Geodesic
Poisson transformation for one-sheeted hyperboloids
Sbornik. Mathematics, Tome 195 (2004) no. 5, pp. 643-667 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

[1] Kashiwara M., Kowata A., Minemura K., Okamoto K., Oshima T., Tanaka M., “Eigenfunctions of invariant differential operators on a symmetric space”, Ann. of Math., 107:1 (1978), 1–39 | DOI | MR | Zbl

[2] Molchanov V. F., “Sfericheskie funktsii na giperboloidakh”, Matem. sb., 99:2 (1976), 139–161 | MR | Zbl

[3] Molchanov V. F., “Garmonicheskii analiz na odnorodnykh prostranstvakh”, Itogi nauki i tekhniki. Sovrem. problemy matem. Fundament. napravleniya, 59, VINITI, M., 1990, 5–144 | MR

[4] Oshima T., “Poisson transformations on affine symmetric spaces”, Proc. Japan Acad. Ser. A Math. Sci., A55:9 (1979), 323–327 | DOI | MR

[5] Sekiguchi J., “Eigenspaces of the Laplace–Beltrami operator on a hyperboloid”, Nagoya Math. J., 79 (1980), 151–185 | MR | Zbl

[6] Harish-Chandra., “Differential equations and semisimple Lie groups”, Collected papers, vol. III, Springer-Verlag, 1983, 57–120

[7] Harish-Chandra., “Spherical functions on a semisimple Lie group”, Amer. J. Math., 80 (1958), 241–310; 553–613 | DOI | MR | Zbl

[8] Casselman W., Milicic D., “Asymptotic behaviour of matrix coefficients of admissible representations”, Duke Math. J., 49 (1982), 869–930 | DOI | MR | Zbl

[9] van den Ban E. P., Schlichtkrull H., “Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces”, J. Reine Angew. Math., 380 (1987), 108–165 | MR | Zbl

[10] van den Ban E. P., Schlichtkrull H., “Local boundary data of eigenfunctions on a Riemannian symmetric space”, Invent. Math., 98 (1989), 639–657 | DOI | MR | Zbl

[11] van den Ban E. P., “Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces”, Indag. Math., 49 (1987), 225–249 | MR | Zbl

[12] van den Ban E. P., Schlichtkrull H., “Asymptotic expansions on symmetric spaces”, Harmonic analysis on reductive groups, Progr. Math., 101, Birkhäuser, Boston, 1991, 79–87 | MR

[13] Molchanov V. F., van Dijk G., Canonical representations and overgroups, Preprint, Math. Inst. Univ. Leiden, MI 2002–05

[14] Artemov A. A., “Asymptotic behaviour of the Poisson transform on a hyperboloid of one sheet”, Lie groups and Lie algebras. Their representations, generalizations, and applications, Math. Appl., 433, eds. Komrakov B. P. et al., Kluwer Acad. Publ., Dordrecht, 1998, 261–284 | MR | Zbl

[15] Artemov A. A., “Poisson transform for hyperboloids”, Vestn. Tamb. un-ta. Ser. estestv. tekhn. nauki, 3:1 (1998), 21–34

[16] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1958

[17] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 | Zbl

[18] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965 | MR

[19] Molchanov V. F., “Predstavleniya psevdoortogonalnoi gruppy, svyazannye s konusom”, Matem. sb., 81:3 (1970), 358–375 | MR | Zbl

[20] Zhelobenko D. P., “O beskonechno differentsiruemykh vektorakh v teorii predstavlenii”, Vestn. MGU. Cer. 1. Matem., mekh., 1965, no. 1, 3–10

[21] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, t. I, II, Nauka, M., 1973, 1974