A new proof of subcritical Trudinger-Moser inequalities on the whole Euclidean space

TL;DR

This paper presents a novel proof of the subcritical Trudinger-Moser inequality on Euclidean space that extends to Riemannian manifolds and the Heisenberg group, avoiding traditional rearrangement techniques.

Contribution

It introduces a new proof method for the inequality that does not rely on rearrangement arguments, enabling broader applicability.

Findings

Proof applicable to Riemannian manifolds

Extension to the Heisenberg group

Simplifies the proof technique

Abstract

In this note, we give a new proof of subcritical Trudinger-Moser inequality on $\mathbb{R}^n$. All the existing proofs on this inequality are based on the rearrangement argument with respect to functions in the Sobolev space $W^{1,n}(\mathbb{R}^n)$. Our method avoids this technique and thus can be used in the Riemannian manifold case and in the entire Heisenberg group.