{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:VX75TPBUXVYXZAN75Z2UV2X4C2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0d58e618f2ae7dce53be4c814a8386a6611b70c68090173b356fc03e79352997","cross_cats_sorted":["math.CV"],"license":"","primary_cat":"math.AP","submitted_at":"2007-12-21T16:04:21Z","title_canon_sha256":"0bd7e10a1675d2fb211245a88e8f6e2abb811bab84aff69c413636493183174d"},"schema_version":"1.0","source":{"id":"0712.3730","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.3730","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"arxiv_version","alias_value":"0712.3730v2","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.3730","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"pith_short_12","alias_value":"VX75TPBUXVYX","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"VX75TPBUXVYXZAN7","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"VX75TPBU","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:5e1d6786f90b84269478cd786e7e5616f2416012382a667456b966fa8e3f0420","target":"graph","created_at":"2026-05-18T00:07:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We show that divergence free vector fields which belong to L^1 on stratified, nilpotent groups are in the dual space of functions whose sub-gradient are L^Q integrable where Q is the homogeneous dimension of the group. This was first obtained on Euclidean space by Bourgain and Brezis.","authors_text":"Jean Van Schaftingen, Sagun Chanillo","cross_cats":["math.CV"],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2007-12-21T16:04:21Z","title":"Subelliptic Bourgain-Brezis Estimates on Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.3730","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b06eb303366cddcbdbeccca74bf1fff64ce8e974d3711076c3a3160eddee1400","target":"record","created_at":"2026-05-18T00:07:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0d58e618f2ae7dce53be4c814a8386a6611b70c68090173b356fc03e79352997","cross_cats_sorted":["math.CV"],"license":"","primary_cat":"math.AP","submitted_at":"2007-12-21T16:04:21Z","title_canon_sha256":"0bd7e10a1675d2fb211245a88e8f6e2abb811bab84aff69c413636493183174d"},"schema_version":"1.0","source":{"id":"0712.3730","kind":"arxiv","version":2}},"canonical_sha256":"adffd9bc34bd717c81bfee754aeafc16b746ddce1fbb3d5b1e041a0c38b28edc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"adffd9bc34bd717c81bfee754aeafc16b746ddce1fbb3d5b1e041a0c38b28edc","first_computed_at":"2026-05-18T00:07:44.243627Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:44.243627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vDs5BbReJ5tqjiyQrh44MMCbMEpI+3l049JHF7pBASwPzpN5inAYN8nUjL0cbjLoi6kS/GG/voCMfQE8ZFZIBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:44.244254Z","signed_message":"canonical_sha256_bytes"},"source_id":"0712.3730","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b06eb303366cddcbdbeccca74bf1fff64ce8e974d3711076c3a3160eddee1400","sha256:5e1d6786f90b84269478cd786e7e5616f2416012382a667456b966fa8e3f0420"],"state_sha256":"bd73f83bbbac4a0b17e462ebf9ff08e9befc1097c3c40f7b2b00a372e3946a0e"}