{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:3U3UQQKW2VAH4JZIIYUYDG6UCO","short_pith_number":"pith:3U3UQQKW","schema_version":"1.0","canonical_sha256":"dd37484156d5407e27284629819bd413abbed6fcfdc618421973acc4dd6c0075","source":{"kind":"arxiv","id":"1111.4653","version":1},"attestation_state":"computed","paper":{"title":"On the boundedness of certain bilinear Fourier integral operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David J. Rule, Salvador Rodriguez-Lopez, Wolfgang Staubach","submitted_at":"2011-11-20T17:39:56Z","abstract_excerpt":"We prove the global $L^2 \\times L^2 \\to L^1$ boundedness of bilinear Fourier integral operators with amplitudes in $S^0_{1,0} (n,2)$. To achieve this, we require that the phase function can be written as $(x,\\xi,\\eta) \\mapsto \\phase_1(x,\\xi) + \\phase_2(x,\\eta)$ where each $\\phase_j$ belongs to the class $\\Phi^2$ and satisfies the strong non-degeneracy condition. This result extends that of R. Coifman and Y. Meyer regarding pseudodifferential operators to the case of Fourier integral operators."},"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":"1111.4653","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-20T17:39:56Z","cross_cats_sorted":[],"title_canon_sha256":"45ba6d4b6ee00dea522ad4d1e49f54fe9ec4a5a924a2324d0fd02b5eb37a6bf3","abstract_canon_sha256":"241747c8c0bf0e69997a79e1109abb0a119dbfa98d939861c6551e3bbdf3db5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:02.909016Z","signature_b64":"D1zmHpOrwGmr62rHubHFUSW68YXDA3tZ7YIDoKi2I7/r8NzfWAo3aTwsD+TOVM0ctzelsNZLI/I2Errb/mGbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd37484156d5407e27284629819bd413abbed6fcfdc618421973acc4dd6c0075","last_reissued_at":"2026-05-18T04:08:02.908539Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:02.908539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the boundedness of certain bilinear Fourier integral operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David J. Rule, Salvador Rodriguez-Lopez, Wolfgang Staubach","submitted_at":"2011-11-20T17:39:56Z","abstract_excerpt":"We prove the global $L^2 \\times L^2 \\to L^1$ boundedness of bilinear Fourier integral operators with amplitudes in $S^0_{1,0} (n,2)$. To achieve this, we require that the phase function can be written as $(x,\\xi,\\eta) \\mapsto \\phase_1(x,\\xi) + \\phase_2(x,\\eta)$ where each $\\phase_j$ belongs to the class $\\Phi^2$ and satisfies the strong non-degeneracy condition. This result extends that of R. Coifman and Y. Meyer regarding pseudodifferential operators to the case of Fourier integral operators."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4653","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":"1111.4653","created_at":"2026-05-18T04:08:02.908607+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.4653v1","created_at":"2026-05-18T04:08:02.908607+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.4653","created_at":"2026-05-18T04:08:02.908607+00:00"},{"alias_kind":"pith_short_12","alias_value":"3U3UQQKW2VAH","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"3U3UQQKW2VAH4JZI","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"3U3UQQKW","created_at":"2026-05-18T12:26:20.644004+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/3U3UQQKW2VAH4JZIIYUYDG6UCO","json":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO.json","graph_json":"https://pith.science/api/pith-number/3U3UQQKW2VAH4JZIIYUYDG6UCO/graph.json","events_json":"https://pith.science/api/pith-number/3U3UQQKW2VAH4JZIIYUYDG6UCO/events.json","paper":"https://pith.science/paper/3U3UQQKW"},"agent_actions":{"view_html":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO","download_json":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO.json","view_paper":"https://pith.science/paper/3U3UQQKW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.4653&json=true","fetch_graph":"https://pith.science/api/pith-number/3U3UQQKW2VAH4JZIIYUYDG6UCO/graph.json","fetch_events":"https://pith.science/api/pith-number/3U3UQQKW2VAH4JZIIYUYDG6UCO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO/action/storage_attestation","attest_author":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO/action/author_attestation","sign_citation":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO/action/citation_signature","submit_replication":"https://pith.science/pith/3U3UQQKW2VAH4JZIIYUYDG6UCO/action/replication_record"}},"created_at":"2026-05-18T04:08:02.908607+00:00","updated_at":"2026-05-18T04:08:02.908607+00:00"}