{"paper":{"title":"Vertical versus horizontal Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"math.CA","authors_text":"Katrin F\\\"assler, Tuomas Orponen","submitted_at":"2019-05-31T14:18:08Z","abstract_excerpt":"Let $\\alpha \\geq 0$, $1 < p < \\infty$, and let $\\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \\colon \\mathbb{H}^{n} \\to \\mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2\\alpha}(\\mathbb{H}^{n})$, then $\\varphi f$ belongs to the Euclidean Sobolev space $S^{p}_{\\alpha}(\\mathbb{R}^{2n + 1})$ for any test function $\\varphi$. In short, $S^{p}_{2\\alpha}(\\mathbb{H}^{n}) \\subset S^{p}_{\\alpha,\\mathrm{loc}}(\\mathbb{R}^{2n + 1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.13630","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"}