Hajłasz gradients are upper gradients

Let (X,d,μ) be a metric measure space, with μ a Borel regular measure. In this article, we prove that, if u∈Lloc1(X) and g is a Hajłasz gradient of u, then there exists u˜ such that u˜=u almost everywhere and 4g is a p-weak upper gradient of u˜. This result avoids a priori assumption on the quasi-continuity of u used in Shanmugalingam (2000) [19]. We also introduce the notion of local Hajłasz gradients, and investigate the relations between the local Hajłasz gradient and the upper gradient.