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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.\""},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.1979","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-09T22:59:12Z","cross_cats_sorted":[],"title_canon_sha256":"6d95629b5c428f44fa4c346ffe64c3efa6e59e9a3dd881a6f8bf3d4424ef84ee","abstract_canon_sha256":"50d2ad718fe94d1d730b2cfe9f796b7fb0e0482e82a088727b4efb4e0adcd2cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:01.073057Z","signature_b64":"sINHx4VD3YiZTrWi8usRK5V2oUFHzedLVA1HA3hdLzQaG7CwwaW+Wkwvv5gIsaU0PCEVsBHFmR/7XymbTUyUCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e32031f96c2cbb6d63ec495cd2acd2bdf27999356bebba57235302c912a11e87","last_reissued_at":"2026-05-18T03:41:01.072485Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:01.072485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Structure of Sobolev Extension Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Arie Israel, Charles L. Fefferman, Garving K. Luli","submitted_at":"2012-06-09T22:59:12Z","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.\""},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1979","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.1979","created_at":"2026-05-18T03:41:01.072583+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.1979v2","created_at":"2026-05-18T03:41:01.072583+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1979","created_at":"2026-05-18T03:41:01.072583+00:00"},{"alias_kind":"pith_short_12","alias_value":"4MQDD6LMFS5W","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"4MQDD6LMFS5W2Y7M","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"4MQDD6LM","created_at":"2026-05-18T12:26:53.410803+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX","json":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX.json","graph_json":"https://pith.science/api/pith-number/4MQDD6LMFS5W2Y7MJFONFLGSXX/graph.json","events_json":"https://pith.science/api/pith-number/4MQDD6LMFS5W2Y7MJFONFLGSXX/events.json","paper":"https://pith.science/paper/4MQDD6LM"},"agent_actions":{"view_html":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX","download_json":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX.json","view_paper":"https://pith.science/paper/4MQDD6LM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.1979&json=true","fetch_graph":"https://pith.science/api/pith-number/4MQDD6LMFS5W2Y7MJFONFLGSXX/graph.json","fetch_events":"https://pith.science/api/pith-number/4MQDD6LMFS5W2Y7MJFONFLGSXX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX/action/storage_attestation","attest_author":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX/action/author_attestation","sign_citation":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX/action/citation_signature","submit_replication":"https://pith.science/pith/4MQDD6LMFS5W2Y7MJFONFLGSXX/action/replication_record"}},"created_at":"2026-05-18T03:41:01.072583+00:00","updated_at":"2026-05-18T03:41:01.072583+00:00"}