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

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.

5 pages