The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces

Abstract

We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.

arXiv admin note: text overlap with arXiv:1310.8101

References in corpus (2)

• The weak Cartan property for the p-fine topology on metric spaces
• Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets

Cited by in corpus (5)

• Existence and almost uniqueness for $p$-harmonic Green functions on bounded domains in metric spaces
• The Dirichlet problem for p-minimizers on finely open sets in metric spaces
• Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces
• Convergence and local-to-global results for $p$-superminimizers on quasiopen sets
• The Perron method associated with finely $p$-harmonic functions on finely open sets