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We show that there exists a bounded linear map $T : L^{m,p}(E) \\rightarrow L^{m,p}(\\R^n)$ such that, for any $f \\in L^{m,p}(E)$, we have $Tf = f$ on $E$. We also give a formula for the order of magnitude of $\\|f\\|_{L^{m,p}(E)}$ for a given $f : E \\rightarrow \\R$ when $E$ is finite."},"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":"1205.2525","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-05-11T13:57:44Z","cross_cats_sorted":[],"title_canon_sha256":"119655e1e3586af013e1faf190585a91e7462f56b44e155db641fe92164dce53","abstract_canon_sha256":"514642e6bdfdd73e3b4442752eaab11430f407bc463b6ac05ea3b72d3ec0d314"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:13.656146Z","signature_b64":"8IDWwTS2fQN8EL3SbfMGrM2Lr97uw70I76zHAlrpSpG+aFJIzvLDqZ2zNdWonENaePUMYA/08DsRMdNu7Ce0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7af7c9e1bd20ee97aba36d73e5cdbd80f4bbf2bc8dbd57f55817c649ae2c3dc","last_reissued_at":"2026-05-18T03:55:13.655652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:13.655652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sobolev Extension By Linear Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Arie Israel, Charles L. 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