Czechoslovak Mathematical Journal, Vol. 70, No. 1, pp. 213-233, 2020
Functional inequalities and manifolds with nonnegative weighted Ricci curvature
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