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

TL;DR

This paper investigates prime end boundaries in metric measure spaces, establishing capacity zero for inaccessible prime ends, proving resolutivity of certain boundary functions, and demonstrating the Kellogg property in this setting.

Contribution

It introduces classifications of prime ends, proves capacity and resolutivity results, and extends the Kellogg property to prime end boundaries in metric spaces.

Findings

Prime end capacity of inaccessible prime ends is zero.

Continuous Lipschitz functions on accessible prime ends are resolutive.

Bounded perturbations in inaccessible prime ends do not change the Perron solution.

Abstract

Prime end boundaries $\partial_P\Omega$ of domains $\Omega$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincar\'e 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 $\partial_P\Omega$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}\Omega$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_P\Omega\setminus\partial_{SP}\Omega$ 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.