On the equivalence between an Onofri-type inequality by Del Pino-Dolbeault and the sharp logarithmic Moser-Trudinger inequality

In this paper we consider the $N$-dimensional Euclidean Onofri inequality proved by del Pino and Dolbeault for smooth compactly supported functions in ${\mathbb{R}}^N$, $N\ge 2$. We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on ${\mathbb{S}}^N$ through stereographic projection is present, except for the planar case. Moreover, in any dimension $N\ge 2$, we show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant proved by Carleson and Chang for balls in ${\mathbb{R}}^N$.