论文标题
Ornstein-Uhlenbeck操作员的最佳Sobolev嵌入
Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator
论文作者
论文摘要
提供了高斯空间中Ornstein-Uhlenbeck操作员的Sobolev型不平等的全面分析。提出了一种统一的方法，为重新排列函数规范等级的有效性提供了标准。相关不平等中的最佳目标和域规范通过还原原理对calderón类型的积分运算符的一维不等式来表征。因此，检测到包括Lebesgue，Lorentz，Lorentz-Zygmund，Orlicz和Marcinkiewicz空间在内的各种特定空间家族中最好的规范。还原原则取决于对高斯空间中的一般方程解决方案的存在和唯一性的初步讨论，对于Ornstein-Uhlenbeck操作员，具有恰好可集成的右侧。这些解决方案的重新排列形式的估计值也可以发挥决定性的作用。
A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space is offered. A unified approach is proposed, providing one with criteria for their validity in the class of rearrangement-invariant function norms. Optimal target and domain norms in the relevant inequalities are characterized via a reduction principle to one-dimensional inequalities for a Calderón type integral operator patterned on the Gaussian isoperimetric function. Consequently, the best possible norms in a variety of specific families of spaces, including Lebesgue, Lorentz, Lorentz-Zygmund, Orlicz and Marcinkiewicz spaces, are detected. The reduction principle hinges on a preliminary discussion of the existence and uniqueness of generalized solutions to equations, in the Gauss space, for the Ornstein-Uhlenbeck operator, with a just integrable right-hand side. A decisive role is also played by a pointwise estimate, in rearrangement form, for these solutions.
