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 (Int Math Res Not IMRN 15:3600–3611, 2013) for smooth compactly supported functions in RN, N≥2. We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on SN through stereographic projection is present, except for the planar case. Moreover, in any dimension N≥2, we show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser–Trudinger inequality with sharp constant proved by Carleson and Chang (Bull Sci Math 110(2):113–127, 1986) for balls in RN.