The subelliptic heat kernel on the three dimensional solvable Lie groups

TL;DR

This paper analyzes the subelliptic heat kernels on three-dimensional solvable Lie groups, classifies their structures, and derives heat kernel expressions and gradient bounds using novel curvature-dimension inequalities.

Contribution

It provides a complete classification of left-invariant sub-Riemannian structures and explicit heat kernel formulas on these groups, introducing new curvature-dimension inequalities for bounds.

Findings

Explicit heat kernel formulas for all classified groups

Gradient bounds derived from new curvature-dimension inequalities

Complete classification of sub-Riemannian structures on these groups

Abstract

We study the subelliptic heat kernels of the CR three dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gradient bounds using a new type of curvature-dimension inequality.