{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:VNMLIMC2JIBWQLLR4OLQOXY7WQ","short_pith_number":"pith:VNMLIMC2","canonical_record":{"source":{"id":"1206.5109","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-06-22T10:59:28Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"42f995ceaa31f4abb5f127e5253066b60d05f053a7212b35bc69e3a6fd8a513a","abstract_canon_sha256":"ba4d8b26e90ecc038e7a0ffd8b55bbe4cb2432ede44e5f840f8075687e9cf33b"},"schema_version":"1.0"},"canonical_sha256":"ab58b4305a4a03682d71e397075f1fb432bc6a01b58271aba6e73bf1b8c9b982","source":{"kind":"arxiv","id":"1206.5109","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5109","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5109v1","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5109","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"pith_short_12","alias_value":"VNMLIMC2JIBW","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VNMLIMC2JIBWQLLR","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VNMLIMC2","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:VNMLIMC2JIBWQLLR4OLQOXY7WQ","target":"record","payload":{"canonical_record":{"source":{"id":"1206.5109","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-06-22T10:59:28Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"42f995ceaa31f4abb5f127e5253066b60d05f053a7212b35bc69e3a6fd8a513a","abstract_canon_sha256":"ba4d8b26e90ecc038e7a0ffd8b55bbe4cb2432ede44e5f840f8075687e9cf33b"},"schema_version":"1.0"},"canonical_sha256":"ab58b4305a4a03682d71e397075f1fb432bc6a01b58271aba6e73bf1b8c9b982","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:52.226907Z","signature_b64":"cHw7jrfMfgMpbuYax16LmUkWkp4F3oG9uv5d7Mr57N5my2xV+XZeQDQa6lhmGM5lJfnPY9/i8WQENT3XLcEyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab58b4305a4a03682d71e397075f1fb432bc6a01b58271aba6e73bf1b8c9b982","last_reissued_at":"2026-05-18T03:52:52.226039Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:52.226039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.5109","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":"2026-05-18T03:52:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g4BuuTLYi9wpjgGZ7wTqkO6M/0VeCJ/kZph8QXKpelu6wfdMBdhvfhBmmx3H3kEgQ00oMp7ewisfPcziYu77Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:21:40.156861Z"},"content_sha256":"de8ee9b0604a30abe7c7b5ac956f6139c73e050546ed77d5e1eac5a364a7858a","schema_version":"1.0","event_id":"sha256:de8ee9b0604a30abe7c7b5ac956f6139c73e050546ed77d5e1eac5a364a7858a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:VNMLIMC2JIBWQLLR4OLQOXY7WQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Weiss conjecture and weak norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OC","authors_text":"Bernhard Hermann Haak","submitted_at":"2012-06-22T10:59:28Z","abstract_excerpt":"In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $Re(\\lambda)^\\einhalb C(\\lambda+A)^{-1}, \\qquad Re(\\lambda)>0$ on the complex right half plane and weak Lebesgue $L^{2,\\infty}$--admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the 'Weiss conjecture' within the scale of Lorentz spaces $L^{p,q}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5109","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":"2026-05-18T03:52:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Ld7JCkQWyJJ+L2k+2DDFcBd7X7z0YYvM94BNSUCCLoKdBmi1m4kXH/8lh/T2gRGkrignd9SrBMzVsY/cBQuAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:21:40.157237Z"},"content_sha256":"9b1de2b33193b433046ebdd4b8b1438e8916e797c4c35e81aeade3d94b5dd8b1","schema_version":"1.0","event_id":"sha256:9b1de2b33193b433046ebdd4b8b1438e8916e797c4c35e81aeade3d94b5dd8b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VNMLIMC2JIBWQLLR4OLQOXY7WQ/bundle.json","state_url":"https://pith.science/pith/VNMLIMC2JIBWQLLR4OLQOXY7WQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VNMLIMC2JIBWQLLR4OLQOXY7WQ/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":"2026-06-25T18:21:40Z","links":{"resolver":"https://pith.science/pith/VNMLIMC2JIBWQLLR4OLQOXY7WQ","bundle":"https://pith.science/pith/VNMLIMC2JIBWQLLR4OLQOXY7WQ/bundle.json","state":"https://pith.science/pith/VNMLIMC2JIBWQLLR4OLQOXY7WQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VNMLIMC2JIBWQLLR4OLQOXY7WQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:VNMLIMC2JIBWQLLR4OLQOXY7WQ","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":"ba4d8b26e90ecc038e7a0ffd8b55bbe4cb2432ede44e5f840f8075687e9cf33b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-06-22T10:59:28Z","title_canon_sha256":"42f995ceaa31f4abb5f127e5253066b60d05f053a7212b35bc69e3a6fd8a513a"},"schema_version":"1.0","source":{"id":"1206.5109","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5109","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5109v1","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5109","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"pith_short_12","alias_value":"VNMLIMC2JIBW","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VNMLIMC2JIBWQLLR","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VNMLIMC2","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:9b1de2b33193b433046ebdd4b8b1438e8916e797c4c35e81aeade3d94b5dd8b1","target":"graph","created_at":"2026-05-18T03:52:52Z","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":"In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $Re(\\lambda)^\\einhalb C(\\lambda+A)^{-1}, \\qquad Re(\\lambda)>0$ on the complex right half plane and weak Lebesgue $L^{2,\\infty}$--admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the 'Weiss conjecture' within the scale of Lorentz spaces $L^{p,q}$.","authors_text":"Bernhard Hermann Haak","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-06-22T10:59:28Z","title":"The Weiss conjecture and weak norms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5109","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:de8ee9b0604a30abe7c7b5ac956f6139c73e050546ed77d5e1eac5a364a7858a","target":"record","created_at":"2026-05-18T03:52:52Z","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":"ba4d8b26e90ecc038e7a0ffd8b55bbe4cb2432ede44e5f840f8075687e9cf33b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2012-06-22T10:59:28Z","title_canon_sha256":"42f995ceaa31f4abb5f127e5253066b60d05f053a7212b35bc69e3a6fd8a513a"},"schema_version":"1.0","source":{"id":"1206.5109","kind":"arxiv","version":1}},"canonical_sha256":"ab58b4305a4a03682d71e397075f1fb432bc6a01b58271aba6e73bf1b8c9b982","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab58b4305a4a03682d71e397075f1fb432bc6a01b58271aba6e73bf1b8c9b982","first_computed_at":"2026-05-18T03:52:52.226039Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:52.226039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cHw7jrfMfgMpbuYax16LmUkWkp4F3oG9uv5d7Mr57N5my2xV+XZeQDQa6lhmGM5lJfnPY9/i8WQENT3XLcEyBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:52.226907Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5109","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de8ee9b0604a30abe7c7b5ac956f6139c73e050546ed77d5e1eac5a364a7858a","sha256:9b1de2b33193b433046ebdd4b8b1438e8916e797c4c35e81aeade3d94b5dd8b1"],"state_sha256":"cefb75bafcbf674930f52e232e02cac9a8174cf4f14c7bcbcdc63543d9fdcb3d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q5gSUJwJdBBFwlHd4nGPz85JaGU1NmovvKdzZfSSo5ifAfFnnneTC59apwj1LQs+BbF8ywzOq9K+6ZLznAUtAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T18:21:40.159934Z","bundle_sha256":"113d34dcb0c71450892134bde19ad5c99c60264718bdb2bdcbc659a842a41fda"}}