{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4MQDD6LMFS5W2Y7MJFONFLGSXX","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":"50d2ad718fe94d1d730b2cfe9f796b7fb0e0482e82a088727b4efb4e0adcd2cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T22:59:12Z","title_canon_sha256":"6d95629b5c428f44fa4c346ffe64c3efa6e59e9a3dd881a6f8bf3d4424ef84ee"},"schema_version":"1.0","source":{"id":"1206.1979","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1979","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1979v2","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1979","created_at":"2026-05-18T03:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"4MQDD6LMFS5W","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4MQDD6LMFS5W2Y7M","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4MQDD6LM","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:de9f85e9d86615864f96aaaa954a6e39bff6dad7dad202e0a815ffa2b55dd878","target":"graph","created_at":"2026-05-18T03:41:01Z","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 $L^{m,p}(\\R^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\\R^n)$, and assume that $p>n$. For $E \\subset \\R^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \\in L^{m,p}(\\R^n)$. It is known that there exist bounded linear maps $T : L^{m,p}(E) \\rightarrow L^{m,p}(\\R^n)$ such that $Tf = f$ on $E$ for any $f \\in L^{m,p}(E)$. We show that $T$ cannot have a simple form called \"bounded depth.\"","authors_text":"Arie Israel, Charles L. Fefferman, Garving K. Luli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T22:59:12Z","title":"The Structure of Sobolev Extension Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1979","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:c496a1a4c05e6a4b0ea781fb73efed8d80ca29dd0f00d2822ac6a9c89f9a1377","target":"record","created_at":"2026-05-18T03:41:01Z","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":"50d2ad718fe94d1d730b2cfe9f796b7fb0e0482e82a088727b4efb4e0adcd2cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T22:59:12Z","title_canon_sha256":"6d95629b5c428f44fa4c346ffe64c3efa6e59e9a3dd881a6f8bf3d4424ef84ee"},"schema_version":"1.0","source":{"id":"1206.1979","kind":"arxiv","version":2}},"canonical_sha256":"e32031f96c2cbb6d63ec495cd2acd2bdf27999356bebba57235302c912a11e87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e32031f96c2cbb6d63ec495cd2acd2bdf27999356bebba57235302c912a11e87","first_computed_at":"2026-05-18T03:41:01.072485Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:01.072485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sINHx4VD3YiZTrWi8usRK5V2oUFHzedLVA1HA3hdLzQaG7CwwaW+Wkwvv5gIsaU0PCEVsBHFmR/7XymbTUyUCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:01.073057Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.1979","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c496a1a4c05e6a4b0ea781fb73efed8d80ca29dd0f00d2822ac6a9c89f9a1377","sha256:de9f85e9d86615864f96aaaa954a6e39bff6dad7dad202e0a815ffa2b55dd878"],"state_sha256":"1b0d82a77a2997d059c90bec5b7a9007ee8a00c43a56de80acecb79cdc357ae4"}