{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:P5562ZWLLDOOYS5EOJDVPDPCYY","short_pith_number":"pith:P5562ZWL","canonical_record":{"source":{"id":"1004.4287","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-24T15:25:32Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"54faf5439940b60cfa6ca04ea026623e4e349817626c20e2719b3e9eb4aa5a89","abstract_canon_sha256":"1412c2c15c2101e87c2aa6b494210d6743a22f7b4cd5677abd5ce236b585e6cf"},"schema_version":"1.0"},"canonical_sha256":"7f7bed66cb58dcec4ba47247578de2c6044a65e29c71bf7b80abf511259ea194","source":{"kind":"arxiv","id":"1004.4287","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.4287","created_at":"2026-05-18T04:30:48Z"},{"alias_kind":"arxiv_version","alias_value":"1004.4287v3","created_at":"2026-05-18T04:30:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.4287","created_at":"2026-05-18T04:30:48Z"},{"alias_kind":"pith_short_12","alias_value":"P5562ZWLLDOO","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"P5562ZWLLDOOYS5E","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"P5562ZWL","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:P5562ZWLLDOOYS5EOJDVPDPCYY","target":"record","payload":{"canonical_record":{"source":{"id":"1004.4287","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-24T15:25:32Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"54faf5439940b60cfa6ca04ea026623e4e349817626c20e2719b3e9eb4aa5a89","abstract_canon_sha256":"1412c2c15c2101e87c2aa6b494210d6743a22f7b4cd5677abd5ce236b585e6cf"},"schema_version":"1.0"},"canonical_sha256":"7f7bed66cb58dcec4ba47247578de2c6044a65e29c71bf7b80abf511259ea194","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:30:48.523363Z","signature_b64":"XgZ72O5luWUOmUGVcAGCFg/VOZZNpL6XeBBpy6QpdJytI1ugczZannwhS8IgyCoCbu5EdzP7ueXDlFwKS5i4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f7bed66cb58dcec4ba47247578de2c6044a65e29c71bf7b80abf511259ea194","last_reissued_at":"2026-05-18T04:30:48.522682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:30:48.522682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.4287","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:30:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z4nW9VC13+uiQQlYX6Pe7g5K2kVrNv+AfDTH7xbwsuRoGrQMsywrhDtE68bG4fjGeXukVDk8iuuZw0zrSgsLCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T07:33:15.316170Z"},"content_sha256":"bf2d30ca026c36baf8511b5e86b3c1c03d9557e25756e0a416d62e11c425eb8b","schema_version":"1.0","event_id":"sha256:bf2d30ca026c36baf8511b5e86b3c1c03d9557e25756e0a416d62e11c425eb8b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:P5562ZWLLDOOYS5EOJDVPDPCYY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Baoxiang Wang, Hichem Hajaiej, Luc Molinet, Tohru Ozawa","submitted_at":"2010-04-24T15:25:32Z","abstract_excerpt":"Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4287","kind":"arxiv","version":3},"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":"2026-05-18T04:30:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fj2mfrOS0255jdRj+H2fd0fyOmBiW952BB3kSlcPSCASCxaIhlgD3+ABgJlvPB7h3qhi4yh6wHIUhbxoy8jNAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T07:33:15.316630Z"},"content_sha256":"9cef4b76df262717f20dac3c226732f6c04d260857b8e4a5ea1a204f64044597","schema_version":"1.0","event_id":"sha256:9cef4b76df262717f20dac3c226732f6c04d260857b8e4a5ea1a204f64044597"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P5562ZWLLDOOYS5EOJDVPDPCYY/bundle.json","state_url":"https://pith.science/pith/P5562ZWLLDOOYS5EOJDVPDPCYY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P5562ZWLLDOOYS5EOJDVPDPCYY/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":"2026-05-25T07:33:15Z","links":{"resolver":"https://pith.science/pith/P5562ZWLLDOOYS5EOJDVPDPCYY","bundle":"https://pith.science/pith/P5562ZWLLDOOYS5EOJDVPDPCYY/bundle.json","state":"https://pith.science/pith/P5562ZWLLDOOYS5EOJDVPDPCYY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P5562ZWLLDOOYS5EOJDVPDPCYY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:P5562ZWLLDOOYS5EOJDVPDPCYY","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":"1412c2c15c2101e87c2aa6b494210d6743a22f7b4cd5677abd5ce236b585e6cf","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-24T15:25:32Z","title_canon_sha256":"54faf5439940b60cfa6ca04ea026623e4e349817626c20e2719b3e9eb4aa5a89"},"schema_version":"1.0","source":{"id":"1004.4287","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.4287","created_at":"2026-05-18T04:30:48Z"},{"alias_kind":"arxiv_version","alias_value":"1004.4287v3","created_at":"2026-05-18T04:30:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.4287","created_at":"2026-05-18T04:30:48Z"},{"alias_kind":"pith_short_12","alias_value":"P5562ZWLLDOO","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"P5562ZWLLDOOYS5E","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"P5562ZWL","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:9cef4b76df262717f20dac3c226732f6c04d260857b8e4a5ea1a204f64044597","target":"graph","created_at":"2026-05-18T04:30:48Z","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":"Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.","authors_text":"Baoxiang Wang, Hichem Hajaiej, Luc Molinet, Tohru Ozawa","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-24T15:25:32Z","title":"Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4287","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:bf2d30ca026c36baf8511b5e86b3c1c03d9557e25756e0a416d62e11c425eb8b","target":"record","created_at":"2026-05-18T04:30:48Z","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":"1412c2c15c2101e87c2aa6b494210d6743a22f7b4cd5677abd5ce236b585e6cf","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-24T15:25:32Z","title_canon_sha256":"54faf5439940b60cfa6ca04ea026623e4e349817626c20e2719b3e9eb4aa5a89"},"schema_version":"1.0","source":{"id":"1004.4287","kind":"arxiv","version":3}},"canonical_sha256":"7f7bed66cb58dcec4ba47247578de2c6044a65e29c71bf7b80abf511259ea194","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7f7bed66cb58dcec4ba47247578de2c6044a65e29c71bf7b80abf511259ea194","first_computed_at":"2026-05-18T04:30:48.522682Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:30:48.522682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XgZ72O5luWUOmUGVcAGCFg/VOZZNpL6XeBBpy6QpdJytI1ugczZannwhS8IgyCoCbu5EdzP7ueXDlFwKS5i4BA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:30:48.523363Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.4287","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf2d30ca026c36baf8511b5e86b3c1c03d9557e25756e0a416d62e11c425eb8b","sha256:9cef4b76df262717f20dac3c226732f6c04d260857b8e4a5ea1a204f64044597"],"state_sha256":"a44afe3c83b80927d7425c7ebf591d6d5259b808fc0878407a6dede2200106cc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WcgUG6197BbloV+fn24RCGcLqI+XBKhARof78oVom2KjCOhTIiTz2flfWub6WqtMVwwidK5vTWsPjMe7DczbAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T07:33:15.319364Z","bundle_sha256":"f41657cfe4905851d39231683061612656e693a29faa4aebe241bd47f3f3dd89"}}