A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality

TL;DR

This paper offers a geometric reinterpretation of the Caffarelli-Kohn-Nirenberg inequality as a Sobolev inequality on weighted Riemannian manifolds, extending conformal invariance and analyzing symmetry-breaking regions.

Contribution

It introduces a conformal geometric perspective on CKN inequalities using weighted models of Euclidean, spherical, and hyperbolic spaces, and develops n-conformal invariants for weighted manifolds.

Findings

Reinterpreted CKN inequality as Sobolev inequality on weighted manifolds

Extended conformal invariance to weighted geometric models

Reproduced optimal symmetry-breaking parameter regions

Abstract

We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short), introduced by these authors in 1984. We explain why the CKN inequality can be viewed as a Sobolev inequality on a weighted Riemannian manifold. More precisely, we prove that the CKN inequality can be interpreted in this way on three different and equivalent models, obtained as weighted versions of the standard Euclidean space, round sphere and hyperbolic space. This result can be viewed as an extension of conformal invariance to the weighted setting. Since the spherical CKN model we introduce has finite measure, the $\Gamma$-calculus introduced by Bakry and Emery provides an easy way to prove the Sobolev inequalities. This method allows us to recover the optimality of the region of parameters describing symmetry-breaking of minimizers of the CKN inequality, introduced by Felli and Schneider and proved by Dolbeault, Esteban and Loss in 2016. Finally, we develop the notion of n-conformal invariants, exhibiting a way to extend the notion of scalar curvature to weighted manifolds such as the CKN models.