{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:2ZOQN5BU46O6XKMWLMLXDSIMIG","short_pith_number":"pith:2ZOQN5BU","canonical_record":{"source":{"id":"math/0703732","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2007-03-25T15:32:16Z","cross_cats_sorted":[],"title_canon_sha256":"54951cb8326070163f65bba15944f89b24fa126853c48b7e888a7661d57a2bcb","abstract_canon_sha256":"7f013fb74dee8c700462376bcfc2767a75633d935c7b4cb9602e4c28f723a262"},"schema_version":"1.0"},"canonical_sha256":"d65d06f434e79deba9965b1771c90c41a86d843dc185eb6d47454a362ab475d7","source":{"kind":"arxiv","id":"math/0703732","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0703732","created_at":"2026-05-18T00:03:42Z"},{"alias_kind":"arxiv_version","alias_value":"math/0703732v2","created_at":"2026-05-18T00:03:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0703732","created_at":"2026-05-18T00:03:42Z"},{"alias_kind":"pith_short_12","alias_value":"2ZOQN5BU46O6","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"2ZOQN5BU46O6XKMW","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"2ZOQN5BU","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:2ZOQN5BU46O6XKMWLMLXDSIMIG","target":"record","payload":{"canonical_record":{"source":{"id":"math/0703732","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2007-03-25T15:32:16Z","cross_cats_sorted":[],"title_canon_sha256":"54951cb8326070163f65bba15944f89b24fa126853c48b7e888a7661d57a2bcb","abstract_canon_sha256":"7f013fb74dee8c700462376bcfc2767a75633d935c7b4cb9602e4c28f723a262"},"schema_version":"1.0"},"canonical_sha256":"d65d06f434e79deba9965b1771c90c41a86d843dc185eb6d47454a362ab475d7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:42.114819Z","signature_b64":"ABEpou4Wwi80HehRh0HjtfDbvFOmtlGRScgMUv3IrCyjst0aPq8Skwo0Pwht+xllnykxkvDhhqJmtxeWwH8RDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d65d06f434e79deba9965b1771c90c41a86d843dc185eb6d47454a362ab475d7","last_reissued_at":"2026-05-18T00:03:42.114132Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:42.114132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0703732","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":"2026-05-18T00:03:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cvjNNgc/xCfo9Fre7omrVi8DYTYj5QyjHXI/hVrUWklvjH1H0mdkm8F/GDaP3tUjQozQ195Xe1PHcQnVLBxnDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:59:46.640821Z"},"content_sha256":"02b07435a74591964754f9e21845177a1b79356451931a6aa631e11da87847ff","schema_version":"1.0","event_id":"sha256:02b07435a74591964754f9e21845177a1b79356451931a6aa631e11da87847ff"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:2ZOQN5BU46O6XKMWLMLXDSIMIG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted norm inequalities for fractional operators]","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Jos\\'e Maria Martell (IMFF), Pascal Auscher (LM-Orsay)","submitted_at":"2007-03-25T15:32:16Z","abstract_excerpt":"We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-$\\lambda$ method that does not use any size or smoothness estimates for the kernels."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703732","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":"2026-05-18T00:03:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v3brRKUgN7WWp81KQ2yqS1SmxR63Qe6r0Zp2MyXtbW8RWk+ri2jeMYbCJIEAqiBnYC0WnmfGktAjkKMdR4fuCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:59:46.641520Z"},"content_sha256":"8d739170543a452f1ef2d680e5f9a0320798628672d4a3567f63135502b3db24","schema_version":"1.0","event_id":"sha256:8d739170543a452f1ef2d680e5f9a0320798628672d4a3567f63135502b3db24"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2ZOQN5BU46O6XKMWLMLXDSIMIG/bundle.json","state_url":"https://pith.science/pith/2ZOQN5BU46O6XKMWLMLXDSIMIG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2ZOQN5BU46O6XKMWLMLXDSIMIG/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":"2026-06-06T19:59:46Z","links":{"resolver":"https://pith.science/pith/2ZOQN5BU46O6XKMWLMLXDSIMIG","bundle":"https://pith.science/pith/2ZOQN5BU46O6XKMWLMLXDSIMIG/bundle.json","state":"https://pith.science/pith/2ZOQN5BU46O6XKMWLMLXDSIMIG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2ZOQN5BU46O6XKMWLMLXDSIMIG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:2ZOQN5BU46O6XKMWLMLXDSIMIG","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":"7f013fb74dee8c700462376bcfc2767a75633d935c7b4cb9602e4c28f723a262","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2007-03-25T15:32:16Z","title_canon_sha256":"54951cb8326070163f65bba15944f89b24fa126853c48b7e888a7661d57a2bcb"},"schema_version":"1.0","source":{"id":"math/0703732","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0703732","created_at":"2026-05-18T00:03:42Z"},{"alias_kind":"arxiv_version","alias_value":"math/0703732v2","created_at":"2026-05-18T00:03:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0703732","created_at":"2026-05-18T00:03:42Z"},{"alias_kind":"pith_short_12","alias_value":"2ZOQN5BU46O6","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"2ZOQN5BU46O6XKMW","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"2ZOQN5BU","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:8d739170543a452f1ef2d680e5f9a0320798628672d4a3567f63135502b3db24","target":"graph","created_at":"2026-05-18T00:03:42Z","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 weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-$\\lambda$ method that does not use any size or smoothness estimates for the kernels.","authors_text":"Jos\\'e Maria Martell (IMFF), Pascal Auscher (LM-Orsay)","cross_cats":[],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"2007-03-25T15:32:16Z","title":"Weighted norm inequalities for fractional operators]"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703732","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":"sha256:02b07435a74591964754f9e21845177a1b79356451931a6aa631e11da87847ff","target":"record","created_at":"2026-05-18T00:03:42Z","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":"7f013fb74dee8c700462376bcfc2767a75633d935c7b4cb9602e4c28f723a262","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2007-03-25T15:32:16Z","title_canon_sha256":"54951cb8326070163f65bba15944f89b24fa126853c48b7e888a7661d57a2bcb"},"schema_version":"1.0","source":{"id":"math/0703732","kind":"arxiv","version":2}},"canonical_sha256":"d65d06f434e79deba9965b1771c90c41a86d843dc185eb6d47454a362ab475d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d65d06f434e79deba9965b1771c90c41a86d843dc185eb6d47454a362ab475d7","first_computed_at":"2026-05-18T00:03:42.114132Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:42.114132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ABEpou4Wwi80HehRh0HjtfDbvFOmtlGRScgMUv3IrCyjst0aPq8Skwo0Pwht+xllnykxkvDhhqJmtxeWwH8RDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:42.114819Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0703732","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02b07435a74591964754f9e21845177a1b79356451931a6aa631e11da87847ff","sha256:8d739170543a452f1ef2d680e5f9a0320798628672d4a3567f63135502b3db24"],"state_sha256":"7bc5aff6b1c7b9beade0323983aeb1bf7052361cf5ee32d6482e39f7f8f96b71"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fv+8MPTkxIVd7mV0uqSXTfc4vzvEs/4wr53YQQLH26n9ZE12L61hFlc4Fqazq1hzl95KT2ZX+Uc1eoiMJK60Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T19:59:46.645256Z","bundle_sha256":"b606e75037910491a906465aa7ed4ce090c398f96f9ad5d31629f7b52975ea88"}}