Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space
Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space
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欧几里德空间和高斯空间中等周不等式的半群证明
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
M. Ledoux
中科院分区：
文献类型：
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作者：
M. Ledoux
。本文是对欧几里得和高斯空间中可用于研究等周不等式的一些半群工具的论述。受N.Varopoulos关于群和流形上等周不等式的泛函方法的启发，我们将在这里特别地观察到Rn中的经典等周不等式等价于说作用于集合的特征函数的热半群的L 2-范数在等周重排下增加.然后，我们考察了Gauss空间中相应的性质，并遵循B.Maurey和G.Pisier对测度集中现象的方法，考察了Ornstein-Uhlenbeck半群的各种性质，如交换性或超缩性，如何以一种简单的方式产生集中现象和Gauss测度的等周不等式本身(形式)。
. — This paper is an exposition of some of the semigroup tools which may be used to investigate the isoperimetric inequality in Euclidean and Gauss space. Inspired by the work of N. Varopoulos in his functional approach to isoperimetric inequalities on groups and manifolds, we will observe here, in particular, that the classical isoperimetric inequality in R n is equivalent to saying that the L 2 -norm of the heat semigroup acting on characteristic functions of sets increases under isoperimetric rearrangement. We then check the corresponding property in Gauss space and, following the approach of B. Maurey and G. Pisier to the concentration of measure phenomenon, we survey how the various properties of the Ornstein-Uhlenbeck semigroup such as the commutation property or hypercontractivity can yield in a simple way both the concentration phenomenon and (a form of) the isoperimetric inequality itself for Gauss measures.