Hajlasz Gradients Are Upper Gradients

TL;DR

This paper proves that Hajlasz gradients are upper gradients in metric measure spaces, removing previous assumptions and establishing new embeddings between related function spaces.

Contribution

It demonstrates that Hajlasz gradients can serve as upper gradients without quasi-continuity assumptions, and introduces local Hajlasz gradients with their properties.

Findings

Hajlasz gradients are shown to be upper gradients under broad conditions.

The paper establishes embeddings between Morrey-type spaces based on Hajlasz and upper gradients.

Introduces and investigates local Hajlasz gradients and their relation to 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.