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arXiv 1106.0709math.FA
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对数凹测度的Brascamp-Lieb型非对称协方差估计及相关不等式

Asymmetric Covariance Estimates of Brascamp-Lieb Type and Related Inequalities for Log-concave Measures

Eric A. Carlen, Dario Cordero-Erausquin, Elliott H. Lieb

备注：
16 pages. This version corrects some misprints, and contains additional comments on the results
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摘要：

Brascamp和Lieb的不等式提供了两个函数关于对数凹测度的协方差的界。该界通过函数梯度的$L^2$范数的乘积来估计协方差，其中梯度的大小使用由对数凹测度的势的逆Hessian矩阵给出的内积来计算。Menz和Otto \cite{OM}证明了一个变体，两个$L^2$范数被$L^1$和$L^\infty$范数取代，但仅适用于$\R^1$。我们证明了推广这些不等式$L^p$和$L^q$规范和$\R^n$，任何$n\geq 1$。证明了函数的均差除积分用梯度积分表示的一个不等式。

英文摘要：

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto \cite{OM} proved a variant of this with the two $L^2$ norms replaced by $L^1$ and $L^\infty$ norms, but only for $\R^1$. We prove a generalization of both by extending these inequalities to $L^p$ and $L^q$ norms and on $\R^n$, for any $n\geq 1$. We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients.

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