{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XEQQ3GGQWDHMW57V7BC3N7B6KQ","short_pith_number":"pith:XEQQ3GGQ","canonical_record":{"source":{"id":"1612.01116","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-12-04T13:17:21Z","cross_cats_sorted":[],"title_canon_sha256":"e8fab266f25bfd5002e1917d8c97c296ce3e2eb7f6378951681f51f2aad0a58b","abstract_canon_sha256":"a149c0ae97b6cd65e956bfc85ebc12604574193f6b12f34a529069545eb89ab7"},"schema_version":"1.0"},"canonical_sha256":"b9210d98d0b0cecb77f5f845b6fc3e541417d68db6f91078b797307e8e2f05a4","source":{"kind":"arxiv","id":"1612.01116","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.01116","created_at":"2026-05-18T00:55:53Z"},{"alias_kind":"arxiv_version","alias_value":"1612.01116v1","created_at":"2026-05-18T00:55:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.01116","created_at":"2026-05-18T00:55:53Z"},{"alias_kind":"pith_short_12","alias_value":"XEQQ3GGQWDHM","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XEQQ3GGQWDHMW57V","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XEQQ3GGQ","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XEQQ3GGQWDHMW57V7BC3N7B6KQ","target":"record","payload":{"canonical_record":{"source":{"id":"1612.01116","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-12-04T13:17:21Z","cross_cats_sorted":[],"title_canon_sha256":"e8fab266f25bfd5002e1917d8c97c296ce3e2eb7f6378951681f51f2aad0a58b","abstract_canon_sha256":"a149c0ae97b6cd65e956bfc85ebc12604574193f6b12f34a529069545eb89ab7"},"schema_version":"1.0"},"canonical_sha256":"b9210d98d0b0cecb77f5f845b6fc3e541417d68db6f91078b797307e8e2f05a4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:53.318362Z","signature_b64":"XOkj5M9Alozx0g98sPZeood9JQ4Le5igU1JKgZQGDlqKti4yuZaP1pzoGSdY2iKu7w+CaHG3F5a8+7ZFRA7zAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9210d98d0b0cecb77f5f845b6fc3e541417d68db6f91078b797307e8e2f05a4","last_reissued_at":"2026-05-18T00:55:53.317920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:53.317920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.01116","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":"2026-05-18T00:55:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JRtRsedufkqx4E8U93n0H3H0eSDS4Z1jn+NOt1p7Kv31OO6DWtMXTiAdjZcKOpXZnfx6/xxF3nG/wvX0XnViBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:39:32.348939Z"},"content_sha256":"93e9c2e3a6b542f0a2bf849f966621e852593caa6458c4e8e62290afb2170eb2","schema_version":"1.0","event_id":"sha256:93e9c2e3a6b542f0a2bf849f966621e852593caa6458c4e8e62290afb2170eb2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XEQQ3GGQWDHMW57V7BC3N7B6KQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Characterization of CMO via compactness of the commutators of bilinear fractional integral operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dinghuai Wang, Jiang Zhou, Wenyi Chen","submitted_at":"2016-12-04T13:17:21Z","abstract_excerpt":"Let $I_{\\alpha}$ be the bilinear fractional integral operator, $B_{\\alpha}$ be a more singular family of bilinear fractional integral operators and $\\vec{b}=(b,b)$. B\\'{e}nyi et al. in \\cite{B1} showed that if $b\\in {\\rm CMO}$, the {\\rm BMO}-closure of $C^{\\infty}_{c}(\\mathbb{R}^n)$, the commutator $[b,B_{\\alpha}]_{i}(i=1,2)$ is a separately compact operator. In this paper, it is proved that $b\\in {\\rm CMO}$ is necessary for $[b,B_{\\alpha}]_{i}(i=1,2)$ is a compact operator. Also, the authors characterize the compactness of the {\\bf iterated} commutator $[\\Pi\\vec{b},I_{\\alpha}]$ of bilinear fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01116","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":""},"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":"2026-05-18T00:55:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cnJiHbsKCVvoUHnrhXQRPj+rZ9x7MOD0OcdPVlgsHD1Il8t+aOB1x2zoheBvI52xsCbLVg1u0DUbA2qiyH2yBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:39:32.349666Z"},"content_sha256":"396081dcacdd747a19bd4df880152978ee8fadf6e53e4e6b6035d989584fda85","schema_version":"1.0","event_id":"sha256:396081dcacdd747a19bd4df880152978ee8fadf6e53e4e6b6035d989584fda85"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XEQQ3GGQWDHMW57V7BC3N7B6KQ/bundle.json","state_url":"https://pith.science/pith/XEQQ3GGQWDHMW57V7BC3N7B6KQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XEQQ3GGQWDHMW57V7BC3N7B6KQ/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":"2026-06-09T06:39:32Z","links":{"resolver":"https://pith.science/pith/XEQQ3GGQWDHMW57V7BC3N7B6KQ","bundle":"https://pith.science/pith/XEQQ3GGQWDHMW57V7BC3N7B6KQ/bundle.json","state":"https://pith.science/pith/XEQQ3GGQWDHMW57V7BC3N7B6KQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XEQQ3GGQWDHMW57V7BC3N7B6KQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XEQQ3GGQWDHMW57V7BC3N7B6KQ","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":"a149c0ae97b6cd65e956bfc85ebc12604574193f6b12f34a529069545eb89ab7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-12-04T13:17:21Z","title_canon_sha256":"e8fab266f25bfd5002e1917d8c97c296ce3e2eb7f6378951681f51f2aad0a58b"},"schema_version":"1.0","source":{"id":"1612.01116","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.01116","created_at":"2026-05-18T00:55:53Z"},{"alias_kind":"arxiv_version","alias_value":"1612.01116v1","created_at":"2026-05-18T00:55:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.01116","created_at":"2026-05-18T00:55:53Z"},{"alias_kind":"pith_short_12","alias_value":"XEQQ3GGQWDHM","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XEQQ3GGQWDHMW57V","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XEQQ3GGQ","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:396081dcacdd747a19bd4df880152978ee8fadf6e53e4e6b6035d989584fda85","target":"graph","created_at":"2026-05-18T00:55:53Z","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":"Let $I_{\\alpha}$ be the bilinear fractional integral operator, $B_{\\alpha}$ be a more singular family of bilinear fractional integral operators and $\\vec{b}=(b,b)$. B\\'{e}nyi et al. in \\cite{B1} showed that if $b\\in {\\rm CMO}$, the {\\rm BMO}-closure of $C^{\\infty}_{c}(\\mathbb{R}^n)$, the commutator $[b,B_{\\alpha}]_{i}(i=1,2)$ is a separately compact operator. In this paper, it is proved that $b\\in {\\rm CMO}$ is necessary for $[b,B_{\\alpha}]_{i}(i=1,2)$ is a compact operator. Also, the authors characterize the compactness of the {\\bf iterated} commutator $[\\Pi\\vec{b},I_{\\alpha}]$ of bilinear fr","authors_text":"Dinghuai Wang, Jiang Zhou, Wenyi Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-12-04T13:17:21Z","title":"Characterization of CMO via compactness of the commutators of bilinear fractional integral operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01116","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":"sha256:93e9c2e3a6b542f0a2bf849f966621e852593caa6458c4e8e62290afb2170eb2","target":"record","created_at":"2026-05-18T00:55:53Z","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":"a149c0ae97b6cd65e956bfc85ebc12604574193f6b12f34a529069545eb89ab7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-12-04T13:17:21Z","title_canon_sha256":"e8fab266f25bfd5002e1917d8c97c296ce3e2eb7f6378951681f51f2aad0a58b"},"schema_version":"1.0","source":{"id":"1612.01116","kind":"arxiv","version":1}},"canonical_sha256":"b9210d98d0b0cecb77f5f845b6fc3e541417d68db6f91078b797307e8e2f05a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9210d98d0b0cecb77f5f845b6fc3e541417d68db6f91078b797307e8e2f05a4","first_computed_at":"2026-05-18T00:55:53.317920Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:53.317920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XOkj5M9Alozx0g98sPZeood9JQ4Le5igU1JKgZQGDlqKti4yuZaP1pzoGSdY2iKu7w+CaHG3F5a8+7ZFRA7zAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:53.318362Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.01116","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93e9c2e3a6b542f0a2bf849f966621e852593caa6458c4e8e62290afb2170eb2","sha256:396081dcacdd747a19bd4df880152978ee8fadf6e53e4e6b6035d989584fda85"],"state_sha256":"8b3ba93c1e5b7dcd3ddf4924b2b67638c5b2a50ccec288eea921605185093ce2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v7+FS95x6hZXnn0ENHqlE+bhnOPmyV530qkgdCq7pXa7hoT89bDpOdiw4PxFk1TZmhhHuhoDq1v6KwDe+feyBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T06:39:32.353350Z","bundle_sha256":"3fcc45ce0c0ba36668c39c872aaa643ee400e9781d03fe758c1159afebd25b43"}}