Hajl asz gradients are upper gradients

JIANG RENJIN; NAGESWARI SHANMUGALINGAM; YANG DACHUN; YUAN WEN

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Hajl asz gradients are upper gradients

JIANG RENJIN 1, NAGESWARI SHANMUGALINGAM 2,YANG DACHUN 1, YUAN WEN 1

1. School of Mathematical Sciences, Beijing Normal University,Laboratory of Mathematics and Complex Systems, Ministry ofEducation, Beijing 100875, People's Republic of China

2. Department of Mathematical Sciences,P.O.Box 210025, University of Cincinnati,Cincinnati, OH 45221--0025, U.S.A.)

*Correspondence author

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Subject:

Funding: the Specialized Research Fund for the Doctoral Program of Higher Educationof China (No. 20120003110003) and the Fundamental Research Funds（No.2012LYB26）, Nageswari Shanmugalingam is partially supported bygrant \\#DMS-1200915 from the National Science Foundation （No.U.S.A.）

Opened online:15 November 2013

Accepted by: none

Citation: JIANG RENJIN, NAGESWARI SHANMUGALINGAM,YANG DACHUN.Hajl asz gradients are upper gradients[OL]. [15 November 2013] http://en.paper.edu.cn/en_releasepaper/content/4568999

Let $(X, d, mu)$ be a metric measure space, with $mu$ a Borel regular measure.In this paper, we prove that, if $uin L^1_loc(X)$ and $g$ is aHajl asz gradient of $u$, then there exists $widetilde u$ such that$widetilde u=u$almost everywhere and $4g$ is a $p$-weakupper gradient of $widetilde u$. This result avoids a priori assumptionon 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 Hajl asz-gradientsinto the corresponding function spaces based on upper gradients is obtained. We also introducethe notion of local Hajl asz gradient, and investigate the relations between local Hajl asz gradientand upper gradient.

Keywords:Hajl asz gradient; upper gradient; metric measure space; Sobolev space

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