The Euclidean Onofri inequality in higher dimensions

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.