The Caffarelli-Kohn-Nirenberg inequalities and manifolds with   nonnegative weighted Ricci curvature
  We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure $n$-space (i.e., the Euclidean metric $n$-space). 
Jing Mao
math.DG