A sharp interpolation between the Hölder and Gaussian Young inequalities

We prove a very general sharp inequality of the Hölder--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 point--wise 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 Poincaré 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 Hölder inequality and Nelson's hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.

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