Invariant differential equations on reductive Lie groups and algebras. II
type: Item
instance of: scholarly article
title: Invariant differential equations on reductive Lie groups and algebras. II
zbMATH Open document ID: 1054.22008
author: Abderrazak Bouaziz
author: Nouri Kamoun
published in: Journal of Functional Analysis
publication date: 2004-10-12T00:00:00Z
review text: [Part I by the authors in Ann. Inst. Fourier 50, 1799--1857 (2000; Zbl 0970.22009).] Let \(D\) be a biinvariant differential operator on a reductive Lie group \(G\) which satisfies the Benabdallah-Rouvière condition. The authors prove that it is possible to solve the equation \(Du=v\) in the space \({\mathcal D}'(G)^G\) of invariant distributions on \(G\). In a previous paper they established the same statement for the space \({\mathcal D}'(G)^G_F\) of invariant distributions with finite order. By using properties of orbital integrals, it amounts to proving the same statement for a Cartan subgroup \(H\). Hence the larger part of the paper is a proof of the following statement: Let \(D\) be an invariant differential operator on \(H={\mathbb T}^p\times {\mathbb R}^n\), and \(\Omega \) a connected open set in \(H\). If \(D\) has a fundamental solution, then \(D{\mathcal D}'(\Omega )={\mathcal D}'(\Omega )\). For \(H={\mathbb R}^n\) it is a classical result by Malgrange and Ehrenpreis. For \(H={\mathbb T}^p\), it has been proved by Cérézo and Rouvière. The proof follows the lines of the one by Malgrange.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 2107271
zbMATH Keywords: reductive Lie group
zbMATH Keywords: differential operator
zbMATH Keywords: fundamental solution
zbMATH Keywords: reductive groups
zbMATH Keywords: orbital integrals
zbMATH Keywords: invariant distributions
zbMATH Keywords: invariant differential operators
reviewed by: Jacques Faraut
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.11.007
OpenAlex ID: W2095048038
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DOI: 10.1016/J.JFA.2003.11.007
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description: scientific article; zbMATH DE number 2107271