In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent $$n(n\ge 2)$$ n ( n ≥ 2 ) , then it has exactly n -dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov spaces, Riemannian manifolds and Finsler manifolds which support a Caffarelli–Kohn–Nirenberg inequality.

中文翻译：

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度量空间上的 Caffarelli-Kohn-Nirenberg 不等式

在本文中，我们证明，如果度量空间满足体积倍增条件和 Caffarelli-Kohn-Nirenberg 不等式且指数相同 $$n(n\ge 2)$$n ( n ≥ 2 ) ，则它有正好是 n 维体积增长。作为应用，我们获得了支持 Caffarelli-Kohn-Nirenberg 不等式的 Alexandrov 空间、黎曼流形和 Finsler 流形的几何和拓扑性质。