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

Prime end boundaries partial derivative(P)Omega of domains Omega are studied in the setting of complete doubling metric measure spaces supporting a p-Poincare 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 (defined by Estep and Shanmugalingam in Potential Anal 42:335-363, 2015) of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continuous functions on partial derivative(P)Omega which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection partial derivative(SP)Omega of all accessible prime ends. Furthermore, bounded perturbations of such functions in partial derivative(P)Omega\partial derivative(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.