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The commutators in the $i$-$th$ entry and the iterated commutators of $T_*$ are defined by $$ T_{\\ast,b,1}(f,g)(x)=\\sup_{\\delta>0}\\bigg|\\iint_{|x-y|+|x-z|>\\delta}K(x,y,z)(b(y)-b(x))f(y)g(z)dydz\\bigg|, $$ $$T_{\\ast,b,2}(f,g)(x)=\\sup_{\\delta>0}\\bigg|\\iint_{|x-y|+|x-z|>\\delta}K(x,y,z)(b(z)-b(x))f(y)g(z)dydz\\bigg|,$$ \\begin{align*} T_{\\ast,(b_1,b_2)}(f,g)(x)=\\sup\\limits_{\\delta>0}\\bigg|\\iint_{|x-y|+|x-z|>\\delta} K(x,y,z)(b_1(y)-b_1(x))(b_2(z)-b_2(x))f(y)g(z)dydz\\bigg|. \\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1310.5787","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-10-22T03:12:51Z","cross_cats_sorted":[],"title_canon_sha256":"dbf3302a2359d99d1c5e7599ebc418fa270ff7205629fef670836199d651268b","abstract_canon_sha256":"66be1c46f34159d37d55fcd4ed2f456ff4dbadf0ee72f6ef35a9d05a88760b26"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:48.226365Z","signature_b64":"rfem22M4fCaZymglmRTM1CrpPU44XQWcs1ZcB5By5lBNAyIFiXEorGiQtxnM45YGmqIRGZ2UMRWR1cFq9z/bDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b3df48abaf7a05a9bc0845b3cf887029b17d9f4dcd82ad5a32458c088b6d7c3","last_reissued_at":"2026-05-18T03:04:48.225682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:48.225682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Compactness of commutators of bilinear maximal Calder\\'{o}n-Zygmund singular integral operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Qingying Xue, Ting Mei, Yong Ding","submitted_at":"2013-10-22T03:12:51Z","abstract_excerpt":"Let $T$ be a bilinear Calder\\'{o}n-Zygmund singular integral operator and $T_*$ be its corresponding truncated maximal operator. The commutators in the $i$-$th$ entry and the iterated commutators of $T_*$ are defined by $$ T_{\\ast,b,1}(f,g)(x)=\\sup_{\\delta>0}\\bigg|\\iint_{|x-y|+|x-z|>\\delta}K(x,y,z)(b(y)-b(x))f(y)g(z)dydz\\bigg|, $$ $$T_{\\ast,b,2}(f,g)(x)=\\sup_{\\delta>0}\\bigg|\\iint_{|x-y|+|x-z|>\\delta}K(x,y,z)(b(z)-b(x))f(y)g(z)dydz\\bigg|,$$ \\begin{align*} T_{\\ast,(b_1,b_2)}(f,g)(x)=\\sup\\limits_{\\delta>0}\\bigg|\\iint_{|x-y|+|x-z|>\\delta} K(x,y,z)(b_1(y)-b_1(x))(b_2(z)-b_2(x))f(y)g(z)dydz\\bigg|. \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.5787","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1310.5787","created_at":"2026-05-18T03:04:48.225810+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.5787v3","created_at":"2026-05-18T03:04:48.225810+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.5787","created_at":"2026-05-18T03:04:48.225810+00:00"},{"alias_kind":"pith_short_12","alias_value":"DM67JCV266QF","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"DM67JCV266QFVG6A","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"DM67JCV2","created_at":"2026-05-18T12:27:43.054852+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK","json":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK.json","graph_json":"https://pith.science/api/pith-number/DM67JCV266QFVG6AQRNTZ6EHAK/graph.json","events_json":"https://pith.science/api/pith-number/DM67JCV266QFVG6AQRNTZ6EHAK/events.json","paper":"https://pith.science/paper/DM67JCV2"},"agent_actions":{"view_html":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK","download_json":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK.json","view_paper":"https://pith.science/paper/DM67JCV2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.5787&json=true","fetch_graph":"https://pith.science/api/pith-number/DM67JCV266QFVG6AQRNTZ6EHAK/graph.json","fetch_events":"https://pith.science/api/pith-number/DM67JCV266QFVG6AQRNTZ6EHAK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK/action/storage_attestation","attest_author":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK/action/author_attestation","sign_citation":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK/action/citation_signature","submit_replication":"https://pith.science/pith/DM67JCV266QFVG6AQRNTZ6EHAK/action/replication_record"}},"created_at":"2026-05-18T03:04:48.225810+00:00","updated_at":"2026-05-18T03:04:48.225810+00:00"}