  In this paper we consider the $N$-dimensional Euclidean Onofri inequality
proved by del Pino and Dolbeault for smooth compactly supported functions in
$\mathbb{R}^N$, $N \geq 2$. 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 $N \geq 2$, 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 $\mathbb{R}^N$.
