Abstract
We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent \(n \ge 3\), then it has exactly the \(n\)-dimensional volume growth. As an application, if an \(n\)-dimensional Finsler manifold of non-negative \(n\)-Ricci curvature satisfies the Caffarelli–Kohn–Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.
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Acknowledgments
The paper was completed when A. Kristály visited the Department of Mathematics of Kyoto University in October 2012. He is supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0241 and partially by Domus Hungarica. S. Ohta is supported by the Grant-in-Aid for Young Scientists (B) 23740048.
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Dedicated to Professor Vicenţiu Rădulescu on the occasion of his 55th birthday.
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Kristály, A., Ohta, Si. Caffarelli–Kohn–Nirenberg inequality on metric measure spaces with applications. Math. Ann. 357, 711–726 (2013). https://doi.org/10.1007/s00208-013-0918-1
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DOI: https://doi.org/10.1007/s00208-013-0918-1