, defined by means of the \p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.Comment: 46 pages, 4 figure

Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular spaces

“…Under the assumption of finite connectedness at the boundary, it can be verified that ∂ M Ω equals ∂ M Ω, together with all the copies of ∞ from different directions. In the terminology of Adamowicz-Björn-Björn-Shanmugalingam [1] and Estep [27], the sequence {Ω k } ∞ k=1 can be identified with a so-called prime end at ∞, see [1, Theorem 9.6 and Corollary 10.9].…”

The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property

“…The first theory of prime ends is due to Carathéodory, who formulated a definition of prime ends from the point of view of conformal mappings in simply-connected planar domains. Subsequently, the theory has extended to more general domains in the plane and in higher dimensional Euclidean spaces, see for instance the works of Freudenthal, Kaufman, Mazurkiewicz, and more recently Epstein and Näkki (for further discussion of the history of prime ends and the literature, we refer to [1,Sections 1,3], see also [2] for an application of the prime end theory in the setting of Heisenberg groups). Here we study prime ends in the more general setting of metric spaces, see [1,12,13].…”

“…Subsequently, the theory has extended to more general domains in the plane and in higher dimensional Euclidean spaces, see for instance the works of Freudenthal, Kaufman, Mazurkiewicz, and more recently Epstein and Näkki (for further discussion of the history of prime ends and the literature, we refer to [1,Sections 1,3], see also [2] for an application of the prime end theory in the setting of Heisenberg groups). Here we study prime ends in the more general setting of metric spaces, see [1,12,13]. The notion of prime ends considered here is from [13], and is a slight modification from that of [1].…”

“…">Notation and preliminariesIn this section we provide descriptions of the basic notions used in the paper. We recommend interested readers to look to the books [6,20] and the papers [1,13,7,4,5] for more information pertaining to these notions.…”

Preservation of bounded geometry under sphericalization and flattening: quasiconvexity and ∞-Poincaré inequality

“…The paper [20] used the sphericalization and flattening techniques to transform bounded uniform domains into unbounded uniform domains, and hence succeeded in extending the results of [5] to bounded uniform domains. As another example, recently the paper [1] proposed a notion of prime end boundary for bounded domains in the metric setting, and such a prime end boundary was the principal focus of the study of Dirichlet problems in the metric setting in [14]. However, the results in [14] needed the domain to be bounded.…”