{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4OZMWCYBU44F5OLNFELVJWYN3D","short_pith_number":"pith:4OZMWCYB","canonical_record":{"source":{"id":"1606.05206","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-16T14:38:35Z","cross_cats_sorted":[],"title_canon_sha256":"0b831cb26f809e3e5993365daf16c2f8b3b7730b0f2b1f79f95d46c5c3518917","abstract_canon_sha256":"0fe5efc03b50fa3b6490aa8976b19332df40d1b57909457f6bc6691c60d5d037"},"schema_version":"1.0"},"canonical_sha256":"e3b2cb0b01a7385eb96d291754db0dd8f2856fb858ffb3983f9a5b4cb288a9bf","source":{"kind":"arxiv","id":"1606.05206","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05206","created_at":"2026-05-18T01:12:21Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05206v1","created_at":"2026-05-18T01:12:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05206","created_at":"2026-05-18T01:12:21Z"},{"alias_kind":"pith_short_12","alias_value":"4OZMWCYBU44F","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4OZMWCYBU44F5OLN","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4OZMWCYB","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4OZMWCYBU44F5OLNFELVJWYN3D","target":"record","payload":{"canonical_record":{"source":{"id":"1606.05206","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-16T14:38:35Z","cross_cats_sorted":[],"title_canon_sha256":"0b831cb26f809e3e5993365daf16c2f8b3b7730b0f2b1f79f95d46c5c3518917","abstract_canon_sha256":"0fe5efc03b50fa3b6490aa8976b19332df40d1b57909457f6bc6691c60d5d037"},"schema_version":"1.0"},"canonical_sha256":"e3b2cb0b01a7385eb96d291754db0dd8f2856fb858ffb3983f9a5b4cb288a9bf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:21.820839Z","signature_b64":"P6pY6+rKOmh2Kbqj83MdZbq1ngZlsdlnU/OPUennPOJuViaCdm589DSLIoJ5RtwB/yy6vNk/nXvpL5ZqzTzYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3b2cb0b01a7385eb96d291754db0dd8f2856fb858ffb3983f9a5b4cb288a9bf","last_reissued_at":"2026-05-18T01:12:21.820520Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:21.820520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.05206","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":"2026-05-18T01:12:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QZW3ENxEP1EfT2BP/oyPRTAF93YINqmmcWnaAr5oCNY1cRnJxUse/36blihEB4eI1RC6pURz26qSEVMVhG9XCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T14:53:06.369866Z"},"content_sha256":"c8f2fdfa13cd4d7b149adb3542bcec51a7c384b32efda505f5a7c32d03c79a35","schema_version":"1.0","event_id":"sha256:c8f2fdfa13cd4d7b149adb3542bcec51a7c384b32efda505f5a7c32d03c79a35"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4OZMWCYBU44F5OLNFELVJWYN3D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a geometric inequality related to fractional integration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ting Chen","submitted_at":"2016-06-16T14:38:35Z","abstract_excerpt":"In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman's paper. Based on it we investigate its multilinear analogue inequalities. Combining with the Gressman's work on multilinear integral, we establish this new kind of geometric inequalities with bilinear form and multilinear form in more general settings. Moreover, in some cases we also find the best constants and optimisers for these geometric inequalities on Euclidean spaces with Lebesgue measure settings with $L^p$ bounds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05206","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":"2026-05-18T01:12:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"axHKNrj3H+Nk+ivErFUrlBOA9Hf6aUd8HJk9/sqzGVv7s3c/h2Q7sbfF3qYlYBZ3Wjtcvo2wjAT9HOuc7WCNDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T14:53:06.370211Z"},"content_sha256":"261ab75b3a6d3141a092dc5b520f0debe10ef45fa4cc18cd9aa9e54a3ed7297a","schema_version":"1.0","event_id":"sha256:261ab75b3a6d3141a092dc5b520f0debe10ef45fa4cc18cd9aa9e54a3ed7297a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4OZMWCYBU44F5OLNFELVJWYN3D/bundle.json","state_url":"https://pith.science/pith/4OZMWCYBU44F5OLNFELVJWYN3D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4OZMWCYBU44F5OLNFELVJWYN3D/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":"2026-06-09T14:53:06Z","links":{"resolver":"https://pith.science/pith/4OZMWCYBU44F5OLNFELVJWYN3D","bundle":"https://pith.science/pith/4OZMWCYBU44F5OLNFELVJWYN3D/bundle.json","state":"https://pith.science/pith/4OZMWCYBU44F5OLNFELVJWYN3D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4OZMWCYBU44F5OLNFELVJWYN3D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4OZMWCYBU44F5OLNFELVJWYN3D","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":"0fe5efc03b50fa3b6490aa8976b19332df40d1b57909457f6bc6691c60d5d037","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-16T14:38:35Z","title_canon_sha256":"0b831cb26f809e3e5993365daf16c2f8b3b7730b0f2b1f79f95d46c5c3518917"},"schema_version":"1.0","source":{"id":"1606.05206","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05206","created_at":"2026-05-18T01:12:21Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05206v1","created_at":"2026-05-18T01:12:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05206","created_at":"2026-05-18T01:12:21Z"},{"alias_kind":"pith_short_12","alias_value":"4OZMWCYBU44F","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4OZMWCYBU44F5OLN","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4OZMWCYB","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:261ab75b3a6d3141a092dc5b520f0debe10ef45fa4cc18cd9aa9e54a3ed7297a","target":"graph","created_at":"2026-05-18T01:12:21Z","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":"In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman's paper. Based on it we investigate its multilinear analogue inequalities. Combining with the Gressman's work on multilinear integral, we establish this new kind of geometric inequalities with bilinear form and multilinear form in more general settings. Moreover, in some cases we also find the best constants and optimisers for these geometric inequalities on Euclidean spaces with Lebesgue measure settings with $L^p$ bounds.","authors_text":"Ting Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-16T14:38:35Z","title":"On a geometric inequality related to fractional integration"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05206","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:c8f2fdfa13cd4d7b149adb3542bcec51a7c384b32efda505f5a7c32d03c79a35","target":"record","created_at":"2026-05-18T01:12:21Z","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":"0fe5efc03b50fa3b6490aa8976b19332df40d1b57909457f6bc6691c60d5d037","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-16T14:38:35Z","title_canon_sha256":"0b831cb26f809e3e5993365daf16c2f8b3b7730b0f2b1f79f95d46c5c3518917"},"schema_version":"1.0","source":{"id":"1606.05206","kind":"arxiv","version":1}},"canonical_sha256":"e3b2cb0b01a7385eb96d291754db0dd8f2856fb858ffb3983f9a5b4cb288a9bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3b2cb0b01a7385eb96d291754db0dd8f2856fb858ffb3983f9a5b4cb288a9bf","first_computed_at":"2026-05-18T01:12:21.820520Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:21.820520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P6pY6+rKOmh2Kbqj83MdZbq1ngZlsdlnU/OPUennPOJuViaCdm589DSLIoJ5RtwB/yy6vNk/nXvpL5ZqzTzYBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:21.820839Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.05206","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8f2fdfa13cd4d7b149adb3542bcec51a7c384b32efda505f5a7c32d03c79a35","sha256:261ab75b3a6d3141a092dc5b520f0debe10ef45fa4cc18cd9aa9e54a3ed7297a"],"state_sha256":"35e7da27f98cdcd24fb8f40de1a0aeb49419ef959bf808e09ec219b0803c9815"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aWNyoiedDP5rHXXLDLA2BIJctkh8Xli9nRrdvIu8q4Y+RfGFjyc/CMaQXR9k1eXgJxmAJZZwreuB4YC+wUGxDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T14:53:06.372144Z","bundle_sha256":"f7bdd0d8be69b7f1144675de3c0a8f15b90056a25d2260fb3cd5e185563788eb"}}