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

TL;DR

This paper establishes a link between the Caffarelli-Kohn-Nirenberg inequality and volume growth in metric measure spaces, with implications for Finsler and Berwald spaces, including characterizations of Minkowski spaces.

Contribution

It proves that satisfying the Caffarelli-Kohn-Nirenberg inequality implies specific geometric properties in metric measure spaces and Finsler manifolds, including volume growth and curvature conditions.

Findings

Spaces satisfying the inequality have exactly n-dimensional volume growth.

Finsler manifolds with non-negative Ricci curvature and sharp inequality constant have zero flag curvature.

Berwald spaces satisfying the inequality are isometric to Minkowski spaces.

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.