{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KHFBOD3L5CNGOLOYTW5CWL3CQM","short_pith_number":"pith:KHFBOD3L","canonical_record":{"source":{"id":"1512.07559","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2015-12-23T17:49:03Z","cross_cats_sorted":[],"title_canon_sha256":"53ba332065f17529c1c86202df71b6d8cfe7308307e135937e0bf0fa46613cd5","abstract_canon_sha256":"8a6c0870094547f0365c64c8b341376315fe0003b83d3958ffd821ee88bfec75"},"schema_version":"1.0"},"canonical_sha256":"51ca170f6be89a672dd89dba2b2f62830bed52daec62b9c24f0a0c4c734be18e","source":{"kind":"arxiv","id":"1512.07559","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.07559","created_at":"2026-05-18T01:21:44Z"},{"alias_kind":"arxiv_version","alias_value":"1512.07559v4","created_at":"2026-05-18T01:21:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07559","created_at":"2026-05-18T01:21:44Z"},{"alias_kind":"pith_short_12","alias_value":"KHFBOD3L5CNG","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KHFBOD3L5CNGOLOY","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KHFBOD3L","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KHFBOD3L5CNGOLOYTW5CWL3CQM","target":"record","payload":{"canonical_record":{"source":{"id":"1512.07559","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2015-12-23T17:49:03Z","cross_cats_sorted":[],"title_canon_sha256":"53ba332065f17529c1c86202df71b6d8cfe7308307e135937e0bf0fa46613cd5","abstract_canon_sha256":"8a6c0870094547f0365c64c8b341376315fe0003b83d3958ffd821ee88bfec75"},"schema_version":"1.0"},"canonical_sha256":"51ca170f6be89a672dd89dba2b2f62830bed52daec62b9c24f0a0c4c734be18e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:44.309814Z","signature_b64":"KHQ5KKO3Y9WSsSLomqQ5RIHvOGeDlGwhzGtwptFf7/iFdeaXnDyjsuxOf+wQGPpT7T+w7oFfjDU6dD+BXky5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51ca170f6be89a672dd89dba2b2f62830bed52daec62b9c24f0a0c4c734be18e","last_reissued_at":"2026-05-18T01:21:44.309184Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:44.309184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.07559","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:21:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wdUyeOxjBYHkcGr82Sy5098mwDwwU8Sv225FIO+ihY5JGn5DI1kl5zntHJx3Ysoe2S+Bj5+9sJ+2GlKy+t/iCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:36:33.313939Z"},"content_sha256":"0d1b585dcc3f1c39f50eac28e39d9b7102572f96d04f30840c02434bde517ed0","schema_version":"1.0","event_id":"sha256:0d1b585dcc3f1c39f50eac28e39d9b7102572f96d04f30840c02434bde517ed0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KHFBOD3L5CNGOLOYTW5CWL3CQM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Hardy Inequality for subelliptic operators with global fundamental solution, and an application to Unique Continuation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Bonfiglioli, Stefano Biagi","submitted_at":"2015-12-23T17:49:03Z","abstract_excerpt":"This is a chapter from PhD Thesis by Stefano Biagi \n(advisor: prof. A. Bonfiglioli).\n\nWe overview existing results showing that it is possible to generalize the classical Hardy's Inequality to more general linear partial differential operators (PDOs, in the sequel), possibly degenerate-elliptic, of the following quasi-divergence form $$ \\mathcal{L} = \\frac{1}{w(x)}\\sum_{i = 1}^N\\frac{\\partial}{\\partial x_i}\n  \\left(\\sum_{j = 1}^Nw(x)a_{ij}(x)\\frac{\\partial}{\\partial x_j}\\right), \\quad x \\in \\mathbb{R}^N, $$ where $w \\in C^{\\infty}(\\mathbb{R}^N,\\mathbb{R})$ is a (smooth and) strictly positive f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07559","kind":"arxiv","version":4},"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":"2026-05-18T01:21:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Aa1kJpoB+9TStBpw7gQR+a9VKjo7AKYRjLY70B3EZ9YUJSK9kLHd6D64KsgSPtmbPQ0C5GpAOJvFYGrZLQufAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:36:33.314278Z"},"content_sha256":"4438390946f79e26258383d917ef42a9c450d615adcaa57dd2b187e85bbcc08b","schema_version":"1.0","event_id":"sha256:4438390946f79e26258383d917ef42a9c450d615adcaa57dd2b187e85bbcc08b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KHFBOD3L5CNGOLOYTW5CWL3CQM/bundle.json","state_url":"https://pith.science/pith/KHFBOD3L5CNGOLOYTW5CWL3CQM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KHFBOD3L5CNGOLOYTW5CWL3CQM/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":"2026-06-20T23:36:33Z","links":{"resolver":"https://pith.science/pith/KHFBOD3L5CNGOLOYTW5CWL3CQM","bundle":"https://pith.science/pith/KHFBOD3L5CNGOLOYTW5CWL3CQM/bundle.json","state":"https://pith.science/pith/KHFBOD3L5CNGOLOYTW5CWL3CQM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KHFBOD3L5CNGOLOYTW5CWL3CQM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KHFBOD3L5CNGOLOYTW5CWL3CQM","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":"8a6c0870094547f0365c64c8b341376315fe0003b83d3958ffd821ee88bfec75","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2015-12-23T17:49:03Z","title_canon_sha256":"53ba332065f17529c1c86202df71b6d8cfe7308307e135937e0bf0fa46613cd5"},"schema_version":"1.0","source":{"id":"1512.07559","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.07559","created_at":"2026-05-18T01:21:44Z"},{"alias_kind":"arxiv_version","alias_value":"1512.07559v4","created_at":"2026-05-18T01:21:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07559","created_at":"2026-05-18T01:21:44Z"},{"alias_kind":"pith_short_12","alias_value":"KHFBOD3L5CNG","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KHFBOD3L5CNGOLOY","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KHFBOD3L","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:4438390946f79e26258383d917ef42a9c450d615adcaa57dd2b187e85bbcc08b","target":"graph","created_at":"2026-05-18T01:21:44Z","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":"This is a chapter from PhD Thesis by Stefano Biagi \n(advisor: prof. A. Bonfiglioli).\n\nWe overview existing results showing that it is possible to generalize the classical Hardy's Inequality to more general linear partial differential operators (PDOs, in the sequel), possibly degenerate-elliptic, of the following quasi-divergence form $$ \\mathcal{L} = \\frac{1}{w(x)}\\sum_{i = 1}^N\\frac{\\partial}{\\partial x_i}\n  \\left(\\sum_{j = 1}^Nw(x)a_{ij}(x)\\frac{\\partial}{\\partial x_j}\\right), \\quad x \\in \\mathbb{R}^N, $$ where $w \\in C^{\\infty}(\\mathbb{R}^N,\\mathbb{R})$ is a (smooth and) strictly positive f","authors_text":"Andrea Bonfiglioli, Stefano Biagi","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2015-12-23T17:49:03Z","title":"A Hardy Inequality for subelliptic operators with global fundamental solution, and an application to Unique Continuation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07559","kind":"arxiv","version":4},"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":"sha256:0d1b585dcc3f1c39f50eac28e39d9b7102572f96d04f30840c02434bde517ed0","target":"record","created_at":"2026-05-18T01:21:44Z","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":"8a6c0870094547f0365c64c8b341376315fe0003b83d3958ffd821ee88bfec75","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2015-12-23T17:49:03Z","title_canon_sha256":"53ba332065f17529c1c86202df71b6d8cfe7308307e135937e0bf0fa46613cd5"},"schema_version":"1.0","source":{"id":"1512.07559","kind":"arxiv","version":4}},"canonical_sha256":"51ca170f6be89a672dd89dba2b2f62830bed52daec62b9c24f0a0c4c734be18e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"51ca170f6be89a672dd89dba2b2f62830bed52daec62b9c24f0a0c4c734be18e","first_computed_at":"2026-05-18T01:21:44.309184Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:44.309184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KHQ5KKO3Y9WSsSLomqQ5RIHvOGeDlGwhzGtwptFf7/iFdeaXnDyjsuxOf+wQGPpT7T+w7oFfjDU6dD+BXky5Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:44.309814Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.07559","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0d1b585dcc3f1c39f50eac28e39d9b7102572f96d04f30840c02434bde517ed0","sha256:4438390946f79e26258383d917ef42a9c450d615adcaa57dd2b187e85bbcc08b"],"state_sha256":"21864cc3c9d00d8f284ce3cc1e7dc905cefb4798f4955b7cd303258d88d9d6e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w9XQ5H2+sc5n22UFiiFeyzzbA2T/3FbAh7FSzmbDjYcIdTwqCuQDQGG/7tQJKeX0DNbPrICe2X5OG4oF74Q+Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T23:36:33.316186Z","bundle_sha256":"3d0f3841b50041f19d268e4a64228576efe791cf136431e91b28bfd6ed9f85ae"}}