The Heat Kernel and the Riesz Transforms on the Quaternionic Heisenberg Groups

As the heat kernel plays an important role in many problems in harmonic analysis, an explicit usable expression is very much desirable. An explicit expression for the heat kernel for the Heisenberg group Hn = C ×R was obtained by Hulanicki [9] and by Gaveau [7]. Gaveau [7] also obtained the heat kernel for free nilpotent Lie groups of step two. Cygan [4] obtained the heat kernel for all nilpotent Lie groups of step two. But neither Gaveau’s expression for free nilpotent Lie groups nor Cygan’s expression for arbitrary nilpotent Lie groups of step two were as explicit as in the case of Heisenberg groups. The Hulanicki-Gaveau’s formula for the heat kernel for the Heisenberg group has many interesting applications: Hueber [8] et al. used it to describe the Martin boundary corresponding to the sublaplacian of the Heisenberg group, Garofalo [6] et al. used it to study the regularity of boundary points in the Dirichlet problem for the heat equation on the Heisenberg group, while Coulhon [3] et al. used it to show the uniform boundedness of Riesz transforms on the Heisenberg group. Although these applications are very impressive, they depend heavily on explicit expressions for the heat kernel. All of these works motivate the following question: Are there other nilpotent Lie groups for which the expressions for the heat kernel are as explicit as in the case of the Heisenberg group? The first aim of this paper is to look for such formulae for the heat kernel for the quaternionic Heisenberg groups. These groups are defined by replacing the complex field C by the field of quaternions H in the definition of Hn. More precisely, we make Hn×R3 into a nilpotent Lie group of step two by suitably defining the group operation. On this group there is a natural sublaplacian with an associated heat kernel. We use the method of Gaveau