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Transformation de Poisson de formes diffĂ©rentielles. Le cas de l’espace hyperbolique. (Poisson transformation of differential forms. The case of hyperbolic space). (French) Zbl 0636.43007
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.
Reviewer: J.Faraut

43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces
58A10 Differential forms in global analysis
Full Text: DOI EuDML