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More exactly, we show that the above operator maps $L^p \\times L^q \\to L^s$ whenever $p, q, s'$ are in the \"local $L^{r'}$\" range, i.e.  $\\displaystyle \\frac{1}{p}+\\frac{1}{q}+"},"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":"1602.01948","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-02-05T08:01:23Z","cross_cats_sorted":[],"title_canon_sha256":"6078ec64e87e67d2445ba87887244c207c111d05580c4295e76fbb765c495169","abstract_canon_sha256":"80cfbed1a9c4d3a54eab7f24c987eea6ad7a1945038b9d4592ab8dfa7bfe50bb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:14.947998Z","signature_b64":"ZFAxRGkh9S9I7UzQyLqqyEAM1odZrPoqTPoHQRKasfpWpwSWqhJD5+lWvJ4mXT18a109wIzQbuJq4T65ySISCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f36f9a5a80b6a42e4f78a1f2b6087c8d976d72a6d920043fe7f416bfbfc12d2d","last_reissued_at":"2026-05-18T01:21:14.947249Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:14.947249Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A bilinear Rubio de Francia inequality for arbitrary squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Cristina Benea (LMJL), Frederic Bernicot (LMJL)","submitted_at":"2016-02-05T08:01:23Z","abstract_excerpt":"We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane\\[\\left(f, g \\right)\\mapsto \\left( \\sum\\_{\\omega \\in \\Omega}\\left| \\int\\_{\\mathbb{R}^2} \\hat{f}(\\xi) \\hat{g}(\\eta) \\Phi\\_{\\omega}(\\xi, \\eta) e^{2 \\pi i x\\left(\\xi+\\eta \\right)} d \\xi d \\eta\\right|^r \\right)^{1/r},\\] provided $r\\textgreater{}2$. More exactly, we show that the above operator maps $L^p \\times L^q \\to L^s$ whenever $p, q, s'$ are in the \"local $L^{r'}$\" range, i.e.  $\\displaystyle \\frac{1}{p}+\\frac{1}{q}+"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01948","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1602.01948","created_at":"2026-05-18T01:21:14.947347+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.01948v1","created_at":"2026-05-18T01:21:14.947347+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.01948","created_at":"2026-05-18T01:21:14.947347+00:00"},{"alias_kind":"pith_short_12","alias_value":"6NXZUWUAW2SC","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"6NXZUWUAW2SC4T3Y","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"6NXZUWUA","created_at":"2026-05-18T12:30:01.593930+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/6NXZUWUAW2SC4T3YUHZLMCD4RW","json":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW.json","graph_json":"https://pith.science/api/pith-number/6NXZUWUAW2SC4T3YUHZLMCD4RW/graph.json","events_json":"https://pith.science/api/pith-number/6NXZUWUAW2SC4T3YUHZLMCD4RW/events.json","paper":"https://pith.science/paper/6NXZUWUA"},"agent_actions":{"view_html":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW","download_json":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW.json","view_paper":"https://pith.science/paper/6NXZUWUA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.01948&json=true","fetch_graph":"https://pith.science/api/pith-number/6NXZUWUAW2SC4T3YUHZLMCD4RW/graph.json","fetch_events":"https://pith.science/api/pith-number/6NXZUWUAW2SC4T3YUHZLMCD4RW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW/action/storage_attestation","attest_author":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW/action/author_attestation","sign_citation":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW/action/citation_signature","submit_replication":"https://pith.science/pith/6NXZUWUAW2SC4T3YUHZLMCD4RW/action/replication_record"}},"created_at":"2026-05-18T01:21:14.947347+00:00","updated_at":"2026-05-18T01:21:14.947347+00:00"}