{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:EL3SZY3JO54CDSAG6SGBLGHEST","short_pith_number":"pith:EL3SZY3J","canonical_record":{"source":{"id":"1202.2183","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-02-10T04:19:04Z","cross_cats_sorted":[],"title_canon_sha256":"0273964976d219b353f44e3cb51b54e307ff8325dd3ba3c11e5b0d2e92d4e854","abstract_canon_sha256":"c4381b686468517c7a6237bb056e064370f64bd57acbc3e7f5163a9076f6c448"},"schema_version":"1.0"},"canonical_sha256":"22f72ce369777821c806f48c1598e494f6af14fea91eaa73a545209913fd94a0","source":{"kind":"arxiv","id":"1202.2183","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.2183","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"arxiv_version","alias_value":"1202.2183v2","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.2183","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"pith_short_12","alias_value":"EL3SZY3JO54C","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EL3SZY3JO54CDSAG","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EL3SZY3J","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:EL3SZY3JO54CDSAG6SGBLGHEST","target":"record","payload":{"canonical_record":{"source":{"id":"1202.2183","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-02-10T04:19:04Z","cross_cats_sorted":[],"title_canon_sha256":"0273964976d219b353f44e3cb51b54e307ff8325dd3ba3c11e5b0d2e92d4e854","abstract_canon_sha256":"c4381b686468517c7a6237bb056e064370f64bd57acbc3e7f5163a9076f6c448"},"schema_version":"1.0"},"canonical_sha256":"22f72ce369777821c806f48c1598e494f6af14fea91eaa73a545209913fd94a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:41.506271Z","signature_b64":"963OojD/jvyte2uo/9mCTPXaUkC9Fc/k3yKHRg+3urMjsBw0f7lhO5/fjqPZbfGqEhuaT/BQxrstFK24eYidAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22f72ce369777821c806f48c1598e494f6af14fea91eaa73a545209913fd94a0","last_reissued_at":"2026-05-18T03:37:41.505614Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:41.505614Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.2183","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:37:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"buOWK85/AefXhED2CBD0q2zfyrtZEstlHqv10H9dMnA/Nn7By2T3hifanTkHaveghZdJPZHWAz58TaW8j7OZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T00:51:39.957105Z"},"content_sha256":"e761b3d5b3d1a9eb80c0d18dd75ba2ff4fe3167ed969c5c344a31eeaa0a3496b","schema_version":"1.0","event_id":"sha256:e761b3d5b3d1a9eb80c0d18dd75ba2ff4fe3167ed969c5c344a31eeaa0a3496b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:EL3SZY3JO54CDSAG6SGBLGHEST","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lipschitz spaces and bounded mean oscillation of harmonic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"M. Vuorinen, SH. Chen, S. Ponnusamy, X. Wang","submitted_at":"2012-02-10T04:19:04Z","abstract_excerpt":"In this paper, we first study the bounded mean oscillation of planar harmonic mappings, then a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings is established. At last, we obtain sharp estimates on Lipschitz number of planar harmonic mappings in terms of bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to $BMO_{2}$ as a Banach space.."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2183","kind":"arxiv","version":2},"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":"2026-05-18T03:37:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eqasMJHc9WbaMSF/q5nvJoEI7UBgo1ftJo387JNrV1IQfG7tXIyqo3hcHGKucc42cwHX1tEsZ3jUFL8Y9pHNCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T00:51:39.957455Z"},"content_sha256":"4d70a038dac15e0c9c2d3a934a9ad0c44e0b32baa8d38cfc896c2fed5d4d3339","schema_version":"1.0","event_id":"sha256:4d70a038dac15e0c9c2d3a934a9ad0c44e0b32baa8d38cfc896c2fed5d4d3339"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EL3SZY3JO54CDSAG6SGBLGHEST/bundle.json","state_url":"https://pith.science/pith/EL3SZY3JO54CDSAG6SGBLGHEST/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EL3SZY3JO54CDSAG6SGBLGHEST/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":"2026-06-27T00:51:39Z","links":{"resolver":"https://pith.science/pith/EL3SZY3JO54CDSAG6SGBLGHEST","bundle":"https://pith.science/pith/EL3SZY3JO54CDSAG6SGBLGHEST/bundle.json","state":"https://pith.science/pith/EL3SZY3JO54CDSAG6SGBLGHEST/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EL3SZY3JO54CDSAG6SGBLGHEST/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:EL3SZY3JO54CDSAG6SGBLGHEST","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":"c4381b686468517c7a6237bb056e064370f64bd57acbc3e7f5163a9076f6c448","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-02-10T04:19:04Z","title_canon_sha256":"0273964976d219b353f44e3cb51b54e307ff8325dd3ba3c11e5b0d2e92d4e854"},"schema_version":"1.0","source":{"id":"1202.2183","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.2183","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"arxiv_version","alias_value":"1202.2183v2","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.2183","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"pith_short_12","alias_value":"EL3SZY3JO54C","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EL3SZY3JO54CDSAG","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EL3SZY3J","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:4d70a038dac15e0c9c2d3a934a9ad0c44e0b32baa8d38cfc896c2fed5d4d3339","target":"graph","created_at":"2026-05-18T03:37:41Z","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 paper, we first study the bounded mean oscillation of planar harmonic mappings, then a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings is established. At last, we obtain sharp estimates on Lipschitz number of planar harmonic mappings in terms of bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to $BMO_{2}$ as a Banach space..","authors_text":"M. Vuorinen, SH. Chen, S. Ponnusamy, X. Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-02-10T04:19:04Z","title":"Lipschitz spaces and bounded mean oscillation of harmonic mappings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2183","kind":"arxiv","version":2},"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:e761b3d5b3d1a9eb80c0d18dd75ba2ff4fe3167ed969c5c344a31eeaa0a3496b","target":"record","created_at":"2026-05-18T03:37:41Z","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":"c4381b686468517c7a6237bb056e064370f64bd57acbc3e7f5163a9076f6c448","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-02-10T04:19:04Z","title_canon_sha256":"0273964976d219b353f44e3cb51b54e307ff8325dd3ba3c11e5b0d2e92d4e854"},"schema_version":"1.0","source":{"id":"1202.2183","kind":"arxiv","version":2}},"canonical_sha256":"22f72ce369777821c806f48c1598e494f6af14fea91eaa73a545209913fd94a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"22f72ce369777821c806f48c1598e494f6af14fea91eaa73a545209913fd94a0","first_computed_at":"2026-05-18T03:37:41.505614Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:41.505614Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"963OojD/jvyte2uo/9mCTPXaUkC9Fc/k3yKHRg+3urMjsBw0f7lhO5/fjqPZbfGqEhuaT/BQxrstFK24eYidAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:41.506271Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.2183","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e761b3d5b3d1a9eb80c0d18dd75ba2ff4fe3167ed969c5c344a31eeaa0a3496b","sha256:4d70a038dac15e0c9c2d3a934a9ad0c44e0b32baa8d38cfc896c2fed5d4d3339"],"state_sha256":"a206ef55ad1006533bf1a5567e9a62d9d60936ef9324a9333918fb80e677dcd5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v4EsgEdGX7OYaO+r5caUf+t3AiCIBCVIPG/lBw0E7MkiBsk5/nEIpm/03LfCyUrS0WtefeeRNpV9qIw9aJjjAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T00:51:39.959230Z","bundle_sha256":"5c501bed7c24d8190e208faf916c8f9e375a44a8a2a6b06c3781d6565856e3e9"}}