Feeling the heat in a group of Heisenberg type

Abstract

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 ${\mathcal{L}}^s$ and ${\mathcal{L}}_s$, $0<s\le 1$. Here, ${\mathcal{L}}^s$ is the fractional power of the horizontal Laplacian, and ${\mathcal{L}}_s$ 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.