{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:SOMTS2GWYCQJOIMZHLIOMZ3GU4","short_pith_number":"pith:SOMTS2GW","canonical_record":{"source":{"id":"1809.01242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-04T21:10:55Z","cross_cats_sorted":[],"title_canon_sha256":"c0cec6d7804e131eaea3ffb667c350d284f08bea087c750a453d468626f8b787","abstract_canon_sha256":"d32d564ed785e09f21485e57ab3679cd02f9d9d464567b28b886ed310459d37d"},"schema_version":"1.0"},"canonical_sha256":"93993968d6c0a09721993ad0e66766a734fa45721c68f40db3ebf87aee7204b4","source":{"kind":"arxiv","id":"1809.01242","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.01242","created_at":"2026-05-18T00:06:28Z"},{"alias_kind":"arxiv_version","alias_value":"1809.01242v1","created_at":"2026-05-18T00:06:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.01242","created_at":"2026-05-18T00:06:28Z"},{"alias_kind":"pith_short_12","alias_value":"SOMTS2GWYCQJ","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"SOMTS2GWYCQJOIMZ","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"SOMTS2GW","created_at":"2026-05-18T12:32:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:SOMTS2GWYCQJOIMZHLIOMZ3GU4","target":"record","payload":{"canonical_record":{"source":{"id":"1809.01242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-04T21:10:55Z","cross_cats_sorted":[],"title_canon_sha256":"c0cec6d7804e131eaea3ffb667c350d284f08bea087c750a453d468626f8b787","abstract_canon_sha256":"d32d564ed785e09f21485e57ab3679cd02f9d9d464567b28b886ed310459d37d"},"schema_version":"1.0"},"canonical_sha256":"93993968d6c0a09721993ad0e66766a734fa45721c68f40db3ebf87aee7204b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:28.254973Z","signature_b64":"aJB91LfWwslj/zPLFAQFVsFMYo7bvcU7yyz2pImRI3QHfOI271ThFYZ2XgFNiXU4/awiTrj3h64/0n7xxeOWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93993968d6c0a09721993ad0e66766a734fa45721c68f40db3ebf87aee7204b4","last_reissued_at":"2026-05-18T00:06:28.254420Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:28.254420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.01242","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gqgik8wz8w36Uv/2nLNd3Uek+EpVQhCvOJO0qaMuJQbBp5vMdAfVa91ILNUdGldz6YVb+gTMjA5BnFjv41e2AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:58:22.146352Z"},"content_sha256":"cdb7624fc3fe9a7a88d1e4a57522e0f9c7f62cec3b78f53b00369fac7f32d918","schema_version":"1.0","event_id":"sha256:cdb7624fc3fe9a7a88d1e4a57522e0f9c7f62cec3b78f53b00369fac7f32d918"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:SOMTS2GWYCQJOIMZHLIOMZ3GU4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some properties of sub-Laplaceans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nicola Garofalo","submitted_at":"2018-09-04T21:10:55Z","abstract_excerpt":"In this note I present some properties of sub-Laplaceans associated with a collection of smooth vector fields satisfying H\\\"ormander's finite rank assumption. One notable aspect of the paper is the development of the fractional powers of sub-Laplaceans as Dirichlet-to-Neumann maps of an extension problem inspired to the famous 2007 work of Caffarelli and Silvestre for the standard Laplacean. A key tool is an extension problem for the fractional heat equation for which I compute the relevant Poisson kernel. I then use the latter to: 1) find the Poisson kernel for the time-independent case; and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.01242","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QqUwZEBm8hkN3dtBAG1oBXLxPt3NDLpyka+xr3ZIE0BE/cZTpNa40mpsSY7iuPNsqByHujt5B06VwTtJs/PGBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:58:22.147011Z"},"content_sha256":"f1322cbe0db4706fadf84c970df776dc7271701ba3afd2ac862c96b689e68f5a","schema_version":"1.0","event_id":"sha256:f1322cbe0db4706fadf84c970df776dc7271701ba3afd2ac862c96b689e68f5a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SOMTS2GWYCQJOIMZHLIOMZ3GU4/bundle.json","state_url":"https://pith.science/pith/SOMTS2GWYCQJOIMZHLIOMZ3GU4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SOMTS2GWYCQJOIMZHLIOMZ3GU4/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":"2026-05-26T11:58:22Z","links":{"resolver":"https://pith.science/pith/SOMTS2GWYCQJOIMZHLIOMZ3GU4","bundle":"https://pith.science/pith/SOMTS2GWYCQJOIMZHLIOMZ3GU4/bundle.json","state":"https://pith.science/pith/SOMTS2GWYCQJOIMZHLIOMZ3GU4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SOMTS2GWYCQJOIMZHLIOMZ3GU4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:SOMTS2GWYCQJOIMZHLIOMZ3GU4","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":"d32d564ed785e09f21485e57ab3679cd02f9d9d464567b28b886ed310459d37d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-04T21:10:55Z","title_canon_sha256":"c0cec6d7804e131eaea3ffb667c350d284f08bea087c750a453d468626f8b787"},"schema_version":"1.0","source":{"id":"1809.01242","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.01242","created_at":"2026-05-18T00:06:28Z"},{"alias_kind":"arxiv_version","alias_value":"1809.01242v1","created_at":"2026-05-18T00:06:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.01242","created_at":"2026-05-18T00:06:28Z"},{"alias_kind":"pith_short_12","alias_value":"SOMTS2GWYCQJ","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"SOMTS2GWYCQJOIMZ","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"SOMTS2GW","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:f1322cbe0db4706fadf84c970df776dc7271701ba3afd2ac862c96b689e68f5a","target":"graph","created_at":"2026-05-18T00:06:28Z","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":"In this note I present some properties of sub-Laplaceans associated with a collection of smooth vector fields satisfying H\\\"ormander's finite rank assumption. One notable aspect of the paper is the development of the fractional powers of sub-Laplaceans as Dirichlet-to-Neumann maps of an extension problem inspired to the famous 2007 work of Caffarelli and Silvestre for the standard Laplacean. A key tool is an extension problem for the fractional heat equation for which I compute the relevant Poisson kernel. I then use the latter to: 1) find the Poisson kernel for the time-independent case; and ","authors_text":"Nicola Garofalo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-04T21:10:55Z","title":"Some properties of sub-Laplaceans"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.01242","kind":"arxiv","version":1},"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:cdb7624fc3fe9a7a88d1e4a57522e0f9c7f62cec3b78f53b00369fac7f32d918","target":"record","created_at":"2026-05-18T00:06:28Z","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":"d32d564ed785e09f21485e57ab3679cd02f9d9d464567b28b886ed310459d37d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-04T21:10:55Z","title_canon_sha256":"c0cec6d7804e131eaea3ffb667c350d284f08bea087c750a453d468626f8b787"},"schema_version":"1.0","source":{"id":"1809.01242","kind":"arxiv","version":1}},"canonical_sha256":"93993968d6c0a09721993ad0e66766a734fa45721c68f40db3ebf87aee7204b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"93993968d6c0a09721993ad0e66766a734fa45721c68f40db3ebf87aee7204b4","first_computed_at":"2026-05-18T00:06:28.254420Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:28.254420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aJB91LfWwslj/zPLFAQFVsFMYo7bvcU7yyz2pImRI3QHfOI271ThFYZ2XgFNiXU4/awiTrj3h64/0n7xxeOWDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:28.254973Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.01242","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cdb7624fc3fe9a7a88d1e4a57522e0f9c7f62cec3b78f53b00369fac7f32d918","sha256:f1322cbe0db4706fadf84c970df776dc7271701ba3afd2ac862c96b689e68f5a"],"state_sha256":"7d50ecb7e14eb792896fd63cdd102cef5ab87cf59292f94553e0d29c3a3bb2b3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FB4R4KjuQcyd0gA1CpFgzdo9yHK0lf8CACFRVHIW3Kg6wEKg7CEN1SzrZyjbZqeUG9S4Yx9MI9ELmS+FAnsyAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T11:58:22.150643Z","bundle_sha256":"311b7676d3fa0d490b1bdc56fb6366aa2b2371914139a5215ce5183f4d1aea84"}}