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Let $0\\leq\\rho\\leq 1,n\\geq 2$ or $0\\leq\\rho<1,n=1$ and $$m_p=\\frac{\\rho-n}{p}+(n-1)\\min\\{\\frac 12,\\rho\\}.$$ If $a$ belongs to the forbidden H\\\"{o}rmander class $S^{m_p}_{\\rho,1}$ and $\\phi\\in \\Phi^{2}$ satisfies the strong non-degeneracy condition, then for any $\\frac {n}{n+1}<p\\leq 1$, we can show that the Fourier integral operator $T_{\\phi,a}$ is bounded from the loca"},"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":"2406.03076","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-06-05T08:59:47Z","cross_cats_sorted":[],"title_canon_sha256":"3c8a29bc523ccfc0b71b9691369c56c5eafc83a3bfedb55c78559af4535d5531","abstract_canon_sha256":"9b739692b55a7a68ecc72c98fb04bde9ebb856eccaea66d148586e1f952bf090"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:59:51.480813Z","signature_b64":"vQtQj20/X58dXs6p17BK/PW6T2Dk12aJxeGB8vh/PxNRXvpYzGUDTMp5dgj6bxtSHmuSvKPXw/wDIj/enhRnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63f9d649d412a6fde1e579d84a9ff3d2d5fbf129dd6868dbbf2961f95cc9fb75","last_reissued_at":"2026-07-05T08:59:51.479406Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:59:51.479406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fourier integral operators on Hardy spaces with Hormander class","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Chunjie Zhang, Xiangrong Zhu, Xiaofeng Ye","submitted_at":"2024-06-05T08:59:47Z","abstract_excerpt":"In this note, we consider a Fourier integral operator defined by \\begin{align*} T_{\\phi,a}f(x) = \\int_{\\mathbb{R}^{n}}e^{i\\phi(x,\\xi)}a(x,\\xi)\\widehat{f} \\xi)d\\xi, \\end{align*}here $a$ is the amplitude, and $\\phi$ is the phase. Let $0\\leq\\rho\\leq 1,n\\geq 2$ or $0\\leq\\rho<1,n=1$ and $$m_p=\\frac{\\rho-n}{p}+(n-1)\\min\\{\\frac 12,\\rho\\}.$$ If $a$ belongs to the forbidden H\\\"{o}rmander class $S^{m_p}_{\\rho,1}$ and $\\phi\\in \\Phi^{2}$ satisfies the strong non-degeneracy condition, then for any $\\frac {n}{n+1}<p\\leq 1$, we can show that the Fourier integral operator $T_{\\phi,a}$ is bounded from the loca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.03076","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.03076/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2406.03076","created_at":"2026-07-05T08:59:51.479579+00:00"},{"alias_kind":"arxiv_version","alias_value":"2406.03076v1","created_at":"2026-07-05T08:59:51.479579+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.03076","created_at":"2026-07-05T08:59:51.479579+00:00"},{"alias_kind":"pith_short_12","alias_value":"MP45MSOUCKTP","created_at":"2026-07-05T08:59:51.479579+00:00"},{"alias_kind":"pith_short_16","alias_value":"MP45MSOUCKTP3YPF","created_at":"2026-07-05T08:59:51.479579+00:00"},{"alias_kind":"pith_short_8","alias_value":"MP45MSOU","created_at":"2026-07-05T08:59:51.479579+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/MP45MSOUCKTP3YPFPHMEVH7T2L","json":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L.json","graph_json":"https://pith.science/api/pith-number/MP45MSOUCKTP3YPFPHMEVH7T2L/graph.json","events_json":"https://pith.science/api/pith-number/MP45MSOUCKTP3YPFPHMEVH7T2L/events.json","paper":"https://pith.science/paper/MP45MSOU"},"agent_actions":{"view_html":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L","download_json":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L.json","view_paper":"https://pith.science/paper/MP45MSOU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2406.03076&json=true","fetch_graph":"https://pith.science/api/pith-number/MP45MSOUCKTP3YPFPHMEVH7T2L/graph.json","fetch_events":"https://pith.science/api/pith-number/MP45MSOUCKTP3YPFPHMEVH7T2L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L/action/storage_attestation","attest_author":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L/action/author_attestation","sign_citation":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L/action/citation_signature","submit_replication":"https://pith.science/pith/MP45MSOUCKTP3YPFPHMEVH7T2L/action/replication_record"}},"created_at":"2026-07-05T08:59:51.479579+00:00","updated_at":"2026-07-05T08:59:51.479579+00:00"}