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

Abstract 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 $$\mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> , $$N \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on $$\mathbb {S}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> through stereographic projection is present, except for the planar case. Moreover, in any dimension $$N \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , 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 $$\mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> .