Precise estimates for the subelliptic heat kernel on H-type groups

We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_1{Q}_t{e}^{-\frac{d^2}{4t}}\le p_t\le C_2{Q}_t{e}^{-\frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Carath'eodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $\mathrm{\mid }\mathrm{\nabla }{p}_t\mathrm{\mid }$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type groups.