A weighted Plancherel formula II. The case of the ball

Genkai Zhang

Studia Mathematica (1992)

  • Volume: 102, Issue: 2, page 103-120
  • ISSN: 0039-3223

Abstract

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The group SU(1,d) acts naturally on the Hilbert space L ² ( B d μ α ) ( α > - 1 ) , where B is the unit ball of ℂ d and d μ α the weighted measure ( 1 - | z | ² ) α d m ( 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.

How to cite

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Zhang, Genkai. "A weighted Plancherel formula II. The case of the ball." Studia Mathematica 102.2 (1992): 103-120. <http://eudml.org/doc/215917>.

@article{Zhang1992,
abstract = {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.},
author = {Zhang, Genkai},
journal = {Studia Mathematica},
keywords = {Plancherel formula; Harish-Chandra c-function; reproducing kernel; orthogonal polynomial; invariant Cauchy-Riemann operator; Hilbert space; unit ball; irreducible decomposition; generalized Harish- Chandra -function; Cauchy-Riemann operators; actions; holomorphic tensor fields},
language = {eng},
number = {2},
pages = {103-120},
title = {A weighted Plancherel formula II. The case of the ball},
url = {http://eudml.org/doc/215917},
volume = {102},
year = {1992},
}

TY - JOUR
AU - Zhang, Genkai
TI - A weighted Plancherel formula II. The case of the ball
JO - Studia Mathematica
PY - 1992
VL - 102
IS - 2
SP - 103
EP - 120
AB - 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.
LA - eng
KW - Plancherel formula; Harish-Chandra c-function; reproducing kernel; orthogonal polynomial; invariant Cauchy-Riemann operator; Hilbert space; unit ball; irreducible decomposition; generalized Harish- Chandra -function; Cauchy-Riemann operators; actions; holomorphic tensor fields
UR - http://eudml.org/doc/215917
ER -

References

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  1. [1] P. Appell et J. Kampé de Fériet, Fonctions hypergéometriques et hypersphériques, Polynomes d'Hermite, Gauthier-Villars, Paris 1926. Zbl52.0361.13
  2. [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. [3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vols. 1, 2, McGraw-Hill, New York 1953. Zbl0051.30303
  4. [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. [5] Harish-Chandra, Plancherel formula for semi-simple Lie groups, Trans. Amer. Math. Soc. 76 (1954) 485-528. 
  6. [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. [7] S. Helgason, Groups and Geometric Analysis, Academic Press, New York 1984. 
  8. [8] S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Progr. in Math. 13, Birkhäuser, Boston 1981. Zbl0467.43001
  9. [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. [10] W. Rudin, Function Theory in the Unit Ball of ℂ n , Springer, New York 1980. Zbl0495.32001
  11. [11] N. Ya. Vilenkin, Special Functions and the Theory of Group Representations, Nauka, Moscow 1965 (in Russian). 

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