Small-Time Asymptotics for Subelliptic Hermite Functions on S U (2) and the CR Sphere
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Small-Time Asymptotics for Subelliptic Hermite Functions on SU (2) and the CR Sphere Joshua Campbell1 · Tai Melcher2 Received: 12 January 2018 / Accepted: 6 September 2019 / © Springer Nature B.V. 2020
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 S3 , 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. Keywords Hypoelliptic · Heat kernel · Hermite functions Mathematics Subject Classification (2010) Primary 58J35 · Secondary 53C17
1 Introduction Heat kernels are a classical object of study and are known to have deep relations to the topological and geometric properties of the space on which they live. In particular, smalltime asymptotics of the heat kernel on a Riemannian manifold can reveal various geometric data about the underlying space. The logarithmic derivatives of the heat kernel, or rather, the Hermite functions, generalize the Hermite polynomials on Rd . Of course, the Hermite polynomials play a key role in the study of the heat kernel on Rd , but also show up in many other parts of analysis; thus Hermite functions are a natural object of interest. In [28] Mitchell studied the small-time behavior of Hermite functions on compact Lie groups. In particular, he showed that, when written in exponential coordinates with a natural re-scaling, these functions converge to the classical Euclidean Hermite polynomials. Later in [29], Mitchell showed that Hermite functions on compact Riemannian manifolds, again Both authors were supported in part by NSF DMS 1255574. Tai Melcher
[email protected] Joshua Campbell
1
Ixonia, WI USA
2
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA
J. Campbell, T. Melcher
written in exponential coordinates with appropriate re-scaling, admit asymptotic expansions √ in powers of t, with a classical Hermite polynomial as the leading coefficient (and the other coefficients are other polynomials). The present paper is concerned with heat kernels related to the natural subRiemannian structure on SU (2) ∼ = S3 and, more generally, on the CR sphere S2d+1 . The aim is to show that a statement analogous to [28] holds for the Hermite functions on the CR sphere, where the limiting objects are now the subRiemannian Hermite functions of the Heisenberg group. The results of [28, 29] may be interpreted as a strong quantification of how compact Riemannian manifolds are locally Euclidean, and the present paper may be viewed as an extension of those results in a particular subRiemannian setting. In general, the tangent cone approximation of a subRiemannian geometry by the appropriate stratified group (its nilpotentization) is much weaker than the tangent space approximation of a Riemannian manifold by its Euclidean tangent space. Thus
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