{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NU2BFQJTUCUZJSFGZTKSU7CM6B","short_pith_number":"pith:NU2BFQJT","canonical_record":{"source":{"id":"2607.18175","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-20T17:13:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"1a6a4b7eb092938f5a443a6d178214a1afc558210737eaff765a64d29bf65086","abstract_canon_sha256":"336582388e90392ad4a891aae1d96b18b2199c81579b8f1c400674fd3528796a"},"schema_version":"1.0"},"canonical_sha256":"6d3412c133a0a994c8a6ccd52a7c4cf07c232e4ea50a35b166acfbb969648db0","source":{"kind":"arxiv","id":"2607.18175","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.18175","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"arxiv_version","alias_value":"2607.18175v1","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.18175","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"pith_short_12","alias_value":"NU2BFQJTUCUZ","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"pith_short_16","alias_value":"NU2BFQJTUCUZJSFG","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"pith_short_8","alias_value":"NU2BFQJT","created_at":"2026-07-21T02:22:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NU2BFQJTUCUZJSFGZTKSU7CM6B","target":"record","payload":{"canonical_record":{"source":{"id":"2607.18175","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-20T17:13:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"1a6a4b7eb092938f5a443a6d178214a1afc558210737eaff765a64d29bf65086","abstract_canon_sha256":"336582388e90392ad4a891aae1d96b18b2199c81579b8f1c400674fd3528796a"},"schema_version":"1.0"},"canonical_sha256":"6d3412c133a0a994c8a6ccd52a7c4cf07c232e4ea50a35b166acfbb969648db0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T02:22:26.097782Z","signature_b64":"5pEI+hu5kHp3iNncPGPQ91us9kEXYMSXkdcwYbvGQXyLuJOIeut38RUh9NsT0qWbtmaDw/6QFwz8X60w9YCaCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d3412c133a0a994c8a6ccd52a7c4cf07c232e4ea50a35b166acfbb969648db0","last_reissued_at":"2026-07-21T02:22:26.096879Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T02:22:26.096879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.18175","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":"2026-07-21T02:22:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FyoZNgQBEJL3o7jT61+bFzPSQaJJZh0CKf417bWjrekoRdTrDs9rJ46esUqxE1fckE3SathsDEmTWeDQqA12Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T14:25:25.507977Z"},"content_sha256":"b87d3b459a0d9ce424b54984172f085016db7427becbbc72c4521cf9fc35d4c7","schema_version":"1.0","event_id":"sha256:b87d3b459a0d9ce424b54984172f085016db7427becbbc72c4521cf9fc35d4c7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NU2BFQJTUCUZJSFGZTKSU7CM6B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On off-diagonal operators in matrix-weighted spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Aapo Laukkarinen, David Cruz-Uribe, Kabe Moen","submitted_at":"2026-07-20T17:13:37Z","abstract_excerpt":"In this paper we prove matrix-weighted inequalities for fractional operators and their commutators. We do so by developing the theory of convex body domination for such operators. Using this approach we prove quantitative estimates for the fractional integral operator (or Riesz potential) and its commutators, and prove matrix-weighted Gagliardo-Nirenberg-Sobolev inequalities for vector-valued functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18175","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.18175/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-21T02:22:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kcPp2sQVuGadt2EBf1p3T6p3sOuOBq7D8vXHbNfLmbQFof0FnwugnzRY9tnUzkoXxytaukDHTETAWocBj06/AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T14:25:25.508265Z"},"content_sha256":"92fb81322334a2c23a3d69566d7f44985b4819b1848ae37c40982b98dad3e9f4","schema_version":"1.0","event_id":"sha256:92fb81322334a2c23a3d69566d7f44985b4819b1848ae37c40982b98dad3e9f4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NU2BFQJTUCUZJSFGZTKSU7CM6B/bundle.json","state_url":"https://pith.science/pith/NU2BFQJTUCUZJSFGZTKSU7CM6B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NU2BFQJTUCUZJSFGZTKSU7CM6B/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":"2026-08-05T14:25:25Z","links":{"resolver":"https://pith.science/pith/NU2BFQJTUCUZJSFGZTKSU7CM6B","bundle":"https://pith.science/pith/NU2BFQJTUCUZJSFGZTKSU7CM6B/bundle.json","state":"https://pith.science/pith/NU2BFQJTUCUZJSFGZTKSU7CM6B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NU2BFQJTUCUZJSFGZTKSU7CM6B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NU2BFQJTUCUZJSFGZTKSU7CM6B","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":"336582388e90392ad4a891aae1d96b18b2199c81579b8f1c400674fd3528796a","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-20T17:13:37Z","title_canon_sha256":"1a6a4b7eb092938f5a443a6d178214a1afc558210737eaff765a64d29bf65086"},"schema_version":"1.0","source":{"id":"2607.18175","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.18175","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"arxiv_version","alias_value":"2607.18175v1","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.18175","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"pith_short_12","alias_value":"NU2BFQJTUCUZ","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"pith_short_16","alias_value":"NU2BFQJTUCUZJSFG","created_at":"2026-07-21T02:22:26Z"},{"alias_kind":"pith_short_8","alias_value":"NU2BFQJT","created_at":"2026-07-21T02:22:26Z"}],"graph_snapshots":[{"event_id":"sha256:92fb81322334a2c23a3d69566d7f44985b4819b1848ae37c40982b98dad3e9f4","target":"graph","created_at":"2026-07-21T02:22:26Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.18175/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we prove matrix-weighted inequalities for fractional operators and their commutators. We do so by developing the theory of convex body domination for such operators. Using this approach we prove quantitative estimates for the fractional integral operator (or Riesz potential) and its commutators, and prove matrix-weighted Gagliardo-Nirenberg-Sobolev inequalities for vector-valued functions.","authors_text":"Aapo Laukkarinen, David Cruz-Uribe, Kabe Moen","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-20T17:13:37Z","title":"On off-diagonal operators in matrix-weighted spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18175","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:b87d3b459a0d9ce424b54984172f085016db7427becbbc72c4521cf9fc35d4c7","target":"record","created_at":"2026-07-21T02:22:26Z","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":"336582388e90392ad4a891aae1d96b18b2199c81579b8f1c400674fd3528796a","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-20T17:13:37Z","title_canon_sha256":"1a6a4b7eb092938f5a443a6d178214a1afc558210737eaff765a64d29bf65086"},"schema_version":"1.0","source":{"id":"2607.18175","kind":"arxiv","version":1}},"canonical_sha256":"6d3412c133a0a994c8a6ccd52a7c4cf07c232e4ea50a35b166acfbb969648db0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d3412c133a0a994c8a6ccd52a7c4cf07c232e4ea50a35b166acfbb969648db0","first_computed_at":"2026-07-21T02:22:26.096879Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-21T02:22:26.096879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5pEI+hu5kHp3iNncPGPQ91us9kEXYMSXkdcwYbvGQXyLuJOIeut38RUh9NsT0qWbtmaDw/6QFwz8X60w9YCaCQ==","signature_status":"signed_v1","signed_at":"2026-07-21T02:22:26.097782Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.18175","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b87d3b459a0d9ce424b54984172f085016db7427becbbc72c4521cf9fc35d4c7","sha256:92fb81322334a2c23a3d69566d7f44985b4819b1848ae37c40982b98dad3e9f4"],"state_sha256":"c398ce3ddaec8f6cf674f6551aea72c84a7f0b22f7083629326d4797612a2845"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5IZG6mArhaEiomOakKxEtGGISSOV1JW1ngLcM3mrlZTlQQ/KFIJ1bJbCRWgDdq19QzmrZseCV8ekX7Zfum7/Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T14:25:25.510513Z","bundle_sha256":"3d8e9183b7af46766cc87c7864ded0d397713c83ebac96412c9a3c9d4d571815"}}