# A weighted Plancherel formula II. The case of the ball

Studia Mathematica (1992)

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

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topZhang, 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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- [3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vols. 1, 2, McGraw-Hill, New York 1953. Zbl0051.30303
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- [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. Zbl0467.43001
- [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 ${\u2102}^{n}$, Springer, New York 1980. Zbl0495.32001
- [11] N. Ya. Vilenkin, Special Functions and the Theory of Group Representations, Nauka, Moscow 1965 (in Russian).

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