{"paper":{"title":"Families of functionals representing Sobolev norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.FA","authors_text":"Andreas Seeger, Haim Brezis, Jean Van Schaftingen, Po-Lam Yung","submitted_at":"2021-09-07T08:31:44Z","abstract_excerpt":"We obtain new characterizations of the Sobolev spaces $\\dot W^{1,p}(\\mathbb{R}^N)$ and the bounded variation space $\\dot{BV}(\\mathbb{R}^N)$. The characterizations are in terms of the functionals $\\nu_{\\gamma} (E_{\\lambda,\\gamma/p}[u])$ where \\[ E_{\\lambda,\\gamma/p}[u]= \\Big\\{(x,y )\\in \\mathbb{R}^N \\times \\mathbb{R}^N \\colon x \\neq y, \\, \\frac{|u(x)-u(y)|}{|x-y|^{1+\\gamma/p}}>\\lambda\\Big\\} \\] and the measure $\\nu_{\\gamma}$ is given by $\\mathrm{d} \\nu_\\gamma(x,y)=|x-y|^{\\gamma-N} \\mathrm{d} x \\mathrm{d} y$. We provide characterizations which involve the $L^{p,\\infty}$-quasi-norms $\\sup_{\\lambda>"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.02930","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.02930/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}