{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:K46G3PG5E3PS5HLMADPYCWH2ZM","short_pith_number":"pith:K46G3PG5","canonical_record":{"source":{"id":"math/0409018","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2004-09-01T14:35:20Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"a95879bac4b6f9c08a5955177cf63e10fb9936dffda11ee2e823d88172e6ec85","abstract_canon_sha256":"858f3dd9ccea64de3a2e9ccb6fb47055c349b9614f319ee32865b51eef5cd480"},"schema_version":"1.0"},"canonical_sha256":"573c6dbcdd26df2e9d6c00df8158facb05f88b380b3cbe849ada0e42ca04968a","source":{"kind":"arxiv","id":"math/0409018","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0409018","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"arxiv_version","alias_value":"math/0409018v1","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0409018","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"pith_short_12","alias_value":"K46G3PG5E3PS","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"pith_short_16","alias_value":"K46G3PG5E3PS5HLM","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"pith_short_8","alias_value":"K46G3PG5","created_at":"2026-07-04T14:39:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:K46G3PG5E3PS5HLMADPYCWH2ZM","target":"record","payload":{"canonical_record":{"source":{"id":"math/0409018","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2004-09-01T14:35:20Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"a95879bac4b6f9c08a5955177cf63e10fb9936dffda11ee2e823d88172e6ec85","abstract_canon_sha256":"858f3dd9ccea64de3a2e9ccb6fb47055c349b9614f319ee32865b51eef5cd480"},"schema_version":"1.0"},"canonical_sha256":"573c6dbcdd26df2e9d6c00df8158facb05f88b380b3cbe849ada0e42ca04968a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:39:02.090593Z","signature_b64":"S2E2LCRwveoSyZBGIRD3AIy8bvAeU0P4jiC5YlRylCGl3yoIUAIPSSLIEWvCQ7YjHiIjqAZFOampaE9aVxV2DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"573c6dbcdd26df2e9d6c00df8158facb05f88b380b3cbe849ada0e42ca04968a","last_reissued_at":"2026-07-04T14:39:02.090179Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:39:02.090179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0409018","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":"2026-07-04T14:39:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WozZCGqywRy+ZXE3L4TOsWzXjgVd7p8rNBfm51CjVXYVdDvsCfYwuch5keGvWtmXyc6VE1gDWSCraguqDzpUDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-21T11:43:38.516632Z"},"content_sha256":"73652b9f9252a2fee023e17cb9ee8a2d07c363067c5de25a7a079b92a577948b","schema_version":"1.0","event_id":"sha256:73652b9f9252a2fee023e17cb9ee8a2d07c363067c5de25a7a079b92a577948b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:K46G3PG5E3PS5HLMADPYCWH2ZM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mixed norm and multidimensional Lorentz spaces","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Anna Kaminska, Javier Soria, Lars-Erik Persson, Sorina Barza","submitted_at":"2004-09-01T14:35:20Z","abstract_excerpt":"In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (\\cite{Sa}, \\cite{CaSoA}). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0409018","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0409018/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T14:39:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CzUW3Xa9iRWF3w2NLFfulINU2f1/Peb2b829WAkRL2fMgZWbTmG3PqKetH34Taz1pDqBh9XzmACxnDJUEU6UDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-21T11:43:38.517014Z"},"content_sha256":"7c9c243e70e88739b706a5d7e7d2427b0b96c2faf8f6fd2878e8c1f0829013af","schema_version":"1.0","event_id":"sha256:7c9c243e70e88739b706a5d7e7d2427b0b96c2faf8f6fd2878e8c1f0829013af"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K46G3PG5E3PS5HLMADPYCWH2ZM/bundle.json","state_url":"https://pith.science/pith/K46G3PG5E3PS5HLMADPYCWH2ZM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K46G3PG5E3PS5HLMADPYCWH2ZM/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":"2026-07-21T11:43:38Z","links":{"resolver":"https://pith.science/pith/K46G3PG5E3PS5HLMADPYCWH2ZM","bundle":"https://pith.science/pith/K46G3PG5E3PS5HLMADPYCWH2ZM/bundle.json","state":"https://pith.science/pith/K46G3PG5E3PS5HLMADPYCWH2ZM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K46G3PG5E3PS5HLMADPYCWH2ZM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:K46G3PG5E3PS5HLMADPYCWH2ZM","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":"858f3dd9ccea64de3a2e9ccb6fb47055c349b9614f319ee32865b51eef5cd480","cross_cats_sorted":["math.FA"],"license":"","primary_cat":"math.CA","submitted_at":"2004-09-01T14:35:20Z","title_canon_sha256":"a95879bac4b6f9c08a5955177cf63e10fb9936dffda11ee2e823d88172e6ec85"},"schema_version":"1.0","source":{"id":"math/0409018","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0409018","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"arxiv_version","alias_value":"math/0409018v1","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0409018","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"pith_short_12","alias_value":"K46G3PG5E3PS","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"pith_short_16","alias_value":"K46G3PG5E3PS5HLM","created_at":"2026-07-04T14:39:02Z"},{"alias_kind":"pith_short_8","alias_value":"K46G3PG5","created_at":"2026-07-04T14:39:02Z"}],"graph_snapshots":[{"event_id":"sha256:7c9c243e70e88739b706a5d7e7d2427b0b96c2faf8f6fd2878e8c1f0829013af","target":"graph","created_at":"2026-07-04T14:39:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0409018/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (\\cite{Sa}, \\cite{CaSoA}). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz s","authors_text":"Anna Kaminska, Javier Soria, Lars-Erik Persson, Sorina Barza","cross_cats":["math.FA"],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"2004-09-01T14:35:20Z","title":"Mixed norm and multidimensional Lorentz spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0409018","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":"sha256:73652b9f9252a2fee023e17cb9ee8a2d07c363067c5de25a7a079b92a577948b","target":"record","created_at":"2026-07-04T14:39:02Z","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":"858f3dd9ccea64de3a2e9ccb6fb47055c349b9614f319ee32865b51eef5cd480","cross_cats_sorted":["math.FA"],"license":"","primary_cat":"math.CA","submitted_at":"2004-09-01T14:35:20Z","title_canon_sha256":"a95879bac4b6f9c08a5955177cf63e10fb9936dffda11ee2e823d88172e6ec85"},"schema_version":"1.0","source":{"id":"math/0409018","kind":"arxiv","version":1}},"canonical_sha256":"573c6dbcdd26df2e9d6c00df8158facb05f88b380b3cbe849ada0e42ca04968a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"573c6dbcdd26df2e9d6c00df8158facb05f88b380b3cbe849ada0e42ca04968a","first_computed_at":"2026-07-04T14:39:02.090179Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:39:02.090179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S2E2LCRwveoSyZBGIRD3AIy8bvAeU0P4jiC5YlRylCGl3yoIUAIPSSLIEWvCQ7YjHiIjqAZFOampaE9aVxV2DQ==","signature_status":"signed_v1","signed_at":"2026-07-04T14:39:02.090593Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0409018","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73652b9f9252a2fee023e17cb9ee8a2d07c363067c5de25a7a079b92a577948b","sha256:7c9c243e70e88739b706a5d7e7d2427b0b96c2faf8f6fd2878e8c1f0829013af"],"state_sha256":"0544a85efcf6eb93b7e642bc77e678611e95bf0c8bc0e862a6dc9ec6682052c4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2gyoOJZ/ER0Ec8ZbPOqOTD9WpCugc5MOAYeUcVOpJmSdaXzW0UhXLoeQ8m4w3lj6fel4WKCYgtgg2evuA2ZUCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-21T11:43:38.519603Z","bundle_sha256":"0cfff52856f26b340a2afed1fd7b74b37465d5633b9a6a1f1af513dc427a68a1"}}