{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:AMRL7LPH434K5ZMWO424RDK4RH","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":"bfa35058d3c9a969ea221cf16bf226430607b090d06c6b47ddc3ed3a374848b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-02-21T02:10:28Z","title_canon_sha256":"3cc70be1e80a5cd521333cb8e9f8f73f54089b2ac76be780d2ccea4991c303f0"},"schema_version":"1.0","source":{"id":"1602.06486","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.06486","created_at":"2026-05-18T01:20:00Z"},{"alias_kind":"arxiv_version","alias_value":"1602.06486v2","created_at":"2026-05-18T01:20:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.06486","created_at":"2026-05-18T01:20:00Z"},{"alias_kind":"pith_short_12","alias_value":"AMRL7LPH434K","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"AMRL7LPH434K5ZMW","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"AMRL7LPH","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:aa024d484f7d0757cc34650ffb0c03f6b0493287d6dec99b0e9777b27ca2448b","target":"graph","created_at":"2026-05-18T01:20:00Z","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":"This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\\mathcal{M}_{\\alpha}$ and the multilinear fractional integral operator $\\mathcal{I}_{\\alpha}$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\\mathcal{M}_\\alpha$ and $\\mathcal{I}_\\alpha$ will be given.","authors_text":"Mingming Cao, Qingying Xue","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-02-21T02:10:28Z","title":"Two-weight entropy Boundedness of Multilinear Fractional Type Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06486","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":"sha256:89aae8d5ce42f15cf42dcaff1be83d6ce40a1b2ecf81494c9a77f360937a5143","target":"record","created_at":"2026-05-18T01:20:00Z","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":"bfa35058d3c9a969ea221cf16bf226430607b090d06c6b47ddc3ed3a374848b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-02-21T02:10:28Z","title_canon_sha256":"3cc70be1e80a5cd521333cb8e9f8f73f54089b2ac76be780d2ccea4991c303f0"},"schema_version":"1.0","source":{"id":"1602.06486","kind":"arxiv","version":2}},"canonical_sha256":"0322bfade7e6f8aee5967735c88d5c89c04efc3ad3db2bcf477debc9fd4cd4ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0322bfade7e6f8aee5967735c88d5c89c04efc3ad3db2bcf477debc9fd4cd4ed","first_computed_at":"2026-05-18T01:20:00.832968Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:00.832968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xx38F1sMp09Mc3gdUtdGoEQUXRm9gXUYJFWiD/A/fEroSstGnDPKb4IU8LY2qvwPCtDsU4LUZmgZVDChHZKUBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:00.833745Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.06486","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89aae8d5ce42f15cf42dcaff1be83d6ce40a1b2ecf81494c9a77f360937a5143","sha256:aa024d484f7d0757cc34650ffb0c03f6b0493287d6dec99b0e9777b27ca2448b"],"state_sha256":"1fb6d0b14f6a87526bc2de09024f69d28ffc0a8191de2d8063beb092f50e8683"}