{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:G4B3XW5AWIESCDB4QG536LZVJZ","short_pith_number":"pith:G4B3XW5A","canonical_record":{"source":{"id":"1408.3642","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-08-15T20:10:24Z","cross_cats_sorted":[],"title_canon_sha256":"95bb8f422b04c89c63ba8b53fedde17fe4b5f355af081dd56c273453de91600c","abstract_canon_sha256":"b656e747ee87f061d42f5b297b3134ae651046e867bacf9a8221e708658996cb"},"schema_version":"1.0"},"canonical_sha256":"3703bbdba0b209210c3c81bbbf2f354e5ada10a9af592f4c1ef91fc32aa6fb32","source":{"kind":"arxiv","id":"1408.3642","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3642","created_at":"2026-05-18T02:19:59Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3642v2","created_at":"2026-05-18T02:19:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3642","created_at":"2026-05-18T02:19:59Z"},{"alias_kind":"pith_short_12","alias_value":"G4B3XW5AWIES","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"G4B3XW5AWIESCDB4","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"G4B3XW5A","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:G4B3XW5AWIESCDB4QG536LZVJZ","target":"record","payload":{"canonical_record":{"source":{"id":"1408.3642","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-08-15T20:10:24Z","cross_cats_sorted":[],"title_canon_sha256":"95bb8f422b04c89c63ba8b53fedde17fe4b5f355af081dd56c273453de91600c","abstract_canon_sha256":"b656e747ee87f061d42f5b297b3134ae651046e867bacf9a8221e708658996cb"},"schema_version":"1.0"},"canonical_sha256":"3703bbdba0b209210c3c81bbbf2f354e5ada10a9af592f4c1ef91fc32aa6fb32","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:59.787864Z","signature_b64":"LBCsd+y/kBAsldTMBEgcQhNwITASMzDzha12gsMmkeQl/Z4Vl94KJMyG9ndOGfaVrOF4QdJkHQJwPa9tTr3WDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3703bbdba0b209210c3c81bbbf2f354e5ada10a9af592f4c1ef91fc32aa6fb32","last_reissued_at":"2026-05-18T02:19:59.787116Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:59.787116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.3642","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":"2026-05-18T02:19:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/5bvgCyFHwTAUNmgluNEsnxO7w0uvS94+Ne5xeZTaXgVmc6TKaGI/4zpKwT6/z3p0BlRzCubnAW6IG+QoCCHBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T03:36:41.195437Z"},"content_sha256":"e0943f934e961e0da42a7a30245e7e912c77d56b987d615d2b4cacee0bab8b1e","schema_version":"1.0","event_id":"sha256:e0943f934e961e0da42a7a30245e7e912c77d56b987d615d2b4cacee0bab8b1e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:G4B3XW5AWIESCDB4QG536LZVJZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sobolev spaces and hyperbolic fillings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Eero Saksman, Mario Bonk","submitted_at":"2014-08-15T20:10:24Z","abstract_excerpt":"Let $Z$ be an Ahlfors $Q$-regular compact metric measure space, where $Q>0$. For $p>1$ we introduce a new (fractional) Sobolev space $A^p(Z)$ consisting of functions whose extensions to the hyperbolic filling of $Z$ satisfies a weak-type gradient condition. If $Z$ supports a $Q$-Poincar\\'e inequality with $Q>1$, then $A^{Q}(Z)$ coincides with the familiar (homogeneous) Haj\\l asz-Sobolev space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3642","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":"2026-05-18T02:19:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AWb62tqcwVpV61tinSmqXARsIi9KTyJTkwSglTiiT2H6VOuv14oU6UzGhPvs7CBoBWQR6btfNLxNzDHkS6XDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T03:36:41.195791Z"},"content_sha256":"c33099f749b25326704306f42efc14ea2a5f2eb1daa6d2497e6e6b994aa1c9db","schema_version":"1.0","event_id":"sha256:c33099f749b25326704306f42efc14ea2a5f2eb1daa6d2497e6e6b994aa1c9db"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G4B3XW5AWIESCDB4QG536LZVJZ/bundle.json","state_url":"https://pith.science/pith/G4B3XW5AWIESCDB4QG536LZVJZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G4B3XW5AWIESCDB4QG536LZVJZ/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":"2026-07-01T03:36:41Z","links":{"resolver":"https://pith.science/pith/G4B3XW5AWIESCDB4QG536LZVJZ","bundle":"https://pith.science/pith/G4B3XW5AWIESCDB4QG536LZVJZ/bundle.json","state":"https://pith.science/pith/G4B3XW5AWIESCDB4QG536LZVJZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G4B3XW5AWIESCDB4QG536LZVJZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:G4B3XW5AWIESCDB4QG536LZVJZ","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":"b656e747ee87f061d42f5b297b3134ae651046e867bacf9a8221e708658996cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-08-15T20:10:24Z","title_canon_sha256":"95bb8f422b04c89c63ba8b53fedde17fe4b5f355af081dd56c273453de91600c"},"schema_version":"1.0","source":{"id":"1408.3642","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3642","created_at":"2026-05-18T02:19:59Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3642v2","created_at":"2026-05-18T02:19:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3642","created_at":"2026-05-18T02:19:59Z"},{"alias_kind":"pith_short_12","alias_value":"G4B3XW5AWIES","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"G4B3XW5AWIESCDB4","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"G4B3XW5A","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:c33099f749b25326704306f42efc14ea2a5f2eb1daa6d2497e6e6b994aa1c9db","target":"graph","created_at":"2026-05-18T02:19:59Z","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":"Let $Z$ be an Ahlfors $Q$-regular compact metric measure space, where $Q>0$. For $p>1$ we introduce a new (fractional) Sobolev space $A^p(Z)$ consisting of functions whose extensions to the hyperbolic filling of $Z$ satisfies a weak-type gradient condition. If $Z$ supports a $Q$-Poincar\\'e inequality with $Q>1$, then $A^{Q}(Z)$ coincides with the familiar (homogeneous) Haj\\l asz-Sobolev space.","authors_text":"Eero Saksman, Mario Bonk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-08-15T20:10:24Z","title":"Sobolev spaces and hyperbolic fillings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3642","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:e0943f934e961e0da42a7a30245e7e912c77d56b987d615d2b4cacee0bab8b1e","target":"record","created_at":"2026-05-18T02:19:59Z","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":"b656e747ee87f061d42f5b297b3134ae651046e867bacf9a8221e708658996cb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-08-15T20:10:24Z","title_canon_sha256":"95bb8f422b04c89c63ba8b53fedde17fe4b5f355af081dd56c273453de91600c"},"schema_version":"1.0","source":{"id":"1408.3642","kind":"arxiv","version":2}},"canonical_sha256":"3703bbdba0b209210c3c81bbbf2f354e5ada10a9af592f4c1ef91fc32aa6fb32","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3703bbdba0b209210c3c81bbbf2f354e5ada10a9af592f4c1ef91fc32aa6fb32","first_computed_at":"2026-05-18T02:19:59.787116Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:59.787116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LBCsd+y/kBAsldTMBEgcQhNwITASMzDzha12gsMmkeQl/Z4Vl94KJMyG9ndOGfaVrOF4QdJkHQJwPa9tTr3WDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:59.787864Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.3642","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0943f934e961e0da42a7a30245e7e912c77d56b987d615d2b4cacee0bab8b1e","sha256:c33099f749b25326704306f42efc14ea2a5f2eb1daa6d2497e6e6b994aa1c9db"],"state_sha256":"5f286703b934be496cd95d830719b50d93e5a50471d15b80c748b27b4b65fde4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ds/MeSeydpYxITaviuoxk3ToLB+kgk8Yr69qmdP22F+oX58b/+Jxap30Myo6HilAIYlAkP7EFiNFrcPHnM1RCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T03:36:41.197706Z","bundle_sha256":"4e685a51c20df6fb504df74eda37b202e4a03457ad8bd591698aaa9ebf563ecc"}}