Conformal harmonic analysis on hyperboloids
The author studies the natural representation of \(SO(p,q)\), when \(p+q\) is even, in the space of functions on the isotropic cone of signature \((p,q)\), which are homogeneous of degree \(-(p+q)/2+1\). This representation is known to be unitarizable and to split into a direct sum of two irreducible representations [see for example \textit{R. Howe} and \textit{Eng-Chye Tan}, Bull. Am. Math. Soc., New Ser. 28, No. 1, 1-74 (1993) (case 00, (iv), p. 13)]. A direct proof of this fact is given; the spaces of the two irreducible representations are described using the distributions \(m_ \pm^ \lambda\) associated to a nonsingular quadratic form \(m\) of signature \((p-1,q-1)\). It is then shown, using former results of the author and \textit{G. M. Henkin} on the inversion of the Radon- Penrose transform of \(\overline{\partial}\)-cohomology, that these spaces are isomorphic to some Hardy spaces of \(\overline{\partial}\)-cohomology in pseudo-Hermitian symmetric manifolds of tube type.
Hardy spaces of \(\overline{\partial}\)-cohomology
pseudo-Hermitian symmetric manifolds of tube type
scientific article; zbMATH DE number 219772