{"paper":{"title":"On the $L^p$-estimates for Beurling-Ahlfors and Riesz transforms on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xiang-Dong Li","submitted_at":"2013-04-03T20:03:14Z","abstract_excerpt":"In our previous papers \\cite{Li2008, Li2011}, we proved some martingale transform representation formulas for the Riesz transforms and the Beurling-Ahlfors transforms on complete Riemannian manifolds, and proved some explicit $L^p$-norm estimates for these operators on complete Riemannian manifolds with suitable curvature conditions. In this paper we correct a gap contained in \\cite{Li2008, Li2011} and prove that the $L^p$-norm of the Riesz transforms $R_a(L)=\\nabla(a-L)^{-1/2}$ can be explicitly bounded by $C(p^*-1)^{3/2}$ if $Ric+\\nabla^2\\phi\\geq -a$ for $a\\geq 0$, and the $L^p$-norm of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1168","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":""},"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"}