{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:MP45MSOUCKTP3YPFPHMEVH7T2L","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9b739692b55a7a68ecc72c98fb04bde9ebb856eccaea66d148586e1f952bf090","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-06-05T08:59:47Z","title_canon_sha256":"3c8a29bc523ccfc0b71b9691369c56c5eafc83a3bfedb55c78559af4535d5531"},"schema_version":"1.0","source":{"id":"2406.03076","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.03076","created_at":"2026-07-05T08:59:51Z"},{"alias_kind":"arxiv_version","alias_value":"2406.03076v1","created_at":"2026-07-05T08:59:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.03076","created_at":"2026-07-05T08:59:51Z"},{"alias_kind":"pith_short_12","alias_value":"MP45MSOUCKTP","created_at":"2026-07-05T08:59:51Z"},{"alias_kind":"pith_short_16","alias_value":"MP45MSOUCKTP3YPF","created_at":"2026-07-05T08:59:51Z"},{"alias_kind":"pith_short_8","alias_value":"MP45MSOU","created_at":"2026-07-05T08:59:51Z"}],"graph_snapshots":[{"event_id":"sha256:5dc6a422b12797269b20503e8ca48ab15402914202688f00a319f39609a78e30","target":"graph","created_at":"2026-07-05T08:59:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2406.03076/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Chunjie Zhang, Xiangrong Zhu, Xiaofeng Ye","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-06-05T08:59:47Z","title":"Fourier integral operators on Hardy spaces with Hormander class"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.03076","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f31d35280da7d214ddca446000abf18bf87c4caefafd9ee8f6fb41e3473b56fe","target":"record","created_at":"2026-07-05T08:59:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9b739692b55a7a68ecc72c98fb04bde9ebb856eccaea66d148586e1f952bf090","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-06-05T08:59:47Z","title_canon_sha256":"3c8a29bc523ccfc0b71b9691369c56c5eafc83a3bfedb55c78559af4535d5531"},"schema_version":"1.0","source":{"id":"2406.03076","kind":"arxiv","version":1}},"canonical_sha256":"63f9d649d412a6fde1e579d84a9ff3d2d5fbf129dd6868dbbf2961f95cc9fb75","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63f9d649d412a6fde1e579d84a9ff3d2d5fbf129dd6868dbbf2961f95cc9fb75","first_computed_at":"2026-07-05T08:59:51.479406Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:59:51.479406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vQtQj20/X58dXs6p17BK/PW6T2Dk12aJxeGB8vh/PxNRXvpYzGUDTMp5dgj6bxtSHmuSvKPXw/wDIj/enhRnCg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:59:51.480813Z","signed_message":"canonical_sha256_bytes"},"source_id":"2406.03076","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f31d35280da7d214ddca446000abf18bf87c4caefafd9ee8f6fb41e3473b56fe","sha256:5dc6a422b12797269b20503e8ca48ab15402914202688f00a319f39609a78e30"],"state_sha256":"f6fde9e54c0a8f06506cccd2f2b70a745bf665e8f70c3af8694d19968f296db9"}