On Newton-Sobolev spaces

TL;DR

This paper extends Newton-Sobolev spaces to metric spaces with non-rectifiable paths, establishing foundational properties and relations to other Sobolev spaces, and explores applications with weighted arc-lengths.

Contribution

It generalizes Newton-Sobolev spaces to settings with non-rectifiable paths, proving their Banach structure, absolute continuity, and establishing connections to Hajlasz spaces.

Findings

Proves Newton-Sobolev spaces are Banach spaces.

Establishes absolute continuity of Sobolev functions over curves.

Demonstrates Sobolev embedding theorems in generalized settings.

Abstract

Newton-Sobolev spaces, as presented by N. Shanmugalingam, describe a way to extend Sobolev spaces to the metric setting via upper gradients, for metric spaces with `sufficient' paths of finite length. Sometimes, as is the case of parabolic metrics, most curves are non-rectifiable. As a course of action to overcome this problem, we generalize some of these results to spaces where paths are not necessarily measured by arc length. In particular, we prove the Banach character of the space and the absolute continuity of these Sobolev functions over curves. Under the assumption of a Poincar\'e-type inequality and an arc-chord property here defined, we obtain the density of some Lipschitz classes, relate Newton-Sobolev spaces to those defined by Hajlasz by means of Hajlasz gradients, and we also get some Sobolev embedding theorems. Finally, we illustrate some non-standard settings where these conditions hold, specifically by adding a weight to arc-length and specifying some conditions over it.