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:

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.

令\({\mathbb {H}}^{n}={\mathbb {C}}^{n}\times {\mathbb {R}}\)为n维海森堡群，\(Q=2n +2\)是\({\mathbb {H}}^{n}\)的齐次维度。我们在本文中建立了以下在 Sobolev 范数的齐次约束下在Heisenberg 群\({\mathbb {H}}^{n} \)上尖锐的 Trudinger-Moser 不等式：

成立当且仅当\(\ \alpha <1\)，其中\(\Phi _Q(t)=e^{t}-\sum _{0}^{Q-2}\frac{t^{j } {j!}\). 与文献中甚至在欧几里得空间中的所有证明不同，我们的证明避免使用与 Moser 泛函相关的欧拉-拉格朗日方程的复杂爆炸分析。事实上，我们的证明揭示了一个令人惊讶的事实，即整个空间上已知的临界 Trudinger-Moser 不等式等效于整个空间上看似更强的锐化 Trudinger-Moser 不等式。此外，我们在整个海森堡群的非齐次约束下获得了临界的 Trudinger-Moser 不等式和浓度-紧致原理。最后，再次使用缩放的方法，我们得到了非齐次约束下的改进的 Trudinger-Moser 不等式。我们的方法非常简单和通用，可以轻松应用于所有分层幂零组和其他设置。