{"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.09234","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.09234/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}