热流下的泛函体积乘积
摘要
我们证明了偶函数的泛函体积乘积沿Fokker-Planck热流是单调递增的。这特别给出了K. Ball以及Artstein-Avidan–Klartag–Milman在偶函数情形下的泛函Blaschke–Santaló不等式的一个新证明。这一结果源于对Ornstein–Uhlenbeck半群正则化性质的新理解。即，我们建立了偶函数的Borell逆超收缩不等式的一个改进，并确定了可容许指数的尖锐范围。作为该不等式尖锐范围成功确定的另一个结果，我们推导了偶函数的Laplace变换的尖锐L^p-L^q不等式。该不等式的最佳常数由中心高斯函数达到，因此这提供了与Beckner尖锐Hausdorff–Young不等式类似的结果。我们在证明中的技术新颖之处在于使用了log-concave测度的Brascamp–Lieb不等式和Cramér–Rao不等式。
引用
@article{arxiv.2401.00427,
title = {The functional volume product under heat flow},
author = {Shohei Nakamura and Hiroshi Tsuji},
journal= {arXiv preprint arXiv:2401.00427},
year = {2024}
}
备注
In this update, we have mentioned about the "detropicalised" approach that has been proposed in the discussion of Klartag and Tao in Tao's blog post as it is closely related this work. We have mentioned works of Berndtsson--Mastrantonis--Rubinstein and Kolesnikov--Werner. Also, we have split the result on the stability from this version. This part will be in the forthcoming paper