Trudinger-Moser inequalities on the entire Heisenberg group

TL;DR

This paper establishes new Trudinger-Moser inequalities on the entire Heisenberg group, identifying optimal constants using a novel approach that combines local estimates with cut-off functions, applicable to other geometric settings.

Contribution

Introduces a new method for proving Trudinger-Moser inequalities on the Heisenberg group, avoiding rearrangement arguments and applicable to broader geometric contexts.

Findings

Derived the best constants for inequalities on the Heisenberg group

Developed a new approach using local estimates and cut-off functions

Method applicable to Euclidean space and Riemannian manifolds

Abstract

Continuing our previous work (Cohn, Lam, Lu, Yang, Nonlinear Analysis (2011), doi: 10.1016 /j.na.2011.09.053), we obtain a class of Trudinger-Moser inequalities on the entire Heisenberg group, which indicate what the best constants are. All the existing proofs of similar inequalities on unbounded domain of the Euclidean space or the Heisenberg group are based on rearrangement argument. In this note, we propose a new approach to solve this problem. Specifically we get the global Trudinger-Moser inequality by gluing local estimates with the help of cut-off functions. Our method still works for similar problems when the Heisenberg group is replaced by the Eclidean space or complete noncompact Riemannian manifolds.