{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XDPJDJEFWCUUI5HCAO77BUCNNE","short_pith_number":"pith:XDPJDJEF","canonical_record":{"source":{"id":"1307.2670","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-07-10T04:48:14Z","cross_cats_sorted":[],"title_canon_sha256":"209a49ec3bd6bf93852ec928cf3236beb7bd554aeae3038fbcc20dbd905b29d0","abstract_canon_sha256":"8a85d6a0488974a0a4b9c5694b0b84bfd53a6e81e2bf1ad8ec62372b4eed9263"},"schema_version":"1.0"},"canonical_sha256":"b8de91a485b0a94474e203bff0d04d693988c21ff087ab93b9b1b4f6d73a7b7e","source":{"kind":"arxiv","id":"1307.2670","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.2670","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"arxiv_version","alias_value":"1307.2670v1","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2670","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"pith_short_12","alias_value":"XDPJDJEFWCUU","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XDPJDJEFWCUUI5HC","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XDPJDJEF","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XDPJDJEFWCUUI5HCAO77BUCNNE","target":"record","payload":{"canonical_record":{"source":{"id":"1307.2670","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-07-10T04:48:14Z","cross_cats_sorted":[],"title_canon_sha256":"209a49ec3bd6bf93852ec928cf3236beb7bd554aeae3038fbcc20dbd905b29d0","abstract_canon_sha256":"8a85d6a0488974a0a4b9c5694b0b84bfd53a6e81e2bf1ad8ec62372b4eed9263"},"schema_version":"1.0"},"canonical_sha256":"b8de91a485b0a94474e203bff0d04d693988c21ff087ab93b9b1b4f6d73a7b7e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:50.774011Z","signature_b64":"ZD4ggeK2eG3kMV22ueFL7uOSwt1vXBEHVgX7CGLBhLGQeguQBbMJ2rNJ4x1kH5YPHBDgS27dSCbWXbsdhCOzBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8de91a485b0a94474e203bff0d04d693988c21ff087ab93b9b1b4f6d73a7b7e","last_reissued_at":"2026-05-18T03:18:50.773088Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:50.773088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.2670","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":"2026-05-18T03:18:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IIcT0u9BgR9DFnUmBKSQvm2hOXClzJwXBLBzeC+bwZWXQTnoCP/Ssk9oe85+970EZFqj9NeDSXs5Z5BI8GPuAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:13:33.474970Z"},"content_sha256":"2466d5da8d917b105586284c0ae95ce9c6c5c602f45bcbb26199a16311647e7b","schema_version":"1.0","event_id":"sha256:2466d5da8d917b105586284c0ae95ce9c6c5c602f45bcbb26199a16311647e7b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XDPJDJEFWCUUI5HCAO77BUCNNE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fock-Sobolev spaces of fractional order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Boo Rim Choe, Hong Rae Cho, Hyungwoon Koo","submitted_at":"2013-07-10T04:48:14Z","abstract_excerpt":"For the full range of index $0<p\\leq\\infty$, real weight $\\alpha$ and real Sobolev order $s$, two types of weighted Fock-Sobolev spaces over $\\mathbb C^n$, $F^p_{\\alpha, s}$ and $\\widetilde F^p_{\\alpha,s}$, are introduced through fractional differentiation and through fractional integration, respectively. We show that they are the same with equivalent norms and, furthermore, that they are identified with the weighted Fock space $F^p_{\\alpha-sp,0}$ for the full range of parameters. So, the study on the weighted Fock-Sobolev spaces is reduced to that on the weighted Fock spaces. We describe expl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2670","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":"2026-05-18T03:18:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jnaCdeaKjM5urBW6KKsQ3e2my92Q1/YVnowYJIN48JFKcdjSRL2c2RxdOfzjCh/Z164BSa7LFRD+N/9q1XhQDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:13:33.475329Z"},"content_sha256":"f27e38576278c71371d757cafe369c4fd8b150b56dc5a1c81888de29bffce87c","schema_version":"1.0","event_id":"sha256:f27e38576278c71371d757cafe369c4fd8b150b56dc5a1c81888de29bffce87c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XDPJDJEFWCUUI5HCAO77BUCNNE/bundle.json","state_url":"https://pith.science/pith/XDPJDJEFWCUUI5HCAO77BUCNNE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XDPJDJEFWCUUI5HCAO77BUCNNE/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":"2026-06-26T01:13:33Z","links":{"resolver":"https://pith.science/pith/XDPJDJEFWCUUI5HCAO77BUCNNE","bundle":"https://pith.science/pith/XDPJDJEFWCUUI5HCAO77BUCNNE/bundle.json","state":"https://pith.science/pith/XDPJDJEFWCUUI5HCAO77BUCNNE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XDPJDJEFWCUUI5HCAO77BUCNNE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XDPJDJEFWCUUI5HCAO77BUCNNE","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":"8a85d6a0488974a0a4b9c5694b0b84bfd53a6e81e2bf1ad8ec62372b4eed9263","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-07-10T04:48:14Z","title_canon_sha256":"209a49ec3bd6bf93852ec928cf3236beb7bd554aeae3038fbcc20dbd905b29d0"},"schema_version":"1.0","source":{"id":"1307.2670","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.2670","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"arxiv_version","alias_value":"1307.2670v1","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.2670","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"pith_short_12","alias_value":"XDPJDJEFWCUU","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XDPJDJEFWCUUI5HC","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XDPJDJEF","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:f27e38576278c71371d757cafe369c4fd8b150b56dc5a1c81888de29bffce87c","target":"graph","created_at":"2026-05-18T03:18:50Z","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":"For the full range of index $0<p\\leq\\infty$, real weight $\\alpha$ and real Sobolev order $s$, two types of weighted Fock-Sobolev spaces over $\\mathbb C^n$, $F^p_{\\alpha, s}$ and $\\widetilde F^p_{\\alpha,s}$, are introduced through fractional differentiation and through fractional integration, respectively. We show that they are the same with equivalent norms and, furthermore, that they are identified with the weighted Fock space $F^p_{\\alpha-sp,0}$ for the full range of parameters. So, the study on the weighted Fock-Sobolev spaces is reduced to that on the weighted Fock spaces. We describe expl","authors_text":"Boo Rim Choe, Hong Rae Cho, Hyungwoon Koo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-07-10T04:48:14Z","title":"Fock-Sobolev spaces of fractional order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2670","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":"sha256:2466d5da8d917b105586284c0ae95ce9c6c5c602f45bcbb26199a16311647e7b","target":"record","created_at":"2026-05-18T03:18:50Z","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":"8a85d6a0488974a0a4b9c5694b0b84bfd53a6e81e2bf1ad8ec62372b4eed9263","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-07-10T04:48:14Z","title_canon_sha256":"209a49ec3bd6bf93852ec928cf3236beb7bd554aeae3038fbcc20dbd905b29d0"},"schema_version":"1.0","source":{"id":"1307.2670","kind":"arxiv","version":1}},"canonical_sha256":"b8de91a485b0a94474e203bff0d04d693988c21ff087ab93b9b1b4f6d73a7b7e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8de91a485b0a94474e203bff0d04d693988c21ff087ab93b9b1b4f6d73a7b7e","first_computed_at":"2026-05-18T03:18:50.773088Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:50.773088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZD4ggeK2eG3kMV22ueFL7uOSwt1vXBEHVgX7CGLBhLGQeguQBbMJ2rNJ4x1kH5YPHBDgS27dSCbWXbsdhCOzBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:50.774011Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.2670","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2466d5da8d917b105586284c0ae95ce9c6c5c602f45bcbb26199a16311647e7b","sha256:f27e38576278c71371d757cafe369c4fd8b150b56dc5a1c81888de29bffce87c"],"state_sha256":"bf8fb393c366db8c74e6a8ec9f6942741532f56c26a817eeeac678636f8513ab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ATksRw2qsGuZPce559ppZldQ1kQyB0BRPmJ2s+59uIQEm5ddYmuYMF+pDiwdElTh/vN2Vf77FeXgxSev046FDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T01:13:33.477347Z","bundle_sha256":"6c121b534aa356a15c0466de930b4e435cfc33a8addefbb9820b2f6c2974d444"}}