Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets

TL;DR

This paper explores various definitions of Sobolev spaces on quasiopen sets within metric spaces, establishing their connections to classical Sobolev theory in Euclidean spaces and analyzing properties like quasicontinuity and the fine topology.

Contribution

It demonstrates the equivalence of Newtonian functions and quasicontinuous Sobolev representatives on quasiopen subsets of R^n, and develops the quasi-Lindel"of principle for the fine topology in metric spaces.

Findings

Newtonian functions coincide with quasicontinuous Sobolev representatives in R^n

Established the quasi-Lindel"of principle for the fine topology

Analyzed variants of local Newtonian and Dirichlet spaces

Abstract

We study different definitions of Sobolev spaces on quasiopen sets in a complete metric space equipped with a doubling measure supporting a p-Poincar\'e inequality with 1<p<\infty, and connect them to the Sobolev theory in R^n. In particular, we show that for quasiopen subsets of R^n the Newtonian functions, which are naturally defined in any metric space, coincide with the quasicontinuous representatives of the Sobolev functions studied by Kilpel\"ainen and Mal\'y in 1992. As a by-product, we establish the quasi-Lindel\"of principle of the fine topology in metric spaces and study several variants of local Newtonian and Dirichlet spaces on quasiopen sets.