On Newton-Sobolev spaces

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 ´sucient´ paths of nite length. Sometimes, as is the case of parabolic metrics, most curves are non-rectiable. We generalize some of these results to spaces where paths are not necessarily measured by arc length. Under the assumption of a Poincaré-type inequality and an arc-chord property here dened, we obtain the density of some Lipschitz classes, relate Newton-Sobolev spaces to those dened by Hajªasz, and we also get some Sobolev embedding theorems. Finally, we illustrate some non-standard settings where these conditions hold, specically by adding a weight to arc-length.