{"paper":{"title":"The Dubovitski\\u{\\i}-Sard Theorem in Sobolev Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Piotr Haj{\\l}asz, Scott Zimmerman","submitted_at":"2015-05-29T20:47:53Z","abstract_excerpt":"The Sard theorem from 1942 requires that a mapping $f:\\mathbb{R}^n \\to \\mathbb{R}^m$ is of class $C^k$, $k > \\max (n-m,0)$. In 1957 Duvovitski\\u{\\i} generalized Sard's theorem to the case of $C^k$ mappings for all $k$. Namely he proved that, for almost all $y\\in \\mathbb{R}^m$, $\\mathcal{H}^{\\ell}(C_f \\cap f^{-1}(y))=0$ where $\\ell = \\max(n-m-k+1,0)$, ${\\mathcal H}^{\\ell}$ denotes the Hausdorff measure, and $C_f$ is the set of critical points of $f$. In 2001 De Pascale proved that the Sard theorem holds true for Sobolev mappings of the class $W_{\\rm loc}^{k,p}(\\mathbb{R}^n,\\mathbb{R}^m)$, $k>\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00025","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":""},"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"}