Perturbation results concerning Gaussian estimates and hypoellipticity for left-invariant Laplacians on compact groups

TL;DR

This paper investigates how Gaussian heat kernel bounds and hypoellipticity properties of bi-invariant Laplacians on compact groups are preserved under certain perturbations, extending understanding of heat kernel behavior in geometric analysis.

Contribution

It demonstrates that Gaussian bounds and hypoellipticity are maintained for form-comparable perturbations of bi-invariant Laplacians on compact groups.

Findings

Gaussian bounds for derivatives of heat kernels are preserved under perturbations.

Parabolic operators with these Laplacians are hypoelliptic in various senses.

Results extend heat kernel estimates to a broader class of operators.

Abstract

In this paper we study left-invariant Laplacians on compact connected groups that are form-comparable perturbations of bi-invariant Laplacians. Our results show that Gaussian bounds for derivatives of heat kernels enjoyed by certain bi-invariant Laplacians hold for their form-comparable perturbations. We further show that the parabolic operators associated with such left-invariant Laplacians, in particular, with the bi-invariant Laplacians, are hypoelliptic in various senses.