{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:NAZMM2MJRBVD5NMRVCKZNJNPLO","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":"02b7e6fa76f321b7cd7a5d3f93fb95c4bcf75c976f1e3bc0315eb16be4be946d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-06T14:28:34Z","title_canon_sha256":"a62d87a2670f1466080b0009bbf67560b5977b7f8599aea59d82ed31ec0e0ef1"},"schema_version":"1.0","source":{"id":"1311.1405","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.1405","created_at":"2026-05-18T01:16:19Z"},{"alias_kind":"arxiv_version","alias_value":"1311.1405v3","created_at":"2026-05-18T01:16:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.1405","created_at":"2026-05-18T01:16:19Z"},{"alias_kind":"pith_short_12","alias_value":"NAZMM2MJRBVD","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"NAZMM2MJRBVD5NMR","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"NAZMM2MJ","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:e8970f781aa48eddb02360356690e618072a1354bc79050b4706e568e86cbab0","target":"graph","created_at":"2026-05-18T01:16:19Z","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 give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to $N=2$ of a result valid for the $N$-fold Weyl product. As a byproduct, we obtain sharp conditions for the twisted convolution to be bounded on Wiener amalgam spaces.","authors_text":"Elena Cordero, Joachim Toft, Patrik Wahlberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-06T14:28:34Z","title":"Sharp results for the Weyl product on modulation spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1405","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":"sha256:8fd722d473ca2ae8472411f527972954f75f0bdb7ae82cede8042df1015d8623","target":"record","created_at":"2026-05-18T01:16:19Z","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":"02b7e6fa76f321b7cd7a5d3f93fb95c4bcf75c976f1e3bc0315eb16be4be946d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-06T14:28:34Z","title_canon_sha256":"a62d87a2670f1466080b0009bbf67560b5977b7f8599aea59d82ed31ec0e0ef1"},"schema_version":"1.0","source":{"id":"1311.1405","kind":"arxiv","version":3}},"canonical_sha256":"6832c66989886a3eb591a89596a5af5b91933f0d62c59bb57b0ed1bda075e96b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6832c66989886a3eb591a89596a5af5b91933f0d62c59bb57b0ed1bda075e96b","first_computed_at":"2026-05-18T01:16:19.228236Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:19.228236Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CWfMhfzoTpS8+zaZjnNHN2ulq/f5gYSWRLbIIldX+Uwo6jpjg1RQn5Z36jJj5v8rIA6kPGpyyvRjRVj1s1sVBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:19.228960Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.1405","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8fd722d473ca2ae8472411f527972954f75f0bdb7ae82cede8042df1015d8623","sha256:e8970f781aa48eddb02360356690e618072a1354bc79050b4706e568e86cbab0"],"state_sha256":"6d1a77e506f2456c776072e46b6550e5194a17186a0d5a31107f79c4f18d8a96"}