{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:YYK4AT52ZJ44QZSNMMXAMSGEXA","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":"39ffd26a5bfa8bd6c41d9a77945dcf4546f909474b529298a10d82763efb1fef","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-07-15T14:16:23Z","title_canon_sha256":"3c8743dffbd2d1a8435de3c8c13083b93a3e2254760ae61a0555c621d55898f7"},"schema_version":"1.0","source":{"id":"0807.2380","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.2380","created_at":"2026-05-18T02:26:53Z"},{"alias_kind":"arxiv_version","alias_value":"0807.2380v2","created_at":"2026-05-18T02:26:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.2380","created_at":"2026-05-18T02:26:53Z"},{"alias_kind":"pith_short_12","alias_value":"YYK4AT52ZJ44","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"YYK4AT52ZJ44QZSN","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"YYK4AT52","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:95bdc9ce795fb0a5593cd8f590a15e0a4779fbc90fae0747c6e7dcce4dadf49c","target":"graph","created_at":"2026-05-18T02:26:53Z","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":"It is known that Fourier integral operators arising when solving Schr\\\"odinger-type operators are bounded on the modulation spaces $\\cM^{p,q}$, for $1\\leq p=q\\leq\\infty$, provided their symbols belong to the Sj\\\"ostrand class $M^{\\infty,1}$. However, they generally fail to be bounded on $\\cM^{p,q}$ for $p\\not=q$. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on $\\cM^{p,q}$ for $p\\not=q$, and between $\\cM^{p,q}\\to\\cM^{q,p}$, $1\\leq q< p\\leq\\infty$. We also study similar problems for operators acting on Wiener a","authors_text":"Elena Cordero, Fabio Nicola","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-07-15T14:16:23Z","title":"Boundedness of Schroedinger type propagators on modulation spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.2380","kind":"arxiv","version":2},"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":"sha256:16da04f58a39c1be3aaa599dc8cf4a213a3ec4456b8826da70f92ef2e01d05d4","target":"record","created_at":"2026-05-18T02:26:53Z","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":"39ffd26a5bfa8bd6c41d9a77945dcf4546f909474b529298a10d82763efb1fef","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-07-15T14:16:23Z","title_canon_sha256":"3c8743dffbd2d1a8435de3c8c13083b93a3e2254760ae61a0555c621d55898f7"},"schema_version":"1.0","source":{"id":"0807.2380","kind":"arxiv","version":2}},"canonical_sha256":"c615c04fbaca79c8664d632e0648c4b82800f025c58ec22c50e6daf454b89512","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c615c04fbaca79c8664d632e0648c4b82800f025c58ec22c50e6daf454b89512","first_computed_at":"2026-05-18T02:26:53.268600Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:53.268600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fYPWez7DnDUPlqRgH8M1oRuHUJnej0Lq93ljEL1df7vrk2qeGNoo8SObWLLBKes/n4tIdDr8FMGoMIUGSmHOCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:53.269018Z","signed_message":"canonical_sha256_bytes"},"source_id":"0807.2380","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16da04f58a39c1be3aaa599dc8cf4a213a3ec4456b8826da70f92ef2e01d05d4","sha256:95bdc9ce795fb0a5593cd8f590a15e0a4779fbc90fae0747c6e7dcce4dadf49c"],"state_sha256":"09043c85b81994b0a6b7da32a08f57bc8ae844f53ffe3decb547da69e22cf6ad"}