Second-order right-invariant partial differential equations on a Lie group
type: Item
instance of: scholarly article
title: Second-order right-invariant partial differential equations on a Lie group
zbMATH Open document ID: 0703.35131
DOI: 10.1016/0022-247X(89)90005-X
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: The author gives a simple, relatively explicit eigenfunction expansion for some Laplacian operators and shows that some more explicit results may be obtained for the sub-Laplacians than is the case for the corresponding elliptic Laplace operators defined from a basis for the Lie algebra. It follows, in particular, that some Laplacians have no singular continuous spectrum when the Lie group G is simply connected nilpotent because the spectrum is (0,\(\infty)\), absolutely continuous and with uniform multiplicity (Theorem 4.1. p. 342). The paper embodies not only all the previous papers by the author on the same subject, but also many results which are listed in the other references. In section 5 the author points out that a spectral representation for the Nelson-Stinespring Laplace operator \(\Delta\) can be constructed by applying the Mackey-Blattner imprimitivity theorem. Finally, this long paper which is nearly a monograph, it contains a large number of remarks, lemmas, assumptions, theorems, examples and proofs which are too complicated to be reproduced here.
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 4153055
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: sub-Laplacians
zbMATH Keywords: singular continuous spectrum
zbMATH Keywords: nilpotent
zbMATH Keywords: absolutely continuous
zbMATH Keywords: multiplicity
zbMATH Keywords: Nelson-Stinespring Laplace operator
Wikidata QID: Q115364517
author: Palle Jorgensen
reviewed by: F. M. Ragab
MaRDI profile type: Publication
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full work available at URL: 0022-247x(89)90005-x
OpenAlex ID: W1978458903
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description: scientific article; zbMATH DE number 4153055