A dual Moser–Onofri inequality and its extensions to higher dimensional spheres

Abstract

We use optimal mass transport to provide a new proof and a dual formula to the Moser-Onofri inequality on S in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani [4] to the Sobolev inequality and of Agueh-Ghoussoub-Kang [1] to more general settings. There are however many hurdles to overcome once a stereographic projection on R is performed: Functions are not necessarily of compact support, hence boundary terms need to be evaluated. Moreover, the corresponding dual free energy of the reference probability density μ2(x) = 1 π(1+|x|2)2 is not finite on the whole space, which requires the introduction of a renormalized free energy into the dual formula. We also extend this duality to higher dimensions and establish an extension of the Onofri inequality to spheres S with n ≥ 2. What is remarkable is that the corresponding free energy is again given by F (ρ) = −nρ1− 1 n , which means that both the prescribed scalar curvature problem and the prescribed Gaussian curvature problem lead essentially to the same dual problem whose extremals are stationary solutions of the fast diffusion equations.