A sharp interpolation between the Holder and Gaussian Young inequalities

作者：Da Pelo Paolo; Lanconelli Alberto; Stan Aurel I^*

来源：Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2016, 19(1): 1650001.

DOI：10.1142/S0219025716500016

• Gaussian T-Wick product
• second quantization operator
• exponential functions
• Holder inequality
• Lieb inequality
• Minkowski inequality
• Jensen inequality

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.

• 出版日期2016-3