Czechoslovak Mathematical Journal, first online, pp. 1-21


Functional inequalities and manifolds with nonnegative weighted Ricci curvature

Jing Mao

Received April 30, 2018.   Published online November 19, 2019.

Abstract:  We show that $n$-dimensional $(n\geq 2)$ 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.
Keywords:  Caffarelli-Kohn-Nirenberg type inequality; weighted Ricci curvature; volume comparison
Classification MSC:  53C21, 31C12
DOI:  10.21136/CMJ.2019.0214-18

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Affiliations:   Jing Mao, Faculty of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, 118 Youyi Ave, Wuchang Qu, Wuhan, Hubei Sheng, 430062, P. R. China, e-mail: jiner120@163.com, jiner120@tom.com


 
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