Abstract:In this paper, we obtain a sharp remainder formula for the Poincaré inequality which implies a simple proof of the sharp Poincaré inequality without using the variational principle. We also extend the idea to general Carnot groups. Thus, we have succeeded in finding a simple proof of the sharp Poincaré inequality in a more general case.

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中文翻译：

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庞加莱不等式的尖锐余数

摘要：在本文中，我们获得了庞加莱不等式的尖锐余数公式，这意味着无需使用变分原理即可简单证明庞加莱不等式。我们还将这一想法扩展到一般的卡诺集团。因此，我们已经成功找到了在更一般的情况下尖锐的庞加莱不等式的简单证明。

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