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Czasopismo
1992 | 102 | 2 | 103-120
Tytuł artykułu

A weighted Plancherel formula II. The case of the ball

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The group SU(1,d) acts naturally on the Hilbert space $L²(B dμ_α) (α > -1)$, where B is the unit ball of $ℂ^d$ and $dμ_α$ the weighted measure $(1-|z|²)^α dm(z)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic tensor fields.
Czasopismo
Rocznik
Tom
102
Numer
2
Strony
103-120
Opis fizyczny
Daty
wydano
1992
otrzymano
1991-02-27
Twórcy
autor
  • Matematiska Institutionen, Stockholms Universitet, Box 6701, S-113 85 Stockholm, Sweden
Bibliografia
  • [1] P. Appell et J. Kampé de Fériet, Fonctions hypergéometriques et hypersphériques, Polynomes d'Hermite, Gauthier-Villars, Paris 1926.
  • [2] J. Arazy, S. Fisher and J. Peetre, Membership in the Schatten-von Neumann classes and Hankel operators on Bergman space, J. London Math. Soc., to appear.
  • [3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vols. 1, 2, McGraw-Hill, New York 1953.
  • [4] I. M. Gel'fand and M. I. Graev, The analogue of Plancherel's theorem for real unimodular groups, Dokl. Akad. Nauk SSSR 92 (1953), 461-464 (in Russian).
  • [5] Harish-Chandra, Plancherel formula for semi-simple Lie groups, Trans. Amer. Math. Soc. 76 (1954) 485-528.
  • [6] D. Hejhal, The Selberg Trace Formula for PSL(2,ℝ), Vol. 1, Lecture Notes in Math. 548; Vol. 2, Lecture Notes in Math. 1001, Springer, Berlin 1976, 1983.
  • [7] S. Helgason, Groups and Geometric Analysis, Academic Press, New York 1984.
  • [8] S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Progr. in Math. 13, Birkhäuser, Boston 1981.
  • [9] J. Peetre, L. Peng and G. Zhang, A weighted Plancherel formula I. The case of the disk. Applications to Hankel operators, technical report, Stockholm.
  • [10] W. Rudin, Function Theory in the Unit Ball of $ℂ^n$, Springer, New York 1980.
  • [11] N. Ya. Vilenkin, Special Functions and the Theory of Group Representations, Nauka, Moscow 1965 (in Russian).
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv102i2p103bwm
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