Sharp exponential inequalities for the Ornstein-Uhlenbeck operator

TL;DR

This paper determines the best constants and extremal functions for exponential inequalities involving the Ornstein-Uhlenbeck operator in Gaussian space, highlighting similarities and differences with Euclidean Laplace operator inequalities.

Contribution

It identifies optimal constants and extremal functions for exponential inequalities related to the Ornstein-Uhlenbeck operator, expanding understanding of Gaussian space inequalities.

Findings

Optimal constants in exponential inequalities are established.

Existence of extremal functions is proven.

Analogies with Adams' inequality are discussed.

Abstract

The optimal constants in a class of exponential type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space are detected. The existence of extremal functions in the relevant inequalities is also established. Our results disclose analogies and dissimilarities in comparison with Adams' inequality for the Laplace operator, a companion of our inequalities in the Euclidean space.