IUMJ

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

Authors: Anders Bjorn, Jana Bjorn and Visa Latvala

Issue: Volume 64 (2015), Issue 3, 915-941

Abstract:

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