{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TGGXYFXGKBZR2TDTNPDAILMERM","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":"657d5eb303b962eabc06c7d2e54ae1d0cebace4e1e492278e78156d06838eb77","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-21T02:56:06Z","title_canon_sha256":"25224a6af86873b35720774f441f618b78d1e2c89f990d512264704d3426c16c"},"schema_version":"1.0","source":{"id":"1208.4177","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4177","created_at":"2026-05-18T03:48:24Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4177v1","created_at":"2026-05-18T03:48:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4177","created_at":"2026-05-18T03:48:24Z"},{"alias_kind":"pith_short_12","alias_value":"TGGXYFXGKBZR","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TGGXYFXGKBZR2TDT","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TGGXYFXG","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:9d14946cc37b739c9e29de019be27998fed6a449162c1ae4843f254b9e3f3390","target":"graph","created_at":"2026-05-18T03:48:24Z","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":"We prove that given any positive integer $k$, for each open set $\\Omega$ and any closed subset $D$ of its closure such that $\\Omega$ is locally an (epsilon,delta)-domain near points in the boundary of $\\Omega$ not contained in $D$ there exists a linear and bounded extension operator $E$ mapping, for each $p\\in[1,\\infty]$, the space $W^{k,p}_D(\\Omega)$ into $W^{k,p}_D({\\mathbb{R}}^n)$. Here, with $O$ denoting either $\\Omega$ or the entire ambient, the space $W^{k,p}_D(O)$ is defined as the completion in the classical Sobolev space $W^{k,p}(O)$ of compactly supported smooth functions whose suppo","authors_text":"Dorina Mitrea, Irina Mitrea, Kevin Brewster, Marius Mitrea","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-21T02:56:06Z","title":"Extending Sobolev Functions with Partially Vanishing Traces from Locally (epsilon,delta)-Domains and Applications to Mixed Boundary Problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4177","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":"sha256:03d17e79651fa454943e37ef539132282dc45e3663ae0a74a2a8306a36695364","target":"record","created_at":"2026-05-18T03:48:24Z","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":"657d5eb303b962eabc06c7d2e54ae1d0cebace4e1e492278e78156d06838eb77","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-21T02:56:06Z","title_canon_sha256":"25224a6af86873b35720774f441f618b78d1e2c89f990d512264704d3426c16c"},"schema_version":"1.0","source":{"id":"1208.4177","kind":"arxiv","version":1}},"canonical_sha256":"998d7c16e650731d4c736bc6042d848b374ddf41f17a5c9bf1ebdf756263745d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"998d7c16e650731d4c736bc6042d848b374ddf41f17a5c9bf1ebdf756263745d","first_computed_at":"2026-05-18T03:48:24.748995Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:24.748995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ReXmq5tl1MSejpgrAiBz802qBRbijt+BQmofJuvk07QDfke2Usjb+iLnEmjPxxi6kaAvzYrmVFoKa2Ksl5O0Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:24.749792Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.4177","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:03d17e79651fa454943e37ef539132282dc45e3663ae0a74a2a8306a36695364","sha256:9d14946cc37b739c9e29de019be27998fed6a449162c1ae4843f254b9e3f3390"],"state_sha256":"17aa8b93b3c0a65e6a269ea0465f50d8fcf701138f1024c5da4c11a7da0c2252"}