Hajlasz gradients are upper gradients

Let (X, d, mu) be a metric measure space, with mu. a Borel regular measure. In this article, we prove that, if u is an element of L-loc(1)(X) and g is a Hajlasz gradient of u, then there exists (u) over tilde such that (u) over tilde = u almost everywhere and 4g is a p-weak upper gradient of (u) over tilde. This result avoids a priori assumption on the quasi-continuity of u used in Shanmugalingam (2000) [19]. We also introduce the notion of local Hajlasz gradients, and investigate the relations between the local Hajlasz gradient and the upper gradient. (C) 2014 Elsevier Inc. All rights reserved.