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

On a stratified Lie group G equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on Lp 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.