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.

References in corpus (5)

• An extension problem related to the fractional Laplacian
• Harnack inequality for fractional sub-Laplacians in Carnot groups
• Some properties of sub-Laplaceans
• Heat kernels for a class of hybrid evolution equations
• An extension problem and trace Hardy inequality for the sublaplacian on $H$-type groups

Cited by in corpus (3)

• Local regularity for nonlocal double phase equations in the Heisenberg group
• Subelliptic Nonlocal Brezis-Nirenberg Problems on Stratified Lie Groups
• Kato-Ponce estimates for fractional sublaplacians