Hajlasz Gradients Are Upper Gradients

Let $(X,d,\mu )$ be a metric measure space, with $\mu$ a Borel regular measure. In this paper, we prove that, if $u\in L^1_{{\mathop$\mathrm{l}\mathrm{o}\mathrm{c}$}}(X)$and$g$isaHaj\text{l}aszgradientof$u$, then there exists $\tilde{u}$ such that $\tilde{u}=u$ almost everywhere and $4g$ is a $p$-weak upper gradient of $\tilde{u}$. 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 Haj{\l}asz-gradients into the corresponding function spaces based on upper gradients is obtained. We also introduce the notion of local Haj{\l}asz gradient, and investigate the relations between local Haj{\l}asz gradient and upper gradient.