Sharpened Trudinger–Moser Inequalities on the Euclidean Space and Heisenberg Group

Abstract

Let ${\mathbb {H}}^{n}={\mathbb {C}}^{n}\times {\mathbb {R}}$ be the n-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of ${\mathbb {H}}^{n}$. We establish in this paper that the following sharpened Trudinger–Moser inequalities on the Heisenberg group ${\mathbb {H}}^{n}$ under the homogeneous constraints of the Sobolev norm:

$$\begin{aligned} \sup _{\Vert \nabla _{{\mathbb {H}}}u\Vert _{Q}^{Q}+\Vert u\Vert _{Q}^{Q}\le 1} \int _{{\mathbb {H}}^{n}}\Phi _Q\left( \alpha _{Q}\left( 1+\alpha \Vert u\Vert _{Q}^{Q} \right) ^{\frac{1}{Q-1}}|u|^{\frac{Q}{Q-1}}\right) d\xi <+\infty , \end{aligned}$$

holds if and only if $\ \alpha <1$, where $\Phi _Q(t)=e^{t}-\sum _{0}^{Q-2}\frac{t^{j} }{j!}$. Unlike all the proofs in the literature even in the Euclidean spaces, our proof avoids using the complicated blow-up analysis of the Euler–Lagrange equation associated with the Moser functional. In fact, our proof reveals a surprising fact that the known critical Trudinger–Moser inequality on the entire space is equivalent to seemingly much stronger sharpened Trudinger–Moser inequality on the entire space. Furthermore, we obtain the critical Trudinger–Moser inequality and the Concentration-Compactness Principle under the inhomogeneous constraints on the entire Heisenberg group. Finally, using the method of scaling again, we obtain improved Trudinger–Moser inequality under the inhomogeneous constraints. Our approach is surprisingly simple and general and can be easily applied to the all stratified nilpotent groups and other settings. In particular, our method also gives an alternative and much simpler proof of the corresponding results in the Euclidean space.