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When $q=\\infty$ we obtain boundedness for $T_\\Omega$ from $L^{p_1}(\\mathbb R^n)\\times L^{p_2}(\\mathbb R^n)$ to $ L^p(\\mathbb R^n) $ when $1<p_1, p_2<\\infty$ and $1/p=1/p_1+1/p_2$. For $q=2$ we obtain that $T_\\Omega$ is bounded from $L^{2}(\\mathbb R^n)\\times L^{ 2}(\\mathbb R^n)$ to $ "},"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":"1509.06099","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-21T03:03:38Z","cross_cats_sorted":[],"title_canon_sha256":"f0182be347182be8ee75796a6b7ba13e810d4c49b3095902de324573752e4e3b","abstract_canon_sha256":"8c9d94450ef7bdbc7e1deb9d7fb4ddf9ab1b606308ea241a3af8cd1bbf260ca0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:20.552021Z","signature_b64":"BgI0RujLBOJlb7XsrF9quoIReVQEwjNkUCCa6AU/RUp2T1OMxe79wUa8Grg5V0E2n6GaAw2H72dZzipxioSSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36e6f345b6a3bc18de1d2fbd37dacb1890fbf91a8952c336cded0383adf6b468","last_reissued_at":"2026-05-18T01:32:20.551386Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:20.551386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rough Bilinear Singular Integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Danqing He, Loukas Grafakos, Petr Honz\\'ik","submitted_at":"2015-09-21T03:03:38Z","abstract_excerpt":"We study the rough bilinear singular integral, introduced by Coifman and Meyer , $$ T_\\Omega(f,g)(x)=p.v. \\! \\int_{\\mathbb R^{n}}\\! \\int_{\\mathbb R^{n}}\\! |(y,z)|^{-2n} \\Omega((y,z)/|(y,z)|)f(x-y)g(x-z) dydz, $$ when $\\Omega $ is a function in $L^q(\\mathbb S^{2n-1})$ with vanishing integral and $2\\le q\\le \\infty$. When $q=\\infty$ we obtain boundedness for $T_\\Omega$ from $L^{p_1}(\\mathbb R^n)\\times L^{p_2}(\\mathbb R^n)$ to $ L^p(\\mathbb R^n) $ when $1<p_1, p_2<\\infty$ and $1/p=1/p_1+1/p_2$. For $q=2$ we obtain that $T_\\Omega$ is bounded from $L^{2}(\\mathbb R^n)\\times L^{ 2}(\\mathbb R^n)$ to $ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06099","kind":"arxiv","version":2},"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":"1509.06099","created_at":"2026-05-18T01:32:20.551482+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.06099v2","created_at":"2026-05-18T01:32:20.551482+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06099","created_at":"2026-05-18T01:32:20.551482+00:00"},{"alias_kind":"pith_short_12","alias_value":"G3TPGRNWUO6B","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"G3TPGRNWUO6BRXQ5","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"G3TPGRNW","created_at":"2026-05-18T12:29:22.688609+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/G3TPGRNWUO6BRXQ5F66TPWWLDC","json":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC.json","graph_json":"https://pith.science/api/pith-number/G3TPGRNWUO6BRXQ5F66TPWWLDC/graph.json","events_json":"https://pith.science/api/pith-number/G3TPGRNWUO6BRXQ5F66TPWWLDC/events.json","paper":"https://pith.science/paper/G3TPGRNW"},"agent_actions":{"view_html":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC","download_json":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC.json","view_paper":"https://pith.science/paper/G3TPGRNW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.06099&json=true","fetch_graph":"https://pith.science/api/pith-number/G3TPGRNWUO6BRXQ5F66TPWWLDC/graph.json","fetch_events":"https://pith.science/api/pith-number/G3TPGRNWUO6BRXQ5F66TPWWLDC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC/action/storage_attestation","attest_author":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC/action/author_attestation","sign_citation":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC/action/citation_signature","submit_replication":"https://pith.science/pith/G3TPGRNWUO6BRXQ5F66TPWWLDC/action/replication_record"}},"created_at":"2026-05-18T01:32:20.551482+00:00","updated_at":"2026-05-18T01:32:20.551482+00:00"}