{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:I36YJLTUA5IQVIN2CPKNNMV2KQ","short_pith_number":"pith:I36YJLTU","canonical_record":{"source":{"id":"1602.00178","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-31T00:39:32Z","cross_cats_sorted":[],"title_canon_sha256":"be3d5c03bd1f926e711cbeaead3d1353f618bed89e2c14356c0dded4657996dc","abstract_canon_sha256":"55871d5881d05e368643d2e4680117ca3deb3ea95a81e7981c4e43e2c1d73f70"},"schema_version":"1.0"},"canonical_sha256":"46fd84ae7407510aa1ba13d4d6b2ba540bb71201c378693de20ddc35ad3e240b","source":{"kind":"arxiv","id":"1602.00178","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.00178","created_at":"2026-05-18T00:10:30Z"},{"alias_kind":"arxiv_version","alias_value":"1602.00178v4","created_at":"2026-05-18T00:10:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00178","created_at":"2026-05-18T00:10:30Z"},{"alias_kind":"pith_short_12","alias_value":"I36YJLTUA5IQ","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"I36YJLTUA5IQVIN2","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"I36YJLTU","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:I36YJLTUA5IQVIN2CPKNNMV2KQ","target":"record","payload":{"canonical_record":{"source":{"id":"1602.00178","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-31T00:39:32Z","cross_cats_sorted":[],"title_canon_sha256":"be3d5c03bd1f926e711cbeaead3d1353f618bed89e2c14356c0dded4657996dc","abstract_canon_sha256":"55871d5881d05e368643d2e4680117ca3deb3ea95a81e7981c4e43e2c1d73f70"},"schema_version":"1.0"},"canonical_sha256":"46fd84ae7407510aa1ba13d4d6b2ba540bb71201c378693de20ddc35ad3e240b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:30.950194Z","signature_b64":"9YaStJ6Iqt0tqwCLXZD190K32s07/jzCmTryLB66uw+mtfXQjUGJKU5HhjCKZUEPSoiWhEwvecSldkSwJSlEBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46fd84ae7407510aa1ba13d4d6b2ba540bb71201c378693de20ddc35ad3e240b","last_reissued_at":"2026-05-18T00:10:30.949641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:30.949641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.00178","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"73aoNEE50TZfCXNtlhGNYsWkYsS03CEvgPGObqY0BPIzkyCEKj2OUya8Ul6ilKnoeTVHb/Hqh6RtJhmnCy+sDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T13:43:38.540441Z"},"content_sha256":"49ace5463f26fd0ad8763432fa9989c05391250fdf7b750187e74600de2713c1","schema_version":"1.0","event_id":"sha256:49ace5463f26fd0ad8763432fa9989c05391250fdf7b750187e74600de2713c1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:I36YJLTUA5IQVIN2CPKNNMV2KQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal exponents for Hardy--Littlewood inequalities for $m$-linear operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"D. M. Serrano-Rodr\\'iguez, D. N\\'u\\~nez-Alarc\\'on, D. Pellegrino, R. M. Aron","submitted_at":"2016-01-31T00:39:32Z","abstract_excerpt":"The Hardy--Littlewood inequalities on $\\ell _{p}$ spaces provide optimal exponents for some classes of inequalities for bilinear forms on $\\ell _{p}$ spaces. In this paper we investigate in detail the exponents involved in Hardy--Littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches. Our first main result asserts that for $q_{1},...,q_{m}>0$ and an infinite-dimensional Banach space $Y$ attaining its cotype $\\cot Y$, if \\begin{equation*} \\frac{1}{p_{1}}+...+\\frac{1}{p_{m}}<\\frac{1}{\\cot Y}, \\end{equation*} then the following assertion"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00178","kind":"arxiv","version":4},"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":"2026-05-18T00:10:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/TWicbzOamViSvzlzvieFKYFdKk1AceAl2M2qZaZsPX+NoCcscdsA/4gOlcXdTPN2OSGOdbLDjRDhcHd8p/wAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T13:43:38.540793Z"},"content_sha256":"95b765436a3acc86d5f1582630e57a4956b8b1acafc89bba49fdbc70d594556b","schema_version":"1.0","event_id":"sha256:95b765436a3acc86d5f1582630e57a4956b8b1acafc89bba49fdbc70d594556b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I36YJLTUA5IQVIN2CPKNNMV2KQ/bundle.json","state_url":"https://pith.science/pith/I36YJLTUA5IQVIN2CPKNNMV2KQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I36YJLTUA5IQVIN2CPKNNMV2KQ/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":"2026-06-25T13:43:38Z","links":{"resolver":"https://pith.science/pith/I36YJLTUA5IQVIN2CPKNNMV2KQ","bundle":"https://pith.science/pith/I36YJLTUA5IQVIN2CPKNNMV2KQ/bundle.json","state":"https://pith.science/pith/I36YJLTUA5IQVIN2CPKNNMV2KQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I36YJLTUA5IQVIN2CPKNNMV2KQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:I36YJLTUA5IQVIN2CPKNNMV2KQ","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":"55871d5881d05e368643d2e4680117ca3deb3ea95a81e7981c4e43e2c1d73f70","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-31T00:39:32Z","title_canon_sha256":"be3d5c03bd1f926e711cbeaead3d1353f618bed89e2c14356c0dded4657996dc"},"schema_version":"1.0","source":{"id":"1602.00178","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.00178","created_at":"2026-05-18T00:10:30Z"},{"alias_kind":"arxiv_version","alias_value":"1602.00178v4","created_at":"2026-05-18T00:10:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00178","created_at":"2026-05-18T00:10:30Z"},{"alias_kind":"pith_short_12","alias_value":"I36YJLTUA5IQ","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"I36YJLTUA5IQVIN2","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"I36YJLTU","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:95b765436a3acc86d5f1582630e57a4956b8b1acafc89bba49fdbc70d594556b","target":"graph","created_at":"2026-05-18T00:10:30Z","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":"The Hardy--Littlewood inequalities on $\\ell _{p}$ spaces provide optimal exponents for some classes of inequalities for bilinear forms on $\\ell _{p}$ spaces. In this paper we investigate in detail the exponents involved in Hardy--Littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches. Our first main result asserts that for $q_{1},...,q_{m}>0$ and an infinite-dimensional Banach space $Y$ attaining its cotype $\\cot Y$, if \\begin{equation*} \\frac{1}{p_{1}}+...+\\frac{1}{p_{m}}<\\frac{1}{\\cot Y}, \\end{equation*} then the following assertion","authors_text":"D. M. Serrano-Rodr\\'iguez, D. N\\'u\\~nez-Alarc\\'on, D. Pellegrino, R. M. Aron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-31T00:39:32Z","title":"Optimal exponents for Hardy--Littlewood inequalities for $m$-linear operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00178","kind":"arxiv","version":4},"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":"sha256:49ace5463f26fd0ad8763432fa9989c05391250fdf7b750187e74600de2713c1","target":"record","created_at":"2026-05-18T00:10:30Z","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":"55871d5881d05e368643d2e4680117ca3deb3ea95a81e7981c4e43e2c1d73f70","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-31T00:39:32Z","title_canon_sha256":"be3d5c03bd1f926e711cbeaead3d1353f618bed89e2c14356c0dded4657996dc"},"schema_version":"1.0","source":{"id":"1602.00178","kind":"arxiv","version":4}},"canonical_sha256":"46fd84ae7407510aa1ba13d4d6b2ba540bb71201c378693de20ddc35ad3e240b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"46fd84ae7407510aa1ba13d4d6b2ba540bb71201c378693de20ddc35ad3e240b","first_computed_at":"2026-05-18T00:10:30.949641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:30.949641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9YaStJ6Iqt0tqwCLXZD190K32s07/jzCmTryLB66uw+mtfXQjUGJKU5HhjCKZUEPSoiWhEwvecSldkSwJSlEBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:30.950194Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.00178","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49ace5463f26fd0ad8763432fa9989c05391250fdf7b750187e74600de2713c1","sha256:95b765436a3acc86d5f1582630e57a4956b8b1acafc89bba49fdbc70d594556b"],"state_sha256":"73a4e6e2e02d837cf171a67b4871880f53259555e8d7b5490598db6e05f7d3b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aAyRNq17mKs34M+1ViMbMmnzhdlcEOXVNk5VNmnNY6NbmwGj5XRBiwds8k5r7yyVXapiv1bhseCKiXxZxUBnAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T13:43:38.542810Z","bundle_sha256":"565f7694f7d0d135dc0cc7d4d2d7eeeee51b72b4b5d609337851e9a98d5da59a"}}