The Euclidean Onofri inequality in higher dimensions

TL;DR

This paper extends the classical Onofri inequality to higher-dimensional Euclidean spaces, introducing a new Sobolev-Orlicz norm and a different probability measure, thus broadening its mathematical framework.

Contribution

It establishes an optimal higher-dimensional version of the Onofri inequality using Gagliardo-Nirenberg interpolation inequalities, involving novel Sobolev-Orlicz norms and measures.

Findings

Derived an optimal inequality in higher dimensions

Introduced a Sobolev-Orlicz norm in the inequality

Connected the inequality to Gagliardo-Nirenberg interpolation

Abstract

The classical Onofri inequality in the two-dimensional sphere assumes a natural form in the plane when transformed via stereographic projection. We establish an optimal version of a generalization of this inequality in the d-dimensional Euclidean space for any d\geq2, by considering the endpoint of a family of optimal Gagliardo-Nirenberg interpolation inequalities. Unlike the two-dimensional case, this extension involves a rather unexpected Sobolev-Orlicz norm, as well as a probability measure no longer related to stereographic projection.