{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:GS5E2YGL42WYA5N4B3CSGE3IQC","short_pith_number":"pith:GS5E2YGL","canonical_record":{"source":{"id":"1907.09953","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-07-23T15:17:47Z","cross_cats_sorted":[],"title_canon_sha256":"3e190a09de20c88db59ce781ad66c1afb5d1fc44e264404a36f37cd2387d3185","abstract_canon_sha256":"d69cc624c956a0dadd62f07a56d7f33f2cabfff4e4975f295f11e1e9dfa8361c"},"schema_version":"1.0"},"canonical_sha256":"34ba4d60cbe6ad8075bc0ec523136880940580e88621617822dc21a646d9f3ea","source":{"kind":"arxiv","id":"1907.09953","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.09953","created_at":"2026-05-17T23:39:29Z"},{"alias_kind":"arxiv_version","alias_value":"1907.09953v2","created_at":"2026-05-17T23:39:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.09953","created_at":"2026-05-17T23:39:29Z"},{"alias_kind":"pith_short_12","alias_value":"GS5E2YGL42WY","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"GS5E2YGL42WYA5N4","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"GS5E2YGL","created_at":"2026-05-18T12:33:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:GS5E2YGL42WYA5N4B3CSGE3IQC","target":"record","payload":{"canonical_record":{"source":{"id":"1907.09953","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-07-23T15:17:47Z","cross_cats_sorted":[],"title_canon_sha256":"3e190a09de20c88db59ce781ad66c1afb5d1fc44e264404a36f37cd2387d3185","abstract_canon_sha256":"d69cc624c956a0dadd62f07a56d7f33f2cabfff4e4975f295f11e1e9dfa8361c"},"schema_version":"1.0"},"canonical_sha256":"34ba4d60cbe6ad8075bc0ec523136880940580e88621617822dc21a646d9f3ea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:29.831309Z","signature_b64":"6vgS2pmPfJpjHQBwOLQ733wIGC6NdI2A6q94rl9/KTjgG6EgAWOdy3xPjx/EVm6ARkYDVFPyHlAo0PWBi+gwAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34ba4d60cbe6ad8075bc0ec523136880940580e88621617822dc21a646d9f3ea","last_reissued_at":"2026-05-17T23:39:29.830806Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:29.830806Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.09953","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":"2026-05-17T23:39:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4LhKnvs4jx+M1COAwcSUBrfK6cxZogYIqBN2FPhoZ+tirgr5qHOGd/8R0FOZ/kKVr5FUFSkMD3aYLdXkKtrpDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:34:46.338735Z"},"content_sha256":"835fb8342ce377537d87203bf17fb3bfe84a37c87abb1c707eb9c295c8fa3a49","schema_version":"1.0","event_id":"sha256:835fb8342ce377537d87203bf17fb3bfe84a37c87abb1c707eb9c295c8fa3a49"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:GS5E2YGL42WYA5N4B3CSGE3IQC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Elodie Pozzi, Qingyan Wu, Zunwei Fu","submitted_at":"2019-07-23T15:17:47Z","abstract_excerpt":"We establish the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows the local version of commutators also has an independent interest."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09953","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":"2026-05-17T23:39:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5yUWHMRPz1xLar05alQpVkXulTKRjn7zHR6fOjz0LrsmlFNuoWgoid9Zd+XBlfJK3K8sW3XpaWmKTqSoY6E8CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:34:46.339074Z"},"content_sha256":"bb545e5026fa8facd20d6e278510fac0edcf93f471bc3b0b026dc1eb8845ebbb","schema_version":"1.0","event_id":"sha256:bb545e5026fa8facd20d6e278510fac0edcf93f471bc3b0b026dc1eb8845ebbb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GS5E2YGL42WYA5N4B3CSGE3IQC/bundle.json","state_url":"https://pith.science/pith/GS5E2YGL42WYA5N4B3CSGE3IQC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GS5E2YGL42WYA5N4B3CSGE3IQC/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":"2026-06-05T13:34:46Z","links":{"resolver":"https://pith.science/pith/GS5E2YGL42WYA5N4B3CSGE3IQC","bundle":"https://pith.science/pith/GS5E2YGL42WYA5N4B3CSGE3IQC/bundle.json","state":"https://pith.science/pith/GS5E2YGL42WYA5N4B3CSGE3IQC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GS5E2YGL42WYA5N4B3CSGE3IQC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:GS5E2YGL42WYA5N4B3CSGE3IQC","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":"d69cc624c956a0dadd62f07a56d7f33f2cabfff4e4975f295f11e1e9dfa8361c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-07-23T15:17:47Z","title_canon_sha256":"3e190a09de20c88db59ce781ad66c1afb5d1fc44e264404a36f37cd2387d3185"},"schema_version":"1.0","source":{"id":"1907.09953","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.09953","created_at":"2026-05-17T23:39:29Z"},{"alias_kind":"arxiv_version","alias_value":"1907.09953v2","created_at":"2026-05-17T23:39:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.09953","created_at":"2026-05-17T23:39:29Z"},{"alias_kind":"pith_short_12","alias_value":"GS5E2YGL42WY","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"GS5E2YGL42WYA5N4","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"GS5E2YGL","created_at":"2026-05-18T12:33:18Z"}],"graph_snapshots":[{"event_id":"sha256:bb545e5026fa8facd20d6e278510fac0edcf93f471bc3b0b026dc1eb8845ebbb","target":"graph","created_at":"2026-05-17T23:39:29Z","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 establish the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows the local version of commutators also has an independent interest.","authors_text":"Elodie Pozzi, Qingyan Wu, Zunwei Fu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-07-23T15:17:47Z","title":"Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09953","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":"sha256:835fb8342ce377537d87203bf17fb3bfe84a37c87abb1c707eb9c295c8fa3a49","target":"record","created_at":"2026-05-17T23:39:29Z","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":"d69cc624c956a0dadd62f07a56d7f33f2cabfff4e4975f295f11e1e9dfa8361c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-07-23T15:17:47Z","title_canon_sha256":"3e190a09de20c88db59ce781ad66c1afb5d1fc44e264404a36f37cd2387d3185"},"schema_version":"1.0","source":{"id":"1907.09953","kind":"arxiv","version":2}},"canonical_sha256":"34ba4d60cbe6ad8075bc0ec523136880940580e88621617822dc21a646d9f3ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"34ba4d60cbe6ad8075bc0ec523136880940580e88621617822dc21a646d9f3ea","first_computed_at":"2026-05-17T23:39:29.830806Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:29.830806Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6vgS2pmPfJpjHQBwOLQ733wIGC6NdI2A6q94rl9/KTjgG6EgAWOdy3xPjx/EVm6ARkYDVFPyHlAo0PWBi+gwAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:29.831309Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.09953","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:835fb8342ce377537d87203bf17fb3bfe84a37c87abb1c707eb9c295c8fa3a49","sha256:bb545e5026fa8facd20d6e278510fac0edcf93f471bc3b0b026dc1eb8845ebbb"],"state_sha256":"deda7fcf30096f69e8b927d0bb83b6516515e324603dccd2b4822d27e45cd2e5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tIuThwscn8PHNDRtKmkx4Hvd8e2Lm5nT+sLraw+7e/WnCPtY5HrXjYR8xYhsV3TPspch02Gxyg7YkUByQn+eCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T13:34:46.341325Z","bundle_sha256":"4f33903d90d5fa8e0d347bfc6db9e690a6d5ec3791418b61597d6b0f0a58e4ec"}}