Optimal Sobolev and Poincaré inequalities with homogeneous weights

Due to its conformal invariance, the sharp Sobolev inequality takes equivalent forms on the three standard model spaces i.e. the Euclidean space, the round sphere and the hyperbolic space. By analogy, we introduce three weighted manifolds named after Caffarelli, Kohn and Nirenberg (CKN) for the following reason: the sharp Caffarelli-Kohn-Nirenberg inequality in the standard Euclidean space can be reformulated as a (sharp) Sobolev inequality written on the CKN Euclidean space. It is equivalent to similar (but new) Sobolev inequalities on the CKN sphere and the CKN hyperbolic space. In addition, the Felli-Schneider condition, that is, the region of parameters for which symmetry breaking occurs in the study of extremals, turns out to have a purely geometric interpretation as an (integrated) curvature-dimension condition. To prove these results, we shall use Bakry's generalization of the notion of scalar curvature, (a weighted version of) Otto's calculus, the reformulation of all the inequalities (and many more) as entropy-entropy production inequalities along appropriate gradient flows in Wasserstein space, and eventually elliptic PDE methods as our best tool for building rigorous and concise proofs. This is joint work with Ivan Gentil (Lyon 1) and Simon Zugmeyer (Paris 5).