A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets

TL;DR

This paper establishes a characterization of p-quasiconnected sets via the constancy of Sobolev functions with zero fine gradient, extending the result to metric measure spaces and p-finely open sets.

Contribution

It provides a new equivalence between p-quasiconnectedness and the constancy of Sobolev functions with zero fine gradient, generalizing previous results to metric spaces.

Findings

Sobolev functions with zero p-fine gradient are a.e.-constant on p-quasiconnected sets.

The equivalence extends to metric measure spaces with doubling measure and Poincaré inequality.

Results apply to p-finely open sets in Euclidean space using Latvala's theorem.

Abstract

We show that every Sobolev function in $W^{1,p}_{\textrm{loc}}(U)$ on a $p$-quasiopen set $U \subset {\bf R}^n$ with a.e.-vanishing $p$-fine gradient is a.e.-constant if and only if $U$ is $p$-quasiconnected. To prove this we use the theory of Newtonian Sobolev spaces on metric measure spaces, and obtain the corresponding equivalence also for complete metric spaces equipped with a doubling measure supporting a $p$-Poincar\'e inequality. On unweighted ${\bf R}^n$, we also obtain the corresponding result for $p$-finely open sets in terms of $p$-fine connectedness, using a deep result by Latvala.