A sharp interpolation between the H\"older and Gaussian Young inequalities

TL;DR

This paper introduces a unified sharp inequality that interpolates between H"older and Gaussian Young inequalities, extending fundamental results in Gaussian analysis through a novel family of products and integral representations.

Contribution

It develops a general family of commutative products interpolating between point-wise and Wick products, leading to a unified framework for H"older and Nelson's hyper-contractive inequalities.

Findings

Established a sharp H"older--Young--type inequality for infinite-dimensional Gaussian spaces.

Unified classical inequalities within a single generalized framework.

Provided integral representations and interpolation techniques for Gaussian analysis.

Abstract

We prove a very general sharp inequality of the H\"older--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 Poincar\'e 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 H\"older inequality and Nelson's hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.