{"paper":{"title":"Planar $W^{1,\\,1}$-extension domains","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Pekka Koskela, Tapio Rajala, Yi Ru-Ya Zhang","submitted_at":"2025-12-09T22:28:21Z","abstract_excerpt":"We show that a bounded planar simply connected domain $\\Omega$ is a $W^{1,\\,1}$-extension domain if and only if for every pair $x,y$ of points in $\\Omega^c$ there exists a curve $\\gamma \\subset \\Omega^c$ connecting $x$ and $y$ with $$ \\int_\\gamma \\frac{1}{\\chi_{\\mathbb R^2\\setminus \\partial\\Omega}(z)}\\,ds(z) \\le C|x-y|.$$ Consequently, a planar Jordan domain $\\Omega$ is a $W^{1,\\,1}$-extension domain if and only if it is a $BV$-extension domain, and if and only if its complementary domain $\\tilde \\Omega$ is a $W^{1,\\,\\infty}$-extension domain."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.09167","kind":"arxiv","version":2},"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/2512.09167/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"}