The subelliptic heat kernels on SL(2,R) and on its universal covering $\widetilde{SL(2,R)}$: integral representations and some functional inequalities

TL;DR

This paper derives integral representations for subelliptic heat kernels on SL(2,R) and its universal cover, enabling analysis of their asymptotics, distances, and functional inequalities like Li-Yau estimates.

Contribution

It provides explicit integral formulas for the heat kernels on SL(2,R) and its universal cover, facilitating new asymptotic and inequality analyses.

Findings

Derived integral representations for heat kernels

Obtained small-time asymptotics and subriemannian distances

Established gradient estimates and functional inequalities

Abstract

In this paper, we study a subelliptic heat kernel on the Lie group SL(2,R) and on its universal covering. The subelliptic structure on SL(2,R) comes from the fibration $SO(2) -> SL(2,R) -> H^2$ and it can be lifted to its universal covering. First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics in small time of the heat kernels and give us a way to compute the subriemannian distances. Then, we establish some gradient estimates and some functional inequalities like a Li-Yau type estimate and a reverse Poincar\'e inequality that are valid for both heat kernels.