Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Abstract

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincaré inequality. We study extensions of Newtonian Sobolev functions to the completion $\hat{X}$ of $X$ and use them to obtain several results on $X$ itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincaré inequalities. We also provide a discussion about possible applications of the completions and extension results to $p$-harmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations.

Second version: with a correction at the end (last three pages). The main paper is identical to the first version

References in corpus (3)

• Sharp capacity estimates for annuli in weighted R^n and in metric spaces
• Quasiopen and p-path open sets, and characterizations of quasicontinuity
• Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets

Cited by in corpus (9)

• The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy
• Locally $p$-admissible measures on $\mathbb{R}$
• Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets
• Classification of metric measure spaces and their ends using $p$-harmonic functions
• Bounded geometry and $p$-harmonic functions under uniformization and hyperbolization
• Poincaré inequalities and $A_p$ weights on bow-ties
• Absolutely continuous mappings on doubling metric measure spaces
• Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted $L^q$ spaces on metric spaces
• Preserving Besov (fractional Sobolev) energies under sphericalization and flattening