{"paper":{"title":"On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anton Lukyanenko, Jeremy Tyson, Noel DeJarnette, Piotr Hajlasz","submitted_at":"2011-09-21T20:02:23Z","abstract_excerpt":"We study the question: when are Lipschitz mappings dense in the Sobolev space $W^{1,p}(M,\\mathbf{H}^n)$? Here $M$ denotes a compact Riemannian manifold with or without boundary, while $\\mathbf{H}^n$ denotes the $n$th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in $W^{1,p}(M,\\mathbf{H}^n)$ for all $1\\le p<\\infty$ if $\\dim M \\le n$, but that Lipschitz maps are not dense in $W^{1,p}(M,\\mathbf{H}^n)$ if $\\dim M \\ge n+1$ and $n\\le p<n+1$. The proofs rely on the construction of smooth horizontal embeddings of the sphere $S^n$ into $\\mathbf{H}^n$. We "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4641","kind":"arxiv","version":3},"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"}