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-Poincaré 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 Kilpeläinen and Malý in 1992. As a by-product, we establish the quasi-Lindelöf principle of the fine topology in metric spaces and study several variants of local Newtonian and Dirichlet spaces on quasiopen sets.

arXiv admin note: text overlap with arXiv:1410.5167

References in corpus (1)

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

Cited by in corpus (7)

• Local and semilocal Poincaré inequalities on metric spaces
• The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces
• Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces
• Quasiopen and p-path open sets, and characterizations of quasicontinuity
• 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