The heat kernel and the Riesz transforms on the quaternionic Heisenberg groups. (Q1880037)

Let \(\mathbb H\) be the field of quaternions, then \({\mathbb H}^n\times {\mathbb R}^3\) becomes a nilpotent Lie group called the quaternionic Heisenberg group. By exploiting stochastic integral methods of \textit{B. Gaveau} [C. R. Acad. Sci., Paris, Sér. A 281, 327--328 (1975; Zbl 0321.35041)] developed for the Heisenberg group \({\mathbb C}^n\times {\mathbb R}\) the author computes the heat kernel for the quaternionic Heisenberg group. Adapting methods of \textit{T. Coulhon, D. Müller} and \textit{J. Zienkiewicz} [Math. Ann. 305, No. 2, 369--379 (1996; Zbl 0859.22006)] it is shown that the Riesz transforms on \({\mathbb H}^n\times {\mathbb R}^3\) are bounded uniformly in \(n\).