{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:VWXZPMLU2YHN25PBLTMMJIWTJM","short_pith_number":"pith:VWXZPMLU","canonical_record":{"source":{"id":"1906.11095","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-26T13:44:13Z","cross_cats_sorted":[],"title_canon_sha256":"dd7ec97b48ed580673a176bfa01fdbc9ed365c89b0ab416f76db25722f8a1d0f","abstract_canon_sha256":"f5bb119464d5d4f687fbe50cb84b9c783ac087270c073a5bc1df66644856fbae"},"schema_version":"1.0"},"canonical_sha256":"adaf97b174d60edd75e15cd8c4a2d34b154802abb0a478866ab062411d186710","source":{"kind":"arxiv","id":"1906.11095","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.11095","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"arxiv_version","alias_value":"1906.11095v1","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.11095","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"pith_short_12","alias_value":"VWXZPMLU2YHN","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VWXZPMLU2YHN25PB","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VWXZPMLU","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:VWXZPMLU2YHN25PBLTMMJIWTJM","target":"record","payload":{"canonical_record":{"source":{"id":"1906.11095","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-26T13:44:13Z","cross_cats_sorted":[],"title_canon_sha256":"dd7ec97b48ed580673a176bfa01fdbc9ed365c89b0ab416f76db25722f8a1d0f","abstract_canon_sha256":"f5bb119464d5d4f687fbe50cb84b9c783ac087270c073a5bc1df66644856fbae"},"schema_version":"1.0"},"canonical_sha256":"adaf97b174d60edd75e15cd8c4a2d34b154802abb0a478866ab062411d186710","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:09.852127Z","signature_b64":"N9y6GN3ME8NHkbzcGEtLJa6UwR30/fQxt6//RIbmn34x7z81SSzH1OunxN8oRgil9JgR1SD+35Hyuyfb6bJSCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"adaf97b174d60edd75e15cd8c4a2d34b154802abb0a478866ab062411d186710","last_reissued_at":"2026-05-17T23:42:09.851431Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:09.851431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.11095","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Gz+1oc6s634e7qqSIyMIN5e8IhOdHlFky+XCVoUGmyiaCpNvwI9qZ1FmJKZnMc6MX/dMCS4Q96wywjG+pMVDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:55:40.133731Z"},"content_sha256":"82119cf4c202324d50b16ffd0558170fbf514106beb6a679a80835b10697d5f5","schema_version":"1.0","event_id":"sha256:82119cf4c202324d50b16ffd0558170fbf514106beb6a679a80835b10697d5f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:VWXZPMLU2YHN25PBLTMMJIWTJM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bilinear pseudo-differential operators with Gevrey-H\\\"ormander symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ahmed Abdeljawad, Nenad Teofanov, Sandro Coriasco","submitted_at":"2019-06-26T13:44:13Z","abstract_excerpt":"We consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.11095","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FHvL9qPUef36cEb75seU1I5iqAeUuy6Z+nMKoRroitHN86qco+DbF78sWGb7roLB9oYiLd+G8Mbp1ynPH5l+DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:55:40.134457Z"},"content_sha256":"0ea58a072649f19576c2c0601e423168e0e2d97ebb6fe2065752559fa8f42dcc","schema_version":"1.0","event_id":"sha256:0ea58a072649f19576c2c0601e423168e0e2d97ebb6fe2065752559fa8f42dcc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VWXZPMLU2YHN25PBLTMMJIWTJM/bundle.json","state_url":"https://pith.science/pith/VWXZPMLU2YHN25PBLTMMJIWTJM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VWXZPMLU2YHN25PBLTMMJIWTJM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T20:55:40Z","links":{"resolver":"https://pith.science/pith/VWXZPMLU2YHN25PBLTMMJIWTJM","bundle":"https://pith.science/pith/VWXZPMLU2YHN25PBLTMMJIWTJM/bundle.json","state":"https://pith.science/pith/VWXZPMLU2YHN25PBLTMMJIWTJM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VWXZPMLU2YHN25PBLTMMJIWTJM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:VWXZPMLU2YHN25PBLTMMJIWTJM","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":"f5bb119464d5d4f687fbe50cb84b9c783ac087270c073a5bc1df66644856fbae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-26T13:44:13Z","title_canon_sha256":"dd7ec97b48ed580673a176bfa01fdbc9ed365c89b0ab416f76db25722f8a1d0f"},"schema_version":"1.0","source":{"id":"1906.11095","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.11095","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"arxiv_version","alias_value":"1906.11095v1","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.11095","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"pith_short_12","alias_value":"VWXZPMLU2YHN","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VWXZPMLU2YHN25PB","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VWXZPMLU","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:0ea58a072649f19576c2c0601e423168e0e2d97ebb6fe2065752559fa8f42dcc","target":"graph","created_at":"2026-05-17T23:42:09Z","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 consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurli","authors_text":"Ahmed Abdeljawad, Nenad Teofanov, Sandro Coriasco","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-26T13:44:13Z","title":"Bilinear pseudo-differential operators with Gevrey-H\\\"ormander symbols"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.11095","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":"sha256:82119cf4c202324d50b16ffd0558170fbf514106beb6a679a80835b10697d5f5","target":"record","created_at":"2026-05-17T23:42:09Z","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":"f5bb119464d5d4f687fbe50cb84b9c783ac087270c073a5bc1df66644856fbae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-26T13:44:13Z","title_canon_sha256":"dd7ec97b48ed580673a176bfa01fdbc9ed365c89b0ab416f76db25722f8a1d0f"},"schema_version":"1.0","source":{"id":"1906.11095","kind":"arxiv","version":1}},"canonical_sha256":"adaf97b174d60edd75e15cd8c4a2d34b154802abb0a478866ab062411d186710","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"adaf97b174d60edd75e15cd8c4a2d34b154802abb0a478866ab062411d186710","first_computed_at":"2026-05-17T23:42:09.851431Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:09.851431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N9y6GN3ME8NHkbzcGEtLJa6UwR30/fQxt6//RIbmn34x7z81SSzH1OunxN8oRgil9JgR1SD+35Hyuyfb6bJSCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:09.852127Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.11095","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82119cf4c202324d50b16ffd0558170fbf514106beb6a679a80835b10697d5f5","sha256:0ea58a072649f19576c2c0601e423168e0e2d97ebb6fe2065752559fa8f42dcc"],"state_sha256":"a841b54312be881cea3f23bc276ac9b53a9962a363e6eef203eabb1fb3ef335a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rFIeWKUDRzSQwlC3qBgKC2hUY72owLEnwqE4b/MAOrL1vZvxlLoEi68TTRaaPUBf6Hjxhd3QcjaILouxRTKkBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:55:40.137916Z","bundle_sha256":"9e52e9c809f176481ca9e0756fa812f447b05d9f8b7e430f93d4526624c9ab1c"}}