Grushin Operator on Infinite Dimensional Homogeneous Lie Groups

TL;DR

This paper studies the properties of a generalized Grushin operator on infinite dimensional Lie groups, establishing key inequalities and heat kernel estimates under Hörmander's condition.

Contribution

It introduces a framework for analyzing Grushin operators in infinite dimensional settings, proving fundamental inequalities and heat kernel bounds.

Findings

Proved Poincaré inequality for the operator

Established Gaussian bounds for heat kernels

Confirmed doubling condition for the metric

Abstract

A collection of infinite dimensional complete vector fields $\left\{V_i\right\}_{i=1}^{\infty}$ acting on a locally convex manifolds $M$ on which a smooth positive measure $\mu$ is defined was considered. It was assumed that the vector fields generates an infinite dimensional Lie algebra $\mathfrak{g}$ and satisfies H$\ddot{o}$rmander's condition. The sum of squares of Grushin operators related to the vector fields was examined and the operator is then considered as the generalized Grushin operator. The paramount proofs were Poincar$\acute{e}$ inequality, Gaussian two-bounded estimate for the related heat kernels and the doubling condition for the metric defined by the underlying vector fields.