Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups

TL;DR

This paper demonstrates that on stratified complex Lie groups, the logarithmic Sobolev inequality implies strong hypercontractivity of certain holomorphic projections of hypoelliptic operators, extending classical results to non-holomorphic settings.

Contribution

It extends Gross's results by establishing strong hypercontractivity for holomorphic projections of hypoelliptic operators on stratified complex Lie groups under the logarithmic Sobolev inequality.

Findings

Logarithmic Sobolev inequality implies strong hypercontractivity.

Extension of Gross's results to non-holomorphic operators.

Application to hypoelliptic Dirichlet form operators on stratified complex Lie groups.

Abstract

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