Geodesic
Functional inequalities and manifolds with nonnegative weighted Ricci curvature
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 213-233

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We show that $n$-dimensional $(n\geqslant 2)$ complete and noncompact metric measure spaces with nonnegative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are isometric to the model metric measure $n$-space (i.e. the Euclidean metric $n$-space). We also show that the Euclidean metric spaces are the only complete and noncompact metric measure spaces of nonnegative weighted Ricci curvature satisfying some prescribed Sobolev type inequality.
DOI : 10.21136/CMJ.2019.0214-18
Classification : 31C12, 53C21
Keywords: Caffarelli-Kohn-Nirenberg type inequality; weighted Ricci curvature; volume comparison 
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Mao, Jing. Functional inequalities and manifolds with nonnegative weighted Ricci curvature. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 213-233. doi: 10.21136/CMJ.2019.0214-18

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