Hajlasz Gradients Are Upper Gradients

Abstract

Let $(X, d, \mu)$ be a metric measure space, with $\mu$ a Borel regular measure. In this paper, we prove that, if $u\in L^1_{{\mathop\mathrm{\,loc\,}}}(X)$ and $g$ is a Haj{\l}asz 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 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.