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Mat. Sb., 2004, Volume 195, Number 5, Pages 33–58 (Mi msb820)  

This article is cited in 5 scientific papers (total in 5 papers)

Poisson transformation for one-sheeted hyperboloids

A. A. Artemov

Tambov State University

Abstract: 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.

DOI: https://doi.org/10.4213/sm820

Full text: PDF file (372 kB)
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English version:
Sbornik: Mathematics, 2004, 195:5, 643–667

Bibliographic databases:

UDC: 517.98
MSC: 43A85, 53C35
Received: 02.06.2003

Citation: A. A. Artemov, “Poisson transformation for one-sheeted hyperboloids”, Mat. Sb., 195:5 (2004), 33–58; Sb. Math., 195:5 (2004), 643–667

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Imamura E., Okamoto K., Tsukamoto M., Yamamori A., “Generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains”, Proc. Japan Acad. Ser. A Math. Sci., 82:9 (2006), 167–172  crossref  mathscinet  zmath  isi  elib
    2. Molchanov V.F., Artemov A.A., Grosheva L.I., “Kanonicheskie i granichnye predstavleniya”, Vestn. Tambovskogo un-ta. Ser.: Estestvennye i tekhnicheskie nauki, 14:6-3 (2009), 1367–1425  elib
    3. A. A. Artemov, “The berezin form and harmonic analysis on a pair of hyperboloids”, Dokl. Math, 84:1 (2011), 437  crossref  mathscinet  zmath  zmath  isi  isi  elib  elib
    4. I A Shilin, A I Nizhnikov, “Some formulas for Legendre functions related to the Poisson transform and Lorentz group representation”, J. Phys.: Conf. Ser, 346 (2012), 012020  crossref  adsnasa  isi
    5. Shilin I.A., Choi J., “Integral and Series Representations of Special Functions Related to the Group So(2,2)”, Ramanujan J., 44:1 (2017), 133–153  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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