{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:DE2O4MQLEDJYKJKMZQLZDZ6WJS","short_pith_number":"pith:DE2O4MQL","canonical_record":{"source":{"id":"1303.6351","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-26T00:07:22Z","cross_cats_sorted":[],"title_canon_sha256":"0208d29cc188de5c2cbe94b90201243e13d4687313ea89b753c9addad64e752a","abstract_canon_sha256":"738c2e2feeb43f91fb51937e015e431ab6e5b957a33a98dca198312a5542eb80"},"schema_version":"1.0"},"canonical_sha256":"1934ee320b20d385254ccc1791e7d64cbb41fb31f1988257addf6ba7db90908f","source":{"kind":"arxiv","id":"1303.6351","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6351","created_at":"2026-05-18T03:29:49Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6351v1","created_at":"2026-05-18T03:29:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6351","created_at":"2026-05-18T03:29:49Z"},{"alias_kind":"pith_short_12","alias_value":"DE2O4MQLEDJY","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DE2O4MQLEDJYKJKM","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DE2O4MQL","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:DE2O4MQLEDJYKJKMZQLZDZ6WJS","target":"record","payload":{"canonical_record":{"source":{"id":"1303.6351","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-26T00:07:22Z","cross_cats_sorted":[],"title_canon_sha256":"0208d29cc188de5c2cbe94b90201243e13d4687313ea89b753c9addad64e752a","abstract_canon_sha256":"738c2e2feeb43f91fb51937e015e431ab6e5b957a33a98dca198312a5542eb80"},"schema_version":"1.0"},"canonical_sha256":"1934ee320b20d385254ccc1791e7d64cbb41fb31f1988257addf6ba7db90908f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:49.674496Z","signature_b64":"+/kc7+RTdxcXJ/gHGAtxxMMWOcXPJXf6nqGA66PvZ6pN2I6uvmIeaOm7atcRPu1GqAalihXa6qzhEDyds92DCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1934ee320b20d385254ccc1791e7d64cbb41fb31f1988257addf6ba7db90908f","last_reissued_at":"2026-05-18T03:29:49.673743Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:49.673743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.6351","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":"2026-05-18T03:29:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8U/Dx1dwP66IrlkSALECsGXbou5g4bBl22wLTwABD8YVmdtjirrqt7MMSBp0HeFiDLMT2P4fzFYSW7D8h5I6AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:12:39.514975Z"},"content_sha256":"fade7817e2bd72134d32c92816e185bb14c7ccb5e1dbef5c1e4cdda9a52af5c3","schema_version":"1.0","event_id":"sha256:fade7817e2bd72134d32c92816e185bb14c7ccb5e1dbef5c1e4cdda9a52af5c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:DE2O4MQLEDJYKJKMZQLZDZ6WJS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David S. McCormick, James C. Robinson, Jose L. Rodrigo","submitted_at":"2013-03-26T00:07:22Z","abstract_excerpt":"Using elementary arguments based on the Fourier transform we prove that for $1 \\leq q < p < \\infty$ and $s \\geq 0$ with $s > n(1/2-1/p)$, if $f \\in L^{q,\\infty}(\\R^n) \\cap \\dot{H}^s(\\R^n)$ then $f \\in L^p(\\R^n)$ and there exists a constant $c_{p,q,s}$ such that\n  \\[ \\|f\\|_{L^p} \\leq c_{p,q,s} \\|f\\|_{L^{q,\\infty}}^\\theta \\|f\\|_{\\dot H^s}^{1-\\theta}, \\] where $1/p = \\theta/q + (1-\\theta)(1/2-s/n)$. In particular, in $\\R^2$ we obtain the generalised Ladyzhenskaya inequality $\\|f\\|_{L^4}\\le c\\|f\\|_{L^{2,\\infty}}^{1/2}\\|f\\|_{\\dot H^1}^{1/2}$. We also show that for $s=n/2$ the norm in $\\|f\\|_{\\dot H"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6351","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":"2026-05-18T03:29:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aihIBc22cq3Kl5wmVosPufNP9lc89079mQ1LDPuD0iRnd1ICm1bYNF7IQtcvESGd75jDn+VHd1s+/I1DGUJtBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:12:39.515352Z"},"content_sha256":"037911cbea0bcd8a44a6d06e97d4f904968386beaed99a92819c257bc63d90f6","schema_version":"1.0","event_id":"sha256:037911cbea0bcd8a44a6d06e97d4f904968386beaed99a92819c257bc63d90f6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DE2O4MQLEDJYKJKMZQLZDZ6WJS/bundle.json","state_url":"https://pith.science/pith/DE2O4MQLEDJYKJKMZQLZDZ6WJS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DE2O4MQLEDJYKJKMZQLZDZ6WJS/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":"2026-06-01T04:12:39Z","links":{"resolver":"https://pith.science/pith/DE2O4MQLEDJYKJKMZQLZDZ6WJS","bundle":"https://pith.science/pith/DE2O4MQLEDJYKJKMZQLZDZ6WJS/bundle.json","state":"https://pith.science/pith/DE2O4MQLEDJYKJKMZQLZDZ6WJS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DE2O4MQLEDJYKJKMZQLZDZ6WJS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:DE2O4MQLEDJYKJKMZQLZDZ6WJS","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":"738c2e2feeb43f91fb51937e015e431ab6e5b957a33a98dca198312a5542eb80","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-26T00:07:22Z","title_canon_sha256":"0208d29cc188de5c2cbe94b90201243e13d4687313ea89b753c9addad64e752a"},"schema_version":"1.0","source":{"id":"1303.6351","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6351","created_at":"2026-05-18T03:29:49Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6351v1","created_at":"2026-05-18T03:29:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6351","created_at":"2026-05-18T03:29:49Z"},{"alias_kind":"pith_short_12","alias_value":"DE2O4MQLEDJY","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DE2O4MQLEDJYKJKM","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DE2O4MQL","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:037911cbea0bcd8a44a6d06e97d4f904968386beaed99a92819c257bc63d90f6","target":"graph","created_at":"2026-05-18T03:29:49Z","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":"Using elementary arguments based on the Fourier transform we prove that for $1 \\leq q < p < \\infty$ and $s \\geq 0$ with $s > n(1/2-1/p)$, if $f \\in L^{q,\\infty}(\\R^n) \\cap \\dot{H}^s(\\R^n)$ then $f \\in L^p(\\R^n)$ and there exists a constant $c_{p,q,s}$ such that\n  \\[ \\|f\\|_{L^p} \\leq c_{p,q,s} \\|f\\|_{L^{q,\\infty}}^\\theta \\|f\\|_{\\dot H^s}^{1-\\theta}, \\] where $1/p = \\theta/q + (1-\\theta)(1/2-s/n)$. In particular, in $\\R^2$ we obtain the generalised Ladyzhenskaya inequality $\\|f\\|_{L^4}\\le c\\|f\\|_{L^{2,\\infty}}^{1/2}\\|f\\|_{\\dot H^1}^{1/2}$. We also show that for $s=n/2$ the norm in $\\|f\\|_{\\dot H","authors_text":"David S. McCormick, James C. Robinson, Jose L. Rodrigo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-26T00:07:22Z","title":"Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6351","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:fade7817e2bd72134d32c92816e185bb14c7ccb5e1dbef5c1e4cdda9a52af5c3","target":"record","created_at":"2026-05-18T03:29:49Z","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":"738c2e2feeb43f91fb51937e015e431ab6e5b957a33a98dca198312a5542eb80","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-03-26T00:07:22Z","title_canon_sha256":"0208d29cc188de5c2cbe94b90201243e13d4687313ea89b753c9addad64e752a"},"schema_version":"1.0","source":{"id":"1303.6351","kind":"arxiv","version":1}},"canonical_sha256":"1934ee320b20d385254ccc1791e7d64cbb41fb31f1988257addf6ba7db90908f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1934ee320b20d385254ccc1791e7d64cbb41fb31f1988257addf6ba7db90908f","first_computed_at":"2026-05-18T03:29:49.673743Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:49.673743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+/kc7+RTdxcXJ/gHGAtxxMMWOcXPJXf6nqGA66PvZ6pN2I6uvmIeaOm7atcRPu1GqAalihXa6qzhEDyds92DCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:49.674496Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.6351","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fade7817e2bd72134d32c92816e185bb14c7ccb5e1dbef5c1e4cdda9a52af5c3","sha256:037911cbea0bcd8a44a6d06e97d4f904968386beaed99a92819c257bc63d90f6"],"state_sha256":"c488de6c0b917733590e6266d952000d6ae9d622a85d34b8e9dfa1d818664f16"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"heAJz24K8XJnjY/RrcQ2DJ+KrgNMIFwZSyEts7o1j4gI6c+7nSXKiIrgqMq8773Q7OV6vHDG+Anqq4xG/nDYDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T04:12:39.518196Z","bundle_sha256":"7d772df051e094069356c07792336f08a9d536414c4a2571b75e247dd394797c"}}