The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces

TL;DR

This paper establishes that metric measure spaces satisfying the Caffarelli-Kohn-Nirenberg inequality and volume doubling condition have exactly n-dimensional volume growth, leading to geometric and topological insights for various spaces.

Contribution

It proves a characterization of n-dimensional volume growth in metric measure spaces supporting the inequality, linking functional inequalities to geometric properties.

Findings

Spaces with the inequality have n-dimensional volume growth

Supports geometric and topological analysis of Alexandrov, Riemannian, and Finsler spaces

Provides conditions under which these spaces exhibit specific geometric behaviors

Abstract

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.