Logarithmic Sobolev Inequalities
Logarithmic Sobolev Inequalities
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DOI：
10.1007/978-3-319-00227-9_5
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
D. Bakry;I. Gentil;M. Ledoux
中科院分区：
文献类型：
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作者：
D. Bakry;I. Gentil;M. Ledoux
继poincar<s:1>不等式之后，对数Sobolev不等式是研究最多的半群泛函不等式之一。它们比庞加莱不等式包含更多的信息，同时也足够普遍，可以在许多感兴趣的情况下使用，特别是在无限维空间中（作为有限维空间上Sobolev不等式的极限）。在给出对数Sobolev不等式的基本定义及其第一个性质之后，本章的第一部分给出了熵的指数衰减以及对数Sobolev不等式与半群的超收缩形式的平滑性质之间的基本等价。其次，讨论了对数Sobolev不等式下特征向量和Lipschitz函数的可积性，并给出了实线上满足对数Sobolev不等式（对于一般梯度）的测度准则。进一步的部分处理曲率条件，首先是热核测度的局部对数Sobolev不等式，然后是具有附加维度信息的不变测度。进一步给出了局部超收缩性和局部对数Sobolev不等式在热核界上的一些应用。无限维曲率条件下的harnack型不等式，与逆局部对数Sobolev不等式相联系，完成了本章。
After Poincaré inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincaré inequalities, and are at the same time sufficiently general to be available in numerous cases of interest, in particular in infinite dimension (as limits of Sobolev inequalities on finite-dimensional spaces). After the basic definition of a logarithmic Sobolev inequality together with its first properties, the first sections of this chapter present the exponential decay in entropy and the fundamental equivalence between the logarithmic Sobolev inequality and smoothing properties of the semigroup in the form of hypercontractivity. Next, integrability properties of eigenvectors and of Lipschitz functions under a logarithmic Sobolev inequality are discussed together with a criterion for measures on the real line to satisfy a logarithmic Sobolev inequality (for the usual gradient). The further sections deal with curvature conditions, first for the local logarithmic Sobolev inequalities for heat kernel measures, then for the invariant measure with an additional dimensional information. Local hypercontractivity and some applications of the local logarithmic Sobolev inequalities towards heat kernel bounds are further presented. Harnack-type inequalities under the infinite-dimensional curvature conditions, linked with reverse local logarithmic Sobolev inequalities complete the chapter.