{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:PFZM3SK7MCHOCCHJS2UXJ2NERW","short_pith_number":"pith:PFZM3SK7","canonical_record":{"source":{"id":"0904.1781","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-11T07:06:05Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"fb53333bfbd496b4caed52b865de2e8dae36636bc513de43d27a497b12d392dc","abstract_canon_sha256":"b4e5413ef2a73026990ed2b2cfa63330319240959b3dbc65ea13ce76c6c82fd8"},"schema_version":"1.0"},"canonical_sha256":"7972cdc95f608ee108e996a974e9a48d899ffff37ed04d391e7469d41e5ce068","source":{"kind":"arxiv","id":"0904.1781","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.1781","created_at":"2026-05-18T02:49:03Z"},{"alias_kind":"arxiv_version","alias_value":"0904.1781v2","created_at":"2026-05-18T02:49:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.1781","created_at":"2026-05-18T02:49:03Z"},{"alias_kind":"pith_short_12","alias_value":"PFZM3SK7MCHO","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"PFZM3SK7MCHOCCHJ","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"PFZM3SK7","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:PFZM3SK7MCHOCCHJS2UXJ2NERW","target":"record","payload":{"canonical_record":{"source":{"id":"0904.1781","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-11T07:06:05Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"fb53333bfbd496b4caed52b865de2e8dae36636bc513de43d27a497b12d392dc","abstract_canon_sha256":"b4e5413ef2a73026990ed2b2cfa63330319240959b3dbc65ea13ce76c6c82fd8"},"schema_version":"1.0"},"canonical_sha256":"7972cdc95f608ee108e996a974e9a48d899ffff37ed04d391e7469d41e5ce068","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:03.568589Z","signature_b64":"1cKZvU+w90tI5tnpnJ1/Qz4sEoOnOuxmd/4VzE+J2hHyY3/vdsgVK4usOSMdyGc1LoTB4CKU23QBdBnhWogBCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7972cdc95f608ee108e996a974e9a48d899ffff37ed04d391e7469d41e5ce068","last_reissued_at":"2026-05-18T02:49:03.567975Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:03.567975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0904.1781","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":"2026-05-18T02:49:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"brCdAYLzmlPYWStRnXCmO3NtNK2gKkA95oxatcZM1wdkCFiE3+foG0RxVTKQOJaJPm3kLhuSS5ftT7B2ZUFrAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:02:47.025687Z"},"content_sha256":"ce6b41a3f9d0bb7d84c1a5de9e206900df63d6b9c6f183581209c1046fb55800","schema_version":"1.0","event_id":"sha256:ce6b41a3f9d0bb7d84c1a5de9e206900df63d6b9c6f183581209c1046fb55800"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:PFZM3SK7MCHOCCHJS2UXJ2NERW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gradient estimates for the subelliptic heat kernel on H-type groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Nathaniel Eldredge","submitted_at":"2009-04-11T07:06:05Z","abstract_excerpt":"We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type:\n  $$|\\nabla P_t f| \\le K P_t(|\\nabla f|)$$ where $P_t$ is the heat semigroup corresponding to the sublaplacian on $G$, $\\nabla$ is the subelliptic gradient, and $K$ is a constant. This extends a result of H.-Q. Li for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry, Baudoin, Bonnefont, and Chafa\\\"i."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.1781","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":"2026-05-18T02:49:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"21V953JYgJWtpG44nN4hU/h3YLLYiDyo2VKWc0JQsD9LJFdZuZ9fzjsOzXc9CemOBGlaHohZr7Quoa0WLgYUBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:02:47.026466Z"},"content_sha256":"749aebebdbc175080b2dcc2b021d44ac1ae07c914127e86b168e7665e8f70b41","schema_version":"1.0","event_id":"sha256:749aebebdbc175080b2dcc2b021d44ac1ae07c914127e86b168e7665e8f70b41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PFZM3SK7MCHOCCHJS2UXJ2NERW/bundle.json","state_url":"https://pith.science/pith/PFZM3SK7MCHOCCHJS2UXJ2NERW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PFZM3SK7MCHOCCHJS2UXJ2NERW/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":"2026-06-11T23:02:47Z","links":{"resolver":"https://pith.science/pith/PFZM3SK7MCHOCCHJS2UXJ2NERW","bundle":"https://pith.science/pith/PFZM3SK7MCHOCCHJS2UXJ2NERW/bundle.json","state":"https://pith.science/pith/PFZM3SK7MCHOCCHJS2UXJ2NERW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PFZM3SK7MCHOCCHJS2UXJ2NERW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:PFZM3SK7MCHOCCHJS2UXJ2NERW","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":"b4e5413ef2a73026990ed2b2cfa63330319240959b3dbc65ea13ce76c6c82fd8","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-11T07:06:05Z","title_canon_sha256":"fb53333bfbd496b4caed52b865de2e8dae36636bc513de43d27a497b12d392dc"},"schema_version":"1.0","source":{"id":"0904.1781","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.1781","created_at":"2026-05-18T02:49:03Z"},{"alias_kind":"arxiv_version","alias_value":"0904.1781v2","created_at":"2026-05-18T02:49:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.1781","created_at":"2026-05-18T02:49:03Z"},{"alias_kind":"pith_short_12","alias_value":"PFZM3SK7MCHO","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"PFZM3SK7MCHOCCHJ","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"PFZM3SK7","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:749aebebdbc175080b2dcc2b021d44ac1ae07c914127e86b168e7665e8f70b41","target":"graph","created_at":"2026-05-18T02:49:03Z","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 the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type:\n  $$|\\nabla P_t f| \\le K P_t(|\\nabla f|)$$ where $P_t$ is the heat semigroup corresponding to the sublaplacian on $G$, $\\nabla$ is the subelliptic gradient, and $K$ is a constant. This extends a result of H.-Q. Li for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry, Baudoin, Bonnefont, and Chafa\\\"i.","authors_text":"Nathaniel Eldredge","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-11T07:06:05Z","title":"Gradient estimates for the subelliptic heat kernel on H-type groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.1781","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:ce6b41a3f9d0bb7d84c1a5de9e206900df63d6b9c6f183581209c1046fb55800","target":"record","created_at":"2026-05-18T02:49:03Z","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":"b4e5413ef2a73026990ed2b2cfa63330319240959b3dbc65ea13ce76c6c82fd8","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-04-11T07:06:05Z","title_canon_sha256":"fb53333bfbd496b4caed52b865de2e8dae36636bc513de43d27a497b12d392dc"},"schema_version":"1.0","source":{"id":"0904.1781","kind":"arxiv","version":2}},"canonical_sha256":"7972cdc95f608ee108e996a974e9a48d899ffff37ed04d391e7469d41e5ce068","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7972cdc95f608ee108e996a974e9a48d899ffff37ed04d391e7469d41e5ce068","first_computed_at":"2026-05-18T02:49:03.567975Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:03.567975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1cKZvU+w90tI5tnpnJ1/Qz4sEoOnOuxmd/4VzE+J2hHyY3/vdsgVK4usOSMdyGc1LoTB4CKU23QBdBnhWogBCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:03.568589Z","signed_message":"canonical_sha256_bytes"},"source_id":"0904.1781","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce6b41a3f9d0bb7d84c1a5de9e206900df63d6b9c6f183581209c1046fb55800","sha256:749aebebdbc175080b2dcc2b021d44ac1ae07c914127e86b168e7665e8f70b41"],"state_sha256":"6353446fe030004bc8060487bfb1bc5488fdab232ea8d85d8b301bd1c3f5bf9e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n8xRYzmbCevZ0IU30RAdWr1dz7YF4gje6/THSfbF/jYS8QEJmQi8IVjXtnRMCANFOoGWS6dOA0rvV4so/Sw+Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:02:47.030415Z","bundle_sha256":"7378e5647f1972bab6ffed0fdcb670fb098ed6d9fb9a38d247e731ad3655d067"}}