{"paper":{"title":"Hajlasz Gradients Are Upper Gradients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Dachun Yang, Nageswari Shanmugalingam, Renjin Jiang, Wen Yuan","submitted_at":"2013-07-19T06:02:10Z","abstract_excerpt":"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 o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5134","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}