Poisson transformation for one-sheeted hyperboloids

This paper is concerned with the study of the Poisson transformation and with finding the complete asymptotic series at infinity for this transformation (not just its dominant term) for an important subclass of the class of semisimple symmetric spaces $G/H$ with non-compact $H$: the real hyperbolic spaces (one-sheeted hyperboloids) $\operatorname{SO}_0(1,n-1)/\operatorname{SO}_0(1,n-2)$. The expansions are obtained for arbitrary (not necessarily $K$-finite) functions. The operators involved in the coefficients of the expansions and acting on functions at the boundary are described. The reducibility and the irreducibility of eigensubspaces of the Laplace–Beltrami operator in function spaces on the hyperboloid are studied. The structure of representations acting in spaces of eigenfunctions is described. Descriptions of the kernel and the closure of the range of the Poisson transformation are presented.