Sharpened Trudinger–Moser Inequalities on the Euclidean Space and Heisenberg Group
Sharpened Trudinger–Moser Inequalities on the Euclidean Space and Heisenberg Group
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DOI：
10.1007/s12220-021-00713-1
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发表时间：
2021-06
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影响因子：
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通讯作者：
Lu Chen;Guozhen Lu;Maochun Zhu
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文献类型：
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作者：
Lu Chen;Guozhen Lu;Maochun Zhu
让H C n×n = R n维海森堡集团Q = 2 n + 2的均匀尺寸H n。我们建立在本文下面磨Trudinger-Moser不等式在海森堡H组n均匀约束下的水列夫规范:一口为∇H u为QQ +为u为QQ≤1∫H nΦ问α问1 +α为u为QQ 1 q1 | | QQ-1 dξ< +∞,当且仅当α< 1,ΦQ (t) = et -∑0 q2 tjj !。与文献中甚至在欧几里得空间中的所有证明不同，我们的证明避免了使用与莫泽泛函相关的欧拉-拉格朗日方程的复杂爆破分析。事实上，我们的证明揭示了一个令人惊讶的事实，即已知的整个空间上的临界Trudinger-Moser不等式等价于整个空间上看起来更强的锐化Trudinger-Moser不等式。进一步，我们得到了整个Heisenberg群在非齐次约束下的临界Trudinger-Moser不等式和集中紧致原理。最后，再次利用尺度变换的方法，得到了非齐次约束下改进的Trudinger-Moser不等式。我们的方法非常简单和通用，可以很容易地应用于所有分层的幂零群体和其他设置。特别地，我们的方法也给出了在欧几里得空间中相应结果的另一种更简单的证明。
Let H n= C n× R be the n-dimensional Heisenberg group, Q= 2 n+ 2 be the homogeneous dimension of H n. We establish in this paper that the following sharpened Trudinger–Moser inequalities on the Heisenberg group H n under the homogeneous constraints of the Sobolev norm: sup‖∇ H u‖ QQ+‖ u‖ QQ≤ 1∫ H n Φ Q α Q 1+ α‖ u‖ QQ 1 Q-1| u| QQ-1 d ξ<+∞, holds if and only if α< 1, where Φ Q (t)= et-∑ 0 Q-2 tjj!. 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.