# Poincar\'e inequalities and Newtonian Sobolev functions on noncomplete metric spaces

## TL;DR

This paper investigates how Newtonian Sobolev functions extend from noncomplete metric spaces to their completions, leading to new insights on function regularity, capacity, and Poincaré inequalities in such spaces.

## Contribution

It introduces methods for extending Sobolev functions to completions of noncomplete spaces and explores their implications for analysis on these spaces.

## Key findings

- Extensions of Sobolev functions to completions are possible under certain conditions.
- Results on minimal weak upper gradients and Lebesgue points are established.
- The study discusses applications to p-harmonic functions and highlights delicate issues in noncomplete spaces.

## Abstract

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincar\'e inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ 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 Poincar\'e 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.