Small-time asymptotics for subelliptic Hermite functions on $SU(2)$ and the CR sphere

TL;DR

This paper investigates the small-time asymptotics of subelliptic heat kernels on $SU(2)$ and higher-dimensional spheres, showing convergence to the corresponding heat kernels on Heisenberg groups, revealing deep geometric-analytic connections.

Contribution

It establishes the convergence of logarithmic derivatives of subelliptic heat kernels on $SU(2)$ and spheres to those on Heisenberg groups under natural scaling, extending previous results to higher dimensions.

Findings

Convergence of heat kernel derivatives from $SU(2)$ to Heisenberg group at small times.

Extension of results to higher-dimensional spheres with subRiemannian structures.

Identification of limiting spaces as higher-dimensional Heisenberg groups.

Abstract

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on $SU(2)$ converges to their analogues on the Heisenberg group at time 1. Realizing $SU(2)$ as $\mathbb{S}^3$, we then generalize these results to higher-order odd-dimensional spheres equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups.