{"paper":{"title":"Haj\\lasz-Sobolev Imbedding and Extension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA"],"primary_cat":"math.CA","authors_text":"Yuan Zhou","submitted_at":"2010-04-29T14:38:16Z","abstract_excerpt":"The author establishes some geometric criteria for a Haj\\lasz-Sobolev $\\dot M^{s,\\,p}_\\ball$-extension (resp. $\\dot M^{s,\\,p}_\\ball$-imbedding) domain of ${\\mathbb R}^n$ with $n\\ge2$,  $s\\in(0,\\,1]$ and $p\\in[n/s,\\,\\infty]$ (resp. $p\\in(n/s,\\,\\infty]$). In particular, the author proves that a bounded finitely connected planar domain $\\boz$ is a weak $\\alpha$-cigar domain with $\\alpha\\in(0,\\,1)$ if and only if $\\dot F^s_{p,\\,\\infty}({\\mathbb R}^2)|_\\boz=\\dot M^{s,\\,p}_\\ball(\\boz)$ for some/all $s\\in[\\alpha,\\,1)$ and $p=(2-\\az)/(s-\\alpha)$, where $\\dot F^s_{p,\\,\\infty}({\\mathbb R}^2)|_\\boz$ deno"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5307","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":""},"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"}