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Hajlasz Gradients Are Upper Gradients

Let $(X, d, μ)$ be a metric measure space, with $μ$ a Borel regular measure. In this paper, we prove that, if $u\in L^1_{\mathop\mathrm{\,loc\,}}(X)$ and $g$ is a Hajłasz gradient of $u$, then there exists $\widetilde u$ such that $\widetilde u=u$ almost everywhere and $4g$ is a $p$-weak upper gradient of $\widetilde u$. This result avoids a priori assumption on the quasi-continuity of $u$ used in [Rev. Mat. Iberoamericana 16 (2000), 243-279]. As an application, an embedding of the Morrey-type function spaces based on Hajłasz-gradients into the corresponding function spaces based on upper gradients is obtained. We also introduce the notion of local Hajłasz gradient, and investigate the relations between local Hajłasz gradient and upper gradient.