Feeling the heat in a group of Heisenberg type

TL;DR

This paper studies fractional and conformal fractional powers of the horizontal Laplacian on Heisenberg type groups using heat equations, providing explicit fundamental solutions through PDE and semigroup methods.

Contribution

It offers a unified PDE-based approach to compute fundamental solutions for fractional operators on Heisenberg groups, extending classical results.

Findings

Explicit fundamental solutions for fractional operators derived.

Unified treatment of different extension problems achieved.

Recovers classical fundamental solutions when s=1.

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 $\mathscr L^s$ 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.