Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on 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.

38 pages; changed title

References in corpus (3)

• On gradient bounds for the heat kernel on the Heisenberg group
• Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups
• Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

Cited by in corpus (2)

• Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups
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