Integrated Harnack inequalities on Lie groups

TL;DR

This paper establishes dimension-independent integrated Harnack inequalities for heat kernels on unimodular Lie groups, linking them to Wang's inequalities and demonstrating implications for infinite-dimensional Lie group measures.

Contribution

It introduces an integrated Harnack inequality for heat kernels on unimodular Lie groups and connects it to Wang's inequality, with applications to infinite-dimensional groups.

Findings

Logarithmic derivatives of heat kernels are exponentially integrable.

The integrated Harnack inequality is equivalent to Wang's inequality.

Implications for quasi-invariance of heat kernel measures in infinite dimensions.

Abstract

We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an "integrated" Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang's Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional "Lie" groups.