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These inequalities take the form \\[ \\|P - \\Pi_{\\mathbb{B}} \\Pi_{\\ker\\mathscr{A}} P\\|_{\\dot{\\mathrm{W}}^{k-1, p^*}(\\mathbb{R}^n)} \\le c \\, (\\|\\mathscr{A}[P]\\|_{\\dot{\\mathrm{W}}^{k-1, p^*}(\\mathbb{R}^n)} + \\|\\mathbb{B} P\\|_{\\mathrm{L}^p(\\mathbb{R}^n)}) \\] for all $ P \\in \\mathrm{C}_c^\\infty(\\mathbb{R}^n; V) $, where $ V $ is a finite-dimensional vector space, $ \\mathscr{A} $ is a linear mapping, and $ \\mathbb{B} $ is a constant coefficient homogeneous differential operator of order $ k $. In particular, we can "},"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":"2412.14866","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-12-19T14:02:17Z","cross_cats_sorted":[],"title_canon_sha256":"f6646c84c9ff31ee84c12064b5550ad912f1efa938faec57c9e06778b56f698f","abstract_canon_sha256":"61cc7137e1df83de4cf65a5ea5d76da52d427d21590c4f0adc32ab65aa4b437b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:51:53.769122Z","signature_b64":"cn0qrrcZulzynMN7YpyphpDJJ6RCTSpOL2dSU9RmNhkDID7hg4ZQO4/4oDjcnqN3TxciHlPoeXpNFZLzIx0wCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40e4eb8c48286ef2e5cacaef57b4afd3ccbc8879f1a2bfa19622c88769f7cb89","last_reissued_at":"2026-07-05T09:51:53.768728Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:51:53.768728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Constant rank operators in Korn-Maxwell-Sobolev inequalities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Paul Stephan, Peter Lewintan","submitted_at":"2024-12-19T14:02:17Z","abstract_excerpt":"We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form \\[ \\|P - \\Pi_{\\mathbb{B}} \\Pi_{\\ker\\mathscr{A}} P\\|_{\\dot{\\mathrm{W}}^{k-1, p^*}(\\mathbb{R}^n)} \\le c \\, (\\|\\mathscr{A}[P]\\|_{\\dot{\\mathrm{W}}^{k-1, p^*}(\\mathbb{R}^n)} + \\|\\mathbb{B} P\\|_{\\mathrm{L}^p(\\mathbb{R}^n)}) \\] for all $ P \\in \\mathrm{C}_c^\\infty(\\mathbb{R}^n; V) $, where $ V $ is a finite-dimensional vector space, $ \\mathscr{A} $ is a linear mapping, and $ \\mathbb{B} $ is a constant coefficient homogeneous differential operator of order $ k $. 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