# Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups

## TL;DR

This paper establishes the equivalence of strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups, extending Euclidean results to a broader geometric setting.

## Contribution

It introduces and proves the equivalence of strong hypercontractivity and strong logarithmic Sobolev inequalities on stratified Lie groups, generalizing known Euclidean results.

## Key findings

- Strong hypercontractivity and strong logarithmic Sobolev inequalities are equivalent on stratified Lie groups.
- If the group satisfies a classical logarithmic Sobolev inequality, then both properties hold.
- The results extend Euclidean inequalities to the setting of stratified Lie groups.

## Abstract

On a stratified Lie group $G$ equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on $L^p$ spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group $G$. Moreover, if $G$ satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.