Fundamental solutions for powers of the Heisenberg sub-Laplacian
type: Item
instance of: scholarly article
title: Fundamental solutions for powers of the Heisenberg sub-Laplacian
zbMATH Open document ID: 0789.22012
author: Chal Benson
author: Gail Ratcliff
author: Anthony H. Dooley
published in: Illinois Journal of Mathematics
publication date: 1994-06-14T00:00:00Z
review text: Let \(\Delta_{H_ n}\) denote the sub-Laplacian operator on the Heisenberg group \(H_ n\) of dimension \(2n+1\). In this paper we compute explicit fundamental solutions for powers \(\Delta^ p_{H_ n}\) of \(\Delta_{H_ n}\) with \(p \leq n\), generalizing a result of G. Folland for the case \(p=1\). We show that \(\Delta^ p_{H_ n}\) has a tempered fundamental solution which is given by a smooth function away from the identity in \(H_ n\). The solution can be written in terms of iterated antiderivatives of elementary functions or obtained as the weak limit of an infinite series. We recover Folland's solution when \(p=1\) and obtain an explicit (finite) form when \(p=2\). The key idea in our derivation exploits the \(U(n)\)-invariance of the operator \(\Delta_{H_ n}\) to express fundamental solutions in terms of \(U(n)\)-spherical functions. Our methods should be applicable to other differential operators on certain nilpotent Lie groups that satisfy strong invariance conditions.
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 35A08
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 561223
zbMATH Keywords: spherical functions
zbMATH Keywords: sub-Laplacian operator
zbMATH Keywords: Heisenberg group
zbMATH Keywords: explicit fundamental solutions
zbMATH Keywords: tempered fundamental solution
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 561223