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.

在本文中，我们将热方程用于一组Heisenberg型 $\mathbb{G}$ 为时间独立的伪微分算子提供对两个非常不同的扩展问题的统一处理 ${\mathcal{大号}}^s$ 和 ${\mathcal{大号}}_s$， $0<s\le \mathrm{1个}$。这里，${\mathcal{大号}}^s$ 是水平拉普拉斯算子的分数幂，并且 ${\mathcal{大号}}_s$ 是水平拉普拉斯算子的共形分数次方 $\mathbb{G}$。我们的主要目标之一是通过专门基于偏微分方程和半群方法的新方法来显式计算这些非局部算子的基本解。什么时候$s=\mathrm{1个}$ 我们的结果重述了Folland发现并由Kaplan推广的著名的基本解决方案。