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A sharp interpolation between the Holder and Gaussian Young inequalities

机译：Holder和高斯杨不等式之间的清晰插值

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摘要

We prove a very general sharp inequality of the Holder-Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong-Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincare inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Holder inequality and Nelson's hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.
机译：对于无限维高斯空间上定义的函数，我们证明了Holder-Young型的非常普遍的尖锐不等式。我们首先考虑一个交换产品系列，这些产品在点积和维克积之间进行插值。该族通过Wong-Zakai型逼近定理在随机微分方程的背景下自然产生，并且在Beckner型Poincare不等式的某些推广中起关键作用。然后，我们获得该产品系列的关键积分表示形式，并利用归因于李布的经典杨不等式的推广来证明我们的主要定理。我们强调，我们的主要不等式包含特定情况下的Holder不等式和Nelson的超收缩估计，因此为高斯分析的两个基本结果提供了统一的框架。

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