{"paper":{"title":"Failure of zero extension in parabolic Sobolev spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Doyoon Kim, Jongkeun Choi, Kwan Woo","submitted_at":"2026-06-20T14:18:55Z","abstract_excerpt":"We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces $\\mathring{\\mathcal{H}}^1_p((0,T) \\times \\Omega)$, which can also be characterized as $$ L_p(0,T;\\mathring{W}^1_p(\\Omega))\\cap W^1_p(0,T; W^{-1}_{p}(\\Omega)). $$ More precisely, for any $p\\in [1, \\infty)$, we construct a function $u\\in \\mathring{\\mathcal{H}}^1_p((0,T)\\times \\mathbb{R}^d_+)$ whose zero extension does not belong to $\\mathcal{H}^1_p((0,T)\\times \\mathbb{R}^d)$. The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22059","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/2606.22059/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"}