{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:5SQSJEPGEROELLEJNFHQQWCG3F","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":"0645a32f6c7836d3e36b1893aac7614f34857f1838947ab28450ea2707d64499","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-10-30T21:29:22Z","title_canon_sha256":"cb1808d69d617849394a30777e0e83b03cd060ba4d823c4855fb59028a8476d5"},"schema_version":"1.0","source":{"id":"2411.05810","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.05810","created_at":"2026-07-05T09:34:39Z"},{"alias_kind":"arxiv_version","alias_value":"2411.05810v2","created_at":"2026-07-05T09:34:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.05810","created_at":"2026-07-05T09:34:39Z"},{"alias_kind":"pith_short_12","alias_value":"5SQSJEPGEROE","created_at":"2026-07-05T09:34:39Z"},{"alias_kind":"pith_short_16","alias_value":"5SQSJEPGEROELLEJ","created_at":"2026-07-05T09:34:39Z"},{"alias_kind":"pith_short_8","alias_value":"5SQSJEPG","created_at":"2026-07-05T09:34:39Z"}],"graph_snapshots":[{"event_id":"sha256:782e3794fb8795ea16abbaf8701a5fb5f5d7f13fe01c9f36e6011c5e52ede7c8","target":"graph","created_at":"2026-07-05T09:34:39Z","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/2411.05810/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This article is devoted to the study of the Schatten class membership of commutators involving singular integral operators. We utilize martingale paraproducts and Hyt\\\"{o}nen's dyadic martingale technique to obtain sufficient conditions on the weak-type and strong-type Schatten class membership of commutators in terms of Sobolev spaces and Besov spaces respectively.\n  We also establish the complex median method, which is applicable to complex-valued functions. We apply it to get the optimal necessary conditions on the weak-type and strong-type Schatten class membership of commutators associate","authors_text":"Hao Zhang, Zhenguo Wei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-10-30T21:29:22Z","title":"Complex median method and Schatten class membership of commutators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.05810","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:b8f6125d50d0de2c205b3ae3a9319f7f7909be50463b718b4bda9df914167c48","target":"record","created_at":"2026-07-05T09:34:39Z","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":"0645a32f6c7836d3e36b1893aac7614f34857f1838947ab28450ea2707d64499","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-10-30T21:29:22Z","title_canon_sha256":"cb1808d69d617849394a30777e0e83b03cd060ba4d823c4855fb59028a8476d5"},"schema_version":"1.0","source":{"id":"2411.05810","kind":"arxiv","version":2}},"canonical_sha256":"eca12491e6245c45ac89694f085846d964d446de0fb6bd0692f1826c3a973416","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eca12491e6245c45ac89694f085846d964d446de0fb6bd0692f1826c3a973416","first_computed_at":"2026-07-05T09:34:39.643641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:34:39.643641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pdSzhaOEoCt2lUJWUFh9tlWgGcl1JwX/L3bRWRl6doa+HZhG9r1uwmVPjKqqDVvRFL/wRzf+4zYQuPygZ5ssAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:34:39.644121Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.05810","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8f6125d50d0de2c205b3ae3a9319f7f7909be50463b718b4bda9df914167c48","sha256:782e3794fb8795ea16abbaf8701a5fb5f5d7f13fe01c9f36e6011c5e52ede7c8"],"state_sha256":"45be1f22d7cbf984462f8ceb50f1c2fad8897ad48c2b04b9dea860dbbc21cc58"}