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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1905.13630","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-05-31T14:18:08Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"016544c059ca18abb3628dc998adfcd5747114893b0b92a3553b818dd42dd669","abstract_canon_sha256":"8d73e027705ebbb641083b123c810ab451ba6654f2022e3760d4c45571aafff3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:34.380676Z","signature_b64":"mxusIrQua5gxkvYJTb3IoNgDXLeDg5io7luNO2BcAAxfzmLv4R635IyzDaQGZDslBnnaYkDqYsLbExZxtnP9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a8255dec4ef99c461cf0f80dc5bcb4efa22a85346d4ac5f4879d7c996731500","last_reissued_at":"2026-05-17T23:44:34.379957Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:34.379957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1905.13630","created_at":"2026-05-17T23:44:34.380088+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.13630v1","created_at":"2026-05-17T23:44:34.380088+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.13630","created_at":"2026-05-17T23:44:34.380088+00:00"},{"alias_kind":"pith_short_12","alias_value":"FKBFLXWE56M4","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_16","alias_value":"FKBFLXWE56M4IYOP","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_8","alias_value":"FKBFLXWE","created_at":"2026-05-18T12:33:15.570797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3","json":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3.json","graph_json":"https://pith.science/api/pith-number/FKBFLXWE56M4IYOPB6ANYW6LJ3/graph.json","events_json":"https://pith.science/api/pith-number/FKBFLXWE56M4IYOPB6ANYW6LJ3/events.json","paper":"https://pith.science/paper/FKBFLXWE"},"agent_actions":{"view_html":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3","download_json":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3.json","view_paper":"https://pith.science/paper/FKBFLXWE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.13630&json=true","fetch_graph":"https://pith.science/api/pith-number/FKBFLXWE56M4IYOPB6ANYW6LJ3/graph.json","fetch_events":"https://pith.science/api/pith-number/FKBFLXWE56M4IYOPB6ANYW6LJ3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3/action/storage_attestation","attest_author":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3/action/author_attestation","sign_citation":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3/action/citation_signature","submit_replication":"https://pith.science/pith/FKBFLXWE56M4IYOPB6ANYW6LJ3/action/replication_record"}},"created_at":"2026-05-17T23:44:34.380088+00:00","updated_at":"2026-05-17T23:44:34.380088+00:00"}