Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets

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.