The Caffarelli-Kohn-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature

Abstract

We prove that $n$-dimensional ($n⩾3$) 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).

18 pages. Similar methods have been used in [J. Mao, The Gagliardo-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature, (2014) submitted] and [J. Mao, Functional inequalities and manifolds with nonnegative weighted Ricci curvature, (2014) submitted] to derive the corresponding rigidity results as in this manuscript. Minor revisions have been made to Remark 1.3 in v1