The Caffarelli-Kohn-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature

TL;DR

This paper demonstrates that complete, non-compact metric measure spaces with non-negative weighted Ricci curvature satisfying certain inequalities are geometrically close to Euclidean space, extending classical inequalities to more general spaces.

Contribution

It establishes a stability result linking Caffarelli-Kohn-Nirenberg inequalities to the geometric structure of metric measure spaces with non-negative weighted Ricci curvature.

Findings

Spaces satisfying the inequalities are close to Euclidean space

The result generalizes classical inequalities to metric measure spaces

Provides a geometric characterization of spaces with these inequalities

Abstract

We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure $n$-space (i.e., the Euclidean metric $n$-space).