The Brascamp-Lieb inequalities: Finiteness, structure and extremals
The Brascamp-Lieb inequalities: Finiteness, structure and extremals
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DOI：
10.1007/s00039-007-0619-6
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发表时间：
2007-01-01
影响因子：
2.2
通讯作者：
Tao, Terence
中科院分区：
文献类型：
--
作者：
Bennett, Jonathan;Carbery, Anthony;Tao, Terence
研究了多维函数积多重线性积分的Brascamp-Lieb不等式。完整地讨论了常数的有限性问题和有中心高斯极值的存在唯一性问题。对于任意极值，我们完全解决了存在的问题，部分解决了唯一性的问题。我们还从结构角度分析了不平等现象。我们的主要工具是线性和多线性环境下热方程正解的单调公式，Carlen， Lieb和Loss [CLL]首先在这种类型的环境中使用了单调公式。本文用热流法得到了关于高斯耗竭的Lieb基本定理的一级情形；我们将这种方法推广到高秩情形，给出了利布定理一般秩情形的两个新的证明。
We consider the Brascamp-Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. Our main tool is a monotonicity formula for positive solutions to heat equations in linear and multilinear settings, which was first used in this type of setting by Carlen, Lieb, and Loss [CLL]. In that paper, the heat flow method was used to obtain the rank-one case of Lieb's fundamental theorem concerning exhaustion by gaussians; we extend the technique to the higher-rank case, giving two new proofs of the general-rank case of Lieb's theorem.