{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OHIT6GPV6NY3Z43NOFB6LAYFZK","short_pith_number":"pith:OHIT6GPV","canonical_record":{"source":{"id":"1601.06854","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-01-26T00:14:11Z","cross_cats_sorted":[],"title_canon_sha256":"d18d1aca232b714ef46d390d6946a9e9f6f81843a8ddfbcf3e8bc271b72a1ab2","abstract_canon_sha256":"abe2fafa9bc3843a7c87fa2b1629d4c84b7213cf7441e26763609d81322d96e7"},"schema_version":"1.0"},"canonical_sha256":"71d13f19f5f371bcf36d7143e58305ca9cb77c5715ecf04a722f6d75372edbe2","source":{"kind":"arxiv","id":"1601.06854","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06854","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06854v2","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06854","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"OHIT6GPV6NY3","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OHIT6GPV6NY3Z43N","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OHIT6GPV","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OHIT6GPV6NY3Z43NOFB6LAYFZK","target":"record","payload":{"canonical_record":{"source":{"id":"1601.06854","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-01-26T00:14:11Z","cross_cats_sorted":[],"title_canon_sha256":"d18d1aca232b714ef46d390d6946a9e9f6f81843a8ddfbcf3e8bc271b72a1ab2","abstract_canon_sha256":"abe2fafa9bc3843a7c87fa2b1629d4c84b7213cf7441e26763609d81322d96e7"},"schema_version":"1.0"},"canonical_sha256":"71d13f19f5f371bcf36d7143e58305ca9cb77c5715ecf04a722f6d75372edbe2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:17.435465Z","signature_b64":"HJCdeBTsG7nOoaWXpQ689AQoWwoScURsGObVLIuS1sH/NLRrxZVYw8i3v+g8ICZQwbcoaahVi8/60v4yFv+4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71d13f19f5f371bcf36d7143e58305ca9cb77c5715ecf04a722f6d75372edbe2","last_reissued_at":"2026-05-18T01:14:17.434781Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:17.434781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.06854","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":"2026-05-18T01:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"upom7IWrVsBdvqdj1rqTkUS/45PTDq4B90QTF4qS5ToA57dSG93OPcY0XPH5rDBHH/Io7arsHjwza+btTmhsBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:42:50.213551Z"},"content_sha256":"7afdacf1b72985e9c169e90e6cbb7dc72f2ae131a7603cdb7b5abfa6bc8d9e71","schema_version":"1.0","event_id":"sha256:7afdacf1b72985e9c169e90e6cbb7dc72f2ae131a7603cdb7b5abfa6bc8d9e71"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OHIT6GPV6NY3Z43NOFB6LAYFZK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kato-Ponce inequalities on weighted and variable Lebesgue spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Cruz-Uribe, Virginia Naibo","submitted_at":"2016-01-26T00:14:11Z","abstract_excerpt":"We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgue spaces of certain bilinear multiplier operators and singular integrals."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06854","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":"2026-05-18T01:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oj8KwbDzF33EWxZri4Z8bqwWKNvHA79Pf3i48AgRJ64d2X1zE1A8sBoTwt/TcnF1KCOGv89HYhkLpPjbF9W0Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:42:50.213883Z"},"content_sha256":"3c28d1b89f4bad2cddb7915319a848a3442a310b41a35ad3d92ad3e94ec22f7b","schema_version":"1.0","event_id":"sha256:3c28d1b89f4bad2cddb7915319a848a3442a310b41a35ad3d92ad3e94ec22f7b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OHIT6GPV6NY3Z43NOFB6LAYFZK/bundle.json","state_url":"https://pith.science/pith/OHIT6GPV6NY3Z43NOFB6LAYFZK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OHIT6GPV6NY3Z43NOFB6LAYFZK/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":"2026-06-27T02:42:50Z","links":{"resolver":"https://pith.science/pith/OHIT6GPV6NY3Z43NOFB6LAYFZK","bundle":"https://pith.science/pith/OHIT6GPV6NY3Z43NOFB6LAYFZK/bundle.json","state":"https://pith.science/pith/OHIT6GPV6NY3Z43NOFB6LAYFZK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OHIT6GPV6NY3Z43NOFB6LAYFZK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OHIT6GPV6NY3Z43NOFB6LAYFZK","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":"abe2fafa9bc3843a7c87fa2b1629d4c84b7213cf7441e26763609d81322d96e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-01-26T00:14:11Z","title_canon_sha256":"d18d1aca232b714ef46d390d6946a9e9f6f81843a8ddfbcf3e8bc271b72a1ab2"},"schema_version":"1.0","source":{"id":"1601.06854","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06854","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06854v2","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06854","created_at":"2026-05-18T01:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"OHIT6GPV6NY3","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OHIT6GPV6NY3Z43N","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OHIT6GPV","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:3c28d1b89f4bad2cddb7915319a848a3442a310b41a35ad3d92ad3e94ec22f7b","target":"graph","created_at":"2026-05-18T01:14:17Z","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 fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgue spaces of certain bilinear multiplier operators and singular integrals.","authors_text":"David Cruz-Uribe, Virginia Naibo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-01-26T00:14:11Z","title":"Kato-Ponce inequalities on weighted and variable Lebesgue spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06854","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":"sha256:7afdacf1b72985e9c169e90e6cbb7dc72f2ae131a7603cdb7b5abfa6bc8d9e71","target":"record","created_at":"2026-05-18T01:14:17Z","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":"abe2fafa9bc3843a7c87fa2b1629d4c84b7213cf7441e26763609d81322d96e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-01-26T00:14:11Z","title_canon_sha256":"d18d1aca232b714ef46d390d6946a9e9f6f81843a8ddfbcf3e8bc271b72a1ab2"},"schema_version":"1.0","source":{"id":"1601.06854","kind":"arxiv","version":2}},"canonical_sha256":"71d13f19f5f371bcf36d7143e58305ca9cb77c5715ecf04a722f6d75372edbe2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71d13f19f5f371bcf36d7143e58305ca9cb77c5715ecf04a722f6d75372edbe2","first_computed_at":"2026-05-18T01:14:17.434781Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:17.434781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HJCdeBTsG7nOoaWXpQ689AQoWwoScURsGObVLIuS1sH/NLRrxZVYw8i3v+g8ICZQwbcoaahVi8/60v4yFv+4Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:17.435465Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.06854","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7afdacf1b72985e9c169e90e6cbb7dc72f2ae131a7603cdb7b5abfa6bc8d9e71","sha256:3c28d1b89f4bad2cddb7915319a848a3442a310b41a35ad3d92ad3e94ec22f7b"],"state_sha256":"fcb66c8113a035887e901931ad009df75618f58199f1cd78dd527415a537845e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M40twyPavyKcOh/yuO1Dk0q6fZl3Be1VtcV517EYXKaQhwamLFvu8tw1RcyvBVkyP/PQlaciiWQRJqeMJMJlAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T02:42:50.215816Z","bundle_sha256":"d68ef30c67f88faabe0d1c8a44d15e3ed18f9c131f02bb5632b0bd86b1da3c26"}}