We show that n -dimensional ( n ⩾ 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.

中文翻译：

---

具有非负加权 Ricci 曲率的函数不等式和流形

我们证明了具有非负加权 Ricci 曲率的 n 维 (n ⩾ 2) 完整和非紧缩度量空间，其中一些 Caffarelli-Kohn-Nirenberg 类型不等式成立，与模型度量度量 n 空间（即欧几里德度量 n -空间）。我们还表明，欧几里得度量空间是唯一满足某些规定的 Sobolev 型不等式的非负加权 Ricci 曲率的完整和非紧度量度空间。