The Euclidean Onofri Inequality in Higher Dimensions

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≥ 2, 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.