Caffarelli–Kohn–Nirenberg inequality on metric measure spaces with applications

Abstract

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent \(n \ge 3\), then it has exactly the \(n\)-dimensional volume growth. As an application, if an \(n\)-dimensional Finsler manifold of non-negative \(n\)-Ricci curvature satisfies the Caffarelli–Kohn–Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.