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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 beviewed as a Sobolev inequality on a weighted Riemannian manifold. Moreprecisely, we prove that the CKN inequality can be interpreted in this way onthree different and equivalent models, obtained as weighted versions of thestandard Euclidean space, round sphere and hyperbolic space. This result can beviewed as an extension of conformal invariance to the weighted setting. Sincethe spherical CKN model we introduce has finite measure, the $\Gamma$-calculusintroduced by Bakry and Emery provides an easy way to prove the Sobolevinequalities. This method allows us to recover the optimality of the region ofparameters describing symmetry-breaking of minimizers of the CKN inequality,introduced by Felli and Schneider and proved by Dolbeault, Esteban and Loss in2016. Finally, we develop the notion of n-conformal invariants, exhibiting away to extend the notion of scalar curvature to weighted manifolds such as theCKN models.

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