{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:XYTEXPXMWZN5YRPIOKLLON2BKP","short_pith_number":"pith:XYTEXPXM","schema_version":"1.0","canonical_sha256":"be264bbeecb65bdc45e87296b7374153d4975a45de03f8b335603b352a81f70d","source":{"kind":"arxiv","id":"1505.06479","version":3},"attestation_state":"computed","paper":{"title":"Cancellation for the multilinear Hilbert transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Terence Tao","submitted_at":"2015-05-24T20:49:18Z","abstract_excerpt":"For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\\dots,f_k )(x) := \\operatorname{p.v.} \\int_{\\bf R} f_1(x+t) \\dots f_k(x+kt)\\ \\frac{dt}{t}$$ for test functions $f_1,\\dots,f_k: {\\bf R} \\to {\\bf C}$. It is conjectured that $H_k$ maps $L^{p_1}({\\bf R}) \\times \\dots \\times L^{p_k}({\\bf R}) \\to L^p({\\bf R})$ whenever $1 < p_1,\\dots,p_k,p < \\infty$ and $\\frac{1}{p} = \\frac{1}{p_1} + \\dots + \\frac{1}{p_k}$. This is proven for $k=1,2$, but remains open for larger $k$.\n  In this paper, we consider the truncated operators $$ H_{k,r,R}( f_1,\\dots,f_k )(x) := \\int_{r \\leq "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.06479","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-05-24T20:49:18Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5db45b942d1581cb9e8ee4dab1bc9cbd7728df9aeea9cca00918c0a340564f0e","abstract_canon_sha256":"3e91031e3cbc85c5d2e6fa00358d9058d977b7c157c1c389e0ab529e1f8a1269"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:00:01.877275Z","signature_b64":"9aYQQhLH506CDf+6J/URSJUtMJ5fgkR7t3l53/wZsoFvmdUWNAXgSfp889R2Fu1WxjfOWqPT4wCRJURMGv08Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be264bbeecb65bdc45e87296b7374153d4975a45de03f8b335603b352a81f70d","last_reissued_at":"2026-05-18T02:00:01.876851Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:00:01.876851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cancellation for the multilinear Hilbert transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Terence Tao","submitted_at":"2015-05-24T20:49:18Z","abstract_excerpt":"For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\\dots,f_k )(x) := \\operatorname{p.v.} \\int_{\\bf R} f_1(x+t) \\dots f_k(x+kt)\\ \\frac{dt}{t}$$ for test functions $f_1,\\dots,f_k: {\\bf R} \\to {\\bf C}$. It is conjectured that $H_k$ maps $L^{p_1}({\\bf R}) \\times \\dots \\times L^{p_k}({\\bf R}) \\to L^p({\\bf R})$ whenever $1 < p_1,\\dots,p_k,p < \\infty$ and $\\frac{1}{p} = \\frac{1}{p_1} + \\dots + \\frac{1}{p_k}$. This is proven for $k=1,2$, but remains open for larger $k$.\n  In this paper, we consider the truncated operators $$ H_{k,r,R}( f_1,\\dots,f_k )(x) := \\int_{r \\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06479","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":"1505.06479","created_at":"2026-05-18T02:00:01.876925+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.06479v3","created_at":"2026-05-18T02:00:01.876925+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06479","created_at":"2026-05-18T02:00:01.876925+00:00"},{"alias_kind":"pith_short_12","alias_value":"XYTEXPXMWZN5","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_16","alias_value":"XYTEXPXMWZN5YRPI","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_8","alias_value":"XYTEXPXM","created_at":"2026-05-18T12:29:50.041715+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/XYTEXPXMWZN5YRPIOKLLON2BKP","json":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP.json","graph_json":"https://pith.science/api/pith-number/XYTEXPXMWZN5YRPIOKLLON2BKP/graph.json","events_json":"https://pith.science/api/pith-number/XYTEXPXMWZN5YRPIOKLLON2BKP/events.json","paper":"https://pith.science/paper/XYTEXPXM"},"agent_actions":{"view_html":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP","download_json":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP.json","view_paper":"https://pith.science/paper/XYTEXPXM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.06479&json=true","fetch_graph":"https://pith.science/api/pith-number/XYTEXPXMWZN5YRPIOKLLON2BKP/graph.json","fetch_events":"https://pith.science/api/pith-number/XYTEXPXMWZN5YRPIOKLLON2BKP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP/action/storage_attestation","attest_author":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP/action/author_attestation","sign_citation":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP/action/citation_signature","submit_replication":"https://pith.science/pith/XYTEXPXMWZN5YRPIOKLLON2BKP/action/replication_record"}},"created_at":"2026-05-18T02:00:01.876925+00:00","updated_at":"2026-05-18T02:00:01.876925+00:00"}