{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:HISQGANXZWOGIDEGOL7ALDGPXX","short_pith_number":"pith:HISQGANX","schema_version":"1.0","canonical_sha256":"3a250301b7cd9c640c8672fe058ccfbdfd5f8d1c5985b2225b49b7dbe92af12a","source":{"kind":"arxiv","id":"2408.09234","version":2},"attestation_state":"computed","paper":{"title":"Attaining the optimal constant for higher-order Sobolev inequalities on manifolds via asymptotic analysis","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lorenzo Carletti","submitted_at":"2024-08-17T15:59:52Z","abstract_excerpt":"Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \\in H^{k}(M)$, \\[\\|u\\|_{L^{2^\\sharp}(M)}^2 \\leq K_0^2 \\int_M |\\Delta_g^{k/2} u|^2 \\,dv_g + B_0 \\|u\\|_{H^{k-1}(M)}^2,\\] where $2^\\sharp = \\frac{2n}{n-2k}$ and $\\Delta_g = -\\operatorname{div}_g(\\nabla\\cdot)$. Here $K_0$ is the optimal constant for the Euclidean Sobolev inequality $\\big(\\int_{\\mathbb{R}^n} |u|^{2^\\sharp}\\big)^{2/2^\\sharp} \\leq K_0^2 \\int_{\\mathbb{R}^n} |\\nabla^k u|^2$ for all $u \\in C_c^\\infty(\\mathbb{R}^n)$. This result i"},"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":"2408.09234","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-08-17T15:59:52Z","cross_cats_sorted":[],"title_canon_sha256":"5bd1e41b34f1283c0cd115fb0a0716846e2ae05c818530a25f432ae01933196c","abstract_canon_sha256":"7cb056dc6509249405e273e87654380a78f15a517aac6dace6becce3cbd570a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:28:10.969968Z","signature_b64":"0esFUaR1XAqOihrDZnbIbdOIgF1VSzJu7AtWOZqYBWjQaFhzdSaslqwq+iAStwmI3CXRYgV/YPXfiVkITLB2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a250301b7cd9c640c8672fe058ccfbdfd5f8d1c5985b2225b49b7dbe92af12a","last_reissued_at":"2026-07-05T11:28:10.969454Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:28:10.969454Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Attaining the optimal constant for higher-order Sobolev inequalities on manifolds via asymptotic analysis","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lorenzo Carletti","submitted_at":"2024-08-17T15:59:52Z","abstract_excerpt":"Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \\in H^{k}(M)$, \\[\\|u\\|_{L^{2^\\sharp}(M)}^2 \\leq K_0^2 \\int_M |\\Delta_g^{k/2} u|^2 \\,dv_g + B_0 \\|u\\|_{H^{k-1}(M)}^2,\\] where $2^\\sharp = \\frac{2n}{n-2k}$ and $\\Delta_g = -\\operatorname{div}_g(\\nabla\\cdot)$. Here $K_0$ is the optimal constant for the Euclidean Sobolev inequality $\\big(\\int_{\\mathbb{R}^n} |u|^{2^\\sharp}\\big)^{2/2^\\sharp} \\leq K_0^2 \\int_{\\mathbb{R}^n} |\\nabla^k u|^2$ for all $u \\in C_c^\\infty(\\mathbb{R}^n)$. 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