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

TL;DR

This paper introduces a new proof and dual formulation of the Moser-Onofri inequality on spheres using optimal mass transport, extends it to higher dimensions, and connects it to curvature problems and fast diffusion equations.

Contribution

It provides a novel optimal transport-based proof, extends the inequality to higher-dimensional spheres, and links the dual problem to curvature and diffusion equations.

Findings

New dual formula for Moser-Onofri inequality on $ ext{S}^2$

Extension of the inequality to spheres $ ext{S}^n$, $n extgreater 2$

Connection between dual problems and curvature/fast diffusion equations

Abstract

We use optimal mass transport to provide a new proof and a dual formula to the Moser-Onofri inequality on $\s^2$ in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani to the Sobolev inequality and of Agueh-Ghoussoub-Kang to more general settings. There are however many hurdles to overcome once a stereographic projection on $\R^2$ 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 $\mu_2(x)=\frac{1}{\pi(1+|x|^2)^2}$ 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 $\s^n$ with $n\geq 2$. What is remarkable is that the corresponding free energy is again given by $F(\rho)=-n\rho^{1-\frac{1}{n}}$, which means that both the {\it prescribed scalar curvature problem} and the {\it prescribed Gaussian curvature problem} lead essentially to the same dual problem whose extremals are stationary solutions of the fast diffusion equations.