The weak Cartan property for the p-fine topology on metric spaces

Abstract

We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 < p< oo. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and allows us to show that the p-fine topology is the coarsest topology making all p-superharmonic functions continuous. Our p-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients and do not rely on a vector-valued differentiable structure.

References in corpus (1)

• Sharp capacity estimates for annuli in weighted R^n and in metric spaces

Cited by in corpus (7)

• The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces
• Quasiopen and p-path open sets, and characterizations of quasicontinuity
• Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets
• Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets
• The Dirichlet problem for p-minimizers on finely open 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