Transformation de Poisson de formes différentielles. Le cas de l'espace hyperbolique. (Poisson transformation of differential forms. The case of hyperbolic space)
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.
scientific article; zbMATH DE number 4036417
scientific article