Sharp Caffarelli-Kohn-Nirenberg inequalities on Riemannian manifolds:\n the influence of curvature

2017/09/18 by Van Hoang Nguyen, Nguyen, Van Hoang
Mathematics · #26D10 #31C12 #53C20 #53C21 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1709.06120

openalex publication_date 2017/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We first establish a family of sharp Caffarelli-Kohn-Nirenberg type\ninequalities on the Euclidean spaces and then extend them to the setting of\nCartan-Hadamard manifolds with the same best constant. The quantitative version\nof these inequalities also are proved by adding a non-negative remainder term\nin terms of the sectional curvature of manifolds. We next prove several\nrigidity results for complete Riemannian manifolds supporting the\nCaffarelli-Kohn-Nirenberg type inequalities with the same sharp constant as in\n mathbb Rⁿ (shortly, sharp CKN inequalities). Our results illustrate the\ninfluence of curvature to the sharp CKN inequalities on the Riemannian\nmanifolds. They extend recent results of Krist 'aly to a larger class of the\nsharp CKN inequalities.\n

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