{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FQHU6PRGKSOLO2OQHUPZUG7WD5","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":"d1bff33266d3b6982dfb4d9bb57fac41b38c661da880575d1418a7153f42f304","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-09T16:44:50Z","title_canon_sha256":"17fa7a6980cede618d82d3047dca01360004654cd45901bed4f0089fa966de03"},"schema_version":"1.0","source":{"id":"1207.2099","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2099","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2099v3","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2099","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"pith_short_12","alias_value":"FQHU6PRGKSOL","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FQHU6PRGKSOLO2OQ","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FQHU6PRG","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:8274c225a127a56b532132ac5e5ade14d35f382c6b74115d1f95bd31678064e1","target":"graph","created_at":"2026-05-18T02:57:53Z","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"},"paper":{"abstract_excerpt":"We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of nondegenerate generalized quadratic forms that includes Schr\\\"odinger propagators and pseudodifferential operators. As a byproduct we obtain a characterization of all exponents $p,q,r_1,r_2,t_1,t_2 \\in [1,\\infty]$ of modulation spaces such that a symbol in $M^{p,q}(\\mathbb R^{2d})$ gives a pseudodifferential operator that is continuous from $M^{r_1,r_2}(\\mathbb R^d)$ into $M^{t_1,t_2}(\\mathbb R^d)$.","authors_text":"Anita Tabacco, Elena Cordero, Patrik Wahlberg","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-09T16:44:50Z","title":"Schr\\\"odinger type propagators, pseudodifferential operators and modulation spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2099","kind":"arxiv","version":3},"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":"sha256:86ed5da7d3429d4d52d83b30c38ade15693ef07e43c2fa766b78b6da1ac85df4","target":"record","created_at":"2026-05-18T02:57:53Z","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":"d1bff33266d3b6982dfb4d9bb57fac41b38c661da880575d1418a7153f42f304","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-09T16:44:50Z","title_canon_sha256":"17fa7a6980cede618d82d3047dca01360004654cd45901bed4f0089fa966de03"},"schema_version":"1.0","source":{"id":"1207.2099","kind":"arxiv","version":3}},"canonical_sha256":"2c0f4f3e26549cb769d03d1f9a1bf61f53b78016cb8a0a150ff737f1d801b195","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c0f4f3e26549cb769d03d1f9a1bf61f53b78016cb8a0a150ff737f1d801b195","first_computed_at":"2026-05-18T02:57:53.388778Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:53.388778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7Iqa597At7+onCwRej8+oUkc4DGMo6fLzphCuwQi3KVC0HcO3WkJXXfikBlXQ5NdjhrEPoTYPdDUEemvIcCtAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:53.389229Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.2099","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86ed5da7d3429d4d52d83b30c38ade15693ef07e43c2fa766b78b6da1ac85df4","sha256:8274c225a127a56b532132ac5e5ade14d35f382c6b74115d1f95bd31678064e1"],"state_sha256":"4963fad62c9487aed27786251ce13f7d93f7a9f078b1fb120a81ceee4fd65fd1"}