Poisson transformation for one-sheeted hyperboloids
Sbornik. Mathematics, Tome 195 (2004) no. 5, pp. 643-667
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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.
@article{SM_2004_195_5_a1,
author = {A. A. Artemov},
title = {Poisson transformation for one-sheeted hyperboloids},
journal = {Sbornik. Mathematics},
pages = {643--667},
publisher = {mathdoc},
volume = {195},
number = {5},
year = {2004},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2004_195_5_a1/}
}
A. A. Artemov. Poisson transformation for one-sheeted hyperboloids. Sbornik. Mathematics, Tome 195 (2004) no. 5, pp. 643-667. http://geodesic.mathdoc.fr/item/SM_2004_195_5_a1/