2021/11/30 by Louis Dupaigne, Dupaigne, Louis, Ivan Gentil +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2111.15383
openalex publication_date 2021/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short),\nintroduced by these authors in 1984. We explain why the CKN inequality can be\nviewed as a Sobolev inequality on a weighted Riemannian manifold. More\nprecisely, we prove that the CKN inequality can be interpreted in this way on\nthree different and equivalent models, obtained as weighted versions of the\nstandard Euclidean space, round sphere and hyperbolic space. This result can be\nviewed as an extension of conformal invariance to the weighted setting. Since\nthe spherical CKN model we introduce has finite measure, the \Γ-calculus\nintroduced by Bakry and Emery provides an easy way to prove the Sobolev\ninequalities. This method allows us to recover the optimality of the region of\nparameters describing symmetry-breaking of minimizers of the CKN inequality,\nintroduced by Felli and Schneider and proved by Dolbeault, Esteban and Loss in\n2016. Finally, we develop the notion of n-conformal invariants, exhibiting a\nway to extend the notion of scalar curvature to weighted manifolds such as the\nCKN models.\n