论文标题
感觉到一群海森堡类型的热量
Feeling the heat in a group of Heisenberg type
论文作者
论文摘要
在本文中，我们在一组Heisenberg类型的$ \ Mathbb {G} $中使用热方程式，以在时间独立的pseudo-differential operators $ \ mathscr l^s $和$ \ mathscr l_s $，$ 0 <s \ s \ s \ leq 1 $中提供对两个非常不同的扩展问题的统一处理。在这里，$ \ mathscr l^s $是水平laplacian的分数力量，而$ \ mathscr l_s $是$ \ mathbb {g} $上的水平laplacian的共形分数力量。我们的主要目标之一是通过新方法专门基于部分微分方程和半群方法来明确计算这些非本地操作员的基本解决方案。当$ s = 1 $时，我们的结果回收了Folland发现的著名基本解决方案，并由Kaplan概括。
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.
