Potential Analysis

, Volume 45, Issue 2, pp 355–386 | Cite as

Hypoelliptic Heat Kernels on Nilpotent Lie Groups

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Abstract

The starting point of our analysis is an old idea of writing an eigenfunction expansion for a heat kernel considered in the case of a hypoelliptic heat kernel on a nilpotent Lie group G. One of the ingredients of this approach is the generalized Fourier transform. The formula one gets using this approach is explicit as long as we can find all unitary irreducible representations of G. In the current paper we consider an n-step nilpotent Lie group G n as an illustration of this technique. First we apply Kirillov’s orbit method to describe these representations for G n . This allows us to write the corresponding hypoelliptic heat kernel using an integral formula over a Euclidean space. As an application, we describe a short-time behavior of the hypoelliptic heat kernel in our case.

Keywords

Nilpotent group Sub-Riemannian manifold Hypoelliptic heat kernel Kirillov’s orbit method 

Mathematics Subject Classification (2010)

Primary 58J35 Secondary 53C17 
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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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