{"paper":{"title":"The Calder\\'on-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Batu G\\\"uneysu, Stefano Pigola","submitted_at":"2014-06-03T15:26:08Z","abstract_excerpt":"We introduce the concept of Calder\\'on-Zygmund inequalities on Riemannian manifolds. For $1<p<\\infty$, these are inequalities of the form $$ \\left\\Vert \\mathrm{Hess}\\left( u\\right) \\right\\Vert _{L^p}\\leq C_{1}\\left\\Vert u\\right\\Vert _{L^p}+C_{2}\\left\\Vert \\Delta u\\right\\Vert _{L^p}, $$ valid a priori for all smooth functions $u$ with compact support, and constants $C_1\\geq 0$, $C_2>0$. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calder\\'on-Zygmund inequality (like new denseness resul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0747","kind":"arxiv","version":1},"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"}