The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov space, Riemannian manifold and Finsler space which support a Caffarelli-Kohn-Nirenberg inequality.