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

TL;DR

This paper demonstrates the equivalence between a Euclidean Onofri inequality and the sharp logarithmic Moser-Trudinger inequality across multiple dimensions, extending the inequality to weighted Sobolev spaces.

Contribution

It extends the Euclidean Onofri inequality to weighted Sobolev spaces and establishes its equivalence with the sharp logarithmic Moser-Trudinger inequality in all dimensions.

Findings

Extended Onofri inequality to weighted Sobolev spaces.

Proved equivalence with the sharp logarithmic Moser-Trudinger inequality.

No direct connection with standard Sobolev spaces on the sphere except in the planar case.

Abstract

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 $\mathbb{R}^N$.