{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:YTSHTE7L4GOJT6BLTYN2QKP6CI","short_pith_number":"pith:YTSHTE7L","canonical_record":{"source":{"id":"1002.1331","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-02-05T22:59:41Z","cross_cats_sorted":[],"title_canon_sha256":"74049e0d48779be72b6563c171cfe17cf9bc0b5a1827cd3da39554bf7a82ed8f","abstract_canon_sha256":"4b36b7ecb4c80f05b99d6fc584a7d359167e6612cb04a4d47c36c135606a59ce"},"schema_version":"1.0"},"canonical_sha256":"c4e47993ebe19c99f82b9e1ba829fe121c259940e5021b8226b869896b3a4624","source":{"kind":"arxiv","id":"1002.1331","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.1331","created_at":"2026-05-18T04:25:50Z"},{"alias_kind":"arxiv_version","alias_value":"1002.1331v2","created_at":"2026-05-18T04:25:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.1331","created_at":"2026-05-18T04:25:50Z"},{"alias_kind":"pith_short_12","alias_value":"YTSHTE7L4GOJ","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YTSHTE7L4GOJT6BL","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YTSHTE7L","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:YTSHTE7L4GOJT6BLTYN2QKP6CI","target":"record","payload":{"canonical_record":{"source":{"id":"1002.1331","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-02-05T22:59:41Z","cross_cats_sorted":[],"title_canon_sha256":"74049e0d48779be72b6563c171cfe17cf9bc0b5a1827cd3da39554bf7a82ed8f","abstract_canon_sha256":"4b36b7ecb4c80f05b99d6fc584a7d359167e6612cb04a4d47c36c135606a59ce"},"schema_version":"1.0"},"canonical_sha256":"c4e47993ebe19c99f82b9e1ba829fe121c259940e5021b8226b869896b3a4624","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:50.644953Z","signature_b64":"fumGPLYsng6+zE8qZGCOgRoTBwSbBqY+fmGao5/HS8vJ+G3Zxo2+149dRF7OpDnGz/0+0KidgcJMSE2QAiEDDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4e47993ebe19c99f82b9e1ba829fe121c259940e5021b8226b869896b3a4624","last_reissued_at":"2026-05-18T04:25:50.644132Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:50.644132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1002.1331","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:25:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sD+q4ZrSQ+8+DE0kc9CKgW86bG3Wxv16jrm7UUnalAioJYYqgVjOuhYf1XlRwuZuurV5oIu3rJS584qL+wyjBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:24:11.167345Z"},"content_sha256":"a09932d482bc4ac1700edaec912dc3ce3a1fea4ede87780f696ead32a6cd2eb8","schema_version":"1.0","event_id":"sha256:a09932d482bc4ac1700edaec912dc3ce3a1fea4ede87780f696ead32a6cd2eb8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:YTSHTE7L4GOJT6BLTYN2QKP6CI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the lifting and approximation theorem for nonsmooth vector fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luca Brandolini, Marco Bramanti, Marco Pedroni","submitted_at":"2010-02-05T22:59:41Z","abstract_excerpt":"We prove a version of Rothschild-Stein's theorem of lifting and approximation and some related results in the context of nonsmooth Hormander's vector fields for which the highest order commutators are only Holder continuous. The theory explicitly covers the case of one vector field having weight two while the others have weight one."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1331","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:25:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/dcT4RBffGnmLB+CPM9kMdU0RlQgrwOA+MVr+nuVw7mda1kQuvYI7aR/e0I5JJ2Y3485TvyAcdwVNAIRYrNeCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:24:11.167697Z"},"content_sha256":"3334cb1742257bbd6929dc158007e51508fff77f0a8afd38d6da584e30d26e48","schema_version":"1.0","event_id":"sha256:3334cb1742257bbd6929dc158007e51508fff77f0a8afd38d6da584e30d26e48"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YTSHTE7L4GOJT6BLTYN2QKP6CI/bundle.json","state_url":"https://pith.science/pith/YTSHTE7L4GOJT6BLTYN2QKP6CI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YTSHTE7L4GOJT6BLTYN2QKP6CI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T03:24:11Z","links":{"resolver":"https://pith.science/pith/YTSHTE7L4GOJT6BLTYN2QKP6CI","bundle":"https://pith.science/pith/YTSHTE7L4GOJT6BLTYN2QKP6CI/bundle.json","state":"https://pith.science/pith/YTSHTE7L4GOJT6BLTYN2QKP6CI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YTSHTE7L4GOJT6BLTYN2QKP6CI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:YTSHTE7L4GOJT6BLTYN2QKP6CI","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":"4b36b7ecb4c80f05b99d6fc584a7d359167e6612cb04a4d47c36c135606a59ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-02-05T22:59:41Z","title_canon_sha256":"74049e0d48779be72b6563c171cfe17cf9bc0b5a1827cd3da39554bf7a82ed8f"},"schema_version":"1.0","source":{"id":"1002.1331","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.1331","created_at":"2026-05-18T04:25:50Z"},{"alias_kind":"arxiv_version","alias_value":"1002.1331v2","created_at":"2026-05-18T04:25:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.1331","created_at":"2026-05-18T04:25:50Z"},{"alias_kind":"pith_short_12","alias_value":"YTSHTE7L4GOJ","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YTSHTE7L4GOJT6BL","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YTSHTE7L","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:3334cb1742257bbd6929dc158007e51508fff77f0a8afd38d6da584e30d26e48","target":"graph","created_at":"2026-05-18T04:25:50Z","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 prove a version of Rothschild-Stein's theorem of lifting and approximation and some related results in the context of nonsmooth Hormander's vector fields for which the highest order commutators are only Holder continuous. The theory explicitly covers the case of one vector field having weight two while the others have weight one.","authors_text":"Luca Brandolini, Marco Bramanti, Marco Pedroni","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-02-05T22:59:41Z","title":"On the lifting and approximation theorem for nonsmooth vector fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1331","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:a09932d482bc4ac1700edaec912dc3ce3a1fea4ede87780f696ead32a6cd2eb8","target":"record","created_at":"2026-05-18T04:25:50Z","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":"4b36b7ecb4c80f05b99d6fc584a7d359167e6612cb04a4d47c36c135606a59ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-02-05T22:59:41Z","title_canon_sha256":"74049e0d48779be72b6563c171cfe17cf9bc0b5a1827cd3da39554bf7a82ed8f"},"schema_version":"1.0","source":{"id":"1002.1331","kind":"arxiv","version":2}},"canonical_sha256":"c4e47993ebe19c99f82b9e1ba829fe121c259940e5021b8226b869896b3a4624","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4e47993ebe19c99f82b9e1ba829fe121c259940e5021b8226b869896b3a4624","first_computed_at":"2026-05-18T04:25:50.644132Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:50.644132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fumGPLYsng6+zE8qZGCOgRoTBwSbBqY+fmGao5/HS8vJ+G3Zxo2+149dRF7OpDnGz/0+0KidgcJMSE2QAiEDDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:50.644953Z","signed_message":"canonical_sha256_bytes"},"source_id":"1002.1331","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a09932d482bc4ac1700edaec912dc3ce3a1fea4ede87780f696ead32a6cd2eb8","sha256:3334cb1742257bbd6929dc158007e51508fff77f0a8afd38d6da584e30d26e48"],"state_sha256":"0cc9593c9d17f898677efbf5f5ac710bdabf1a2532318e3355b8221f30e97478"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pvUHajbJUzbFuZvfiDsvOiKFx4bjOd56bmwucvkKvM+EI2ykhF8UZ4jYjkdVM7qHXI9oFEa5KshvHKEGvOrACg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T03:24:11.169654Z","bundle_sha256":"7583485da1ea38c8b221da309df1f9dd1ebf176b7ea083da4ed43717f6ca1417"}}