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Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

Part of: Global differential geometry Inequalities Other generalizations

Published online by Cambridge University Press:  22 January 2021

Van Hoang Nguyen Open the ORCID record for Van Hoang Nguyen [Opens in a new window]
Van Hoang Nguyen*
Affiliation:
Department of Mathematics, FPT University, Ha Noi, Viet Nam (vanhoang0610@yahoo.com; hoangnv47@fe.edu.vn)
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Abstract

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

Keywords

Caffarelli–Kohn–Nirenberg inequalitiessharp constantRiemannian manifoldcurvature

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 31C12: Potential theory on Riemannian manifolds 53C20: Global Riemannian geometry, including pinching 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 102 - 127
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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