Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups

Abstract

We show that for a hypoelliptic Dirichlet form operator A A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A A is not holomorphic.