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

TL;DR

The paper investigates the equivalence between the $N$-dimensional Euclidean Onofri inequality of Del Pino–Dolbeault and the sharp logarithmic Moser–Trudinger inequality on balls in $\mathbb{R}^N$, and extends the framework to a weighted Sobolev space. It introduces the weighted measure $d\mu_N$, the associated $v_N=\ln\mu_N$, and the $N$-Laplacian-based functional $H_N$, establishing a weighted space $W_{\mu_N}(\mathbb{R}^N)$ with norm $\|\cdot\|_{\mu_N}$ and proving density of smooth compactly supported functions. A central result shows the identity $\inf_{W_{\mu_N}(\mathbb{R}^N)} I = \inf_{W^{1,N}_0(B_1)} J + \sum_{k=1}^{N-1}\frac{1}{k}$, linking the sharp Onofri inequality to Carleson–Chang's sharp logarithmic Moser–Trudinger inequality and implying a simple alternative proof of the Onofri inequality in higher dimensions. In the special case $N=2$, the findings illuminate the connection between Onofri's inequality on $\mathbb{S}^2$ and the sharp planar Carleson–Chang inequality, thereby unifying Euclidean and spherical inequalities within a weighted variational framework.

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 $\mathbb{R}^N$.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof