The optimal constants in a class of exponential type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space are detected. The existence of extremal functions in the relevant inequalities is also established. Our results disclose analogies and dissimilarities in comparison with Adams' inequality for the Laplace operator, a companion of our inequalities in the Euclidean space.

中文翻译：

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Ornstein-Uhlenbeck 算子的尖锐指数不等式

检测高斯空间中 Ornstein-Uhlenbeck 算子的一类指数型不等式中的最优常数。还成立了相关不等式中极值函数的存在。我们的结果揭示了与拉普拉斯算子的亚当斯不等式相比的类比和不同之处，拉普拉斯算子是我们在欧几里德空间中不等式的伴侣。