Feeling the heat in a group of Heisenberg type

In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators $\mathscr L^s$ and $\mathscr L_s$, $0< s\leq 1$. Here, $\mathscr L^s$ is the fractional power of the horizontal Laplacian, and $\mathscr L_s$ is the conformal fractional power of the horizontal Laplacian on $\mathbb{G}$. One of our main objective is compute explicitly the fundamental solutions of these nonlocal operators by a new approach exclusively based on partial differential equations and semigroup methods. When $s=1$ our results recapture the famous fundamental solution found by Folland and generalised by Kaplan.