Transformation de Poisson de formes différentielles. Le cas de l'espace hyperbolique. (Poisson transformation of differential forms. The case of hyperbolic space)
type: Item
instance of: scholarly article
title: Transformation de Poisson de formes différentielles. Le cas de l'espace hyperbolique. (Poisson transformation of differential forms. The case of hyperbolic space)
zbMATH Open document ID: 0636.43007
DOI: 10.1007/BF02621934
author: Pierre-Yves Gaillard
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140070
review text: It is known that, on the real hyperbolic space \(H^ n\) any eigenfunction of the Laplace operator is the Poisson transform of a hyperfunction on its boundary \(\vartheta H^ n\). Instead of functions the author considers differential forms. He defines the Poisson transform \(\Phi_ p\) mapping the space of p-forms on \(\vartheta H^ n\) into the space of coclosed harmonic p-forms on \(H^ n\), and proves that, for \(p\neq (n- 1)/2\), it is an isomorphism. The hyperform is a courant if and only if the corresponding form is slowly increasing. Finally a Fatou-type theorem is proved. The proofs use Fourier expansions with respect to the orthogonal group \(O_ n\) and properties of the hypergeometric function.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 58A10
zbMATH DE Number: 4036417
zbMATH Keywords: real hyperbolic space
zbMATH Keywords: eigenfunction
zbMATH Keywords: Laplace operator
zbMATH Keywords: Poisson transform
zbMATH Keywords: hyperfunction
zbMATH Keywords: differential forms
zbMATH Keywords: space of p-forms
zbMATH Keywords: coclosed harmonic p-forms
zbMATH Keywords: courant
zbMATH Keywords: Fatou-type theorem
reviewed by: Jacques Faraut
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4036417