# Moser inequalities in Gauss space

## TL;DR

This paper establishes sharp constants for exponential Sobolev inequalities in Gauss space, analogous to Moser inequalities in Euclidean space, highlighting both similarities and differences.

## Contribution

It provides the first explicit determination of sharp constants for Gaussian Moser-type inequalities, extending classical Euclidean results to Gauss space.

## Key findings

- Sharp constants for Gaussian inequalities are explicitly characterized.
- Gaussian inequalities share features with Euclidean Moser inequalities.
- Distinct differences between Gaussian and Euclidean cases are identified.

## Abstract

The sharp constants in a family of exponential Sobolev type inequalities in Gauss space are exhibited. They constitute the Gaussian analogues of the Moser inequality in the borderline case of the Sobolev embedding in the Euclidean space. Interestingly, the Gaussian results have features in common with the Euclidean ones, but also reveal marked diversities.