\(L^2\)-concrete spectral analysis of the invariant Laplacian \(\Delta_{\alpha\beta}\) in the unit complex ball \(B^n\)
type: Item
instance of: scholarly article
title: \(L^2\)-concrete spectral analysis of the invariant Laplacian \(\Delta_{\alpha\beta}\) in the unit complex ball \(B^n\)
zbMATH Open document ID: 0924.43009
author: Abdelhamid Boussejra
author: A. Intissar
published in: Journal of Functional Analysis
publication date: 1999-05-18T00:00:00Z
review text: The operator \(\Delta_{\alpha\beta}\) in the title is \[ 4(1-| z| ^2) \biggl\{\sum^n_{i,j=1} (\delta_{ij}-z_i\overline z _j)\frac{\partial^2} {\partial z_j\partial\overline z _j} +\alpha\sum^n_{j=1} z_j \frac{\partial}{\partial z_j} +\beta\sum^n_{j=1}\overline z_j \frac{\partial} {\partial\overline z _j} -\alpha\beta\biggr\}, \] where \(\alpha\) and \(\beta\) are real numbers, and \(\delta_{ij}\) is the Kronecker symbol; and the ``concrete'' spectral theory is in the sense of \textit{R. S. Strichartz} [J. Funct. Anal. 87, No.1, 51-148 (1989; Zbl 0694.43008)]. The authors describe the eigenspaces of \(\Delta_{\alpha\beta}\) and characterize the eigenfunctions that are generalized Poisson transforms of square-integrable functions on the boundary of the unit ball in \(\mathbf{C}^n\). They previously announced the result in [C. R. Acad. Sci., Paris, Sér. I 315, No. 13, 1353-1357 (1992; Zbl 0772.32001)] for the special case when \(\alpha=\beta=0\).
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1246379
zbMATH Keywords: Poisson integral
zbMATH Keywords: Plancherel formula
zbMATH Keywords: eigenfunctions
zbMATH Keywords: spherical harmonics
zbMATH Keywords: Jacobi functions
zbMATH Keywords: Calderon-Zygmund theory
reviewed by: Harold P. Boas
MaRDI profile type: Publication
full work available at URL: jfan.1998.3318
OpenAlex ID: W2012564862
cites work: Q4697143
cites work: Q4296129
cites work: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. (Calderón-Zygmund operators, para-accretive functions and interpolation)
cites work: Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I
cites work: Spherical Harmonic Expansion of the Poisson-Szego Kernel for the Ball
cites work: Some results in \(H^ p\) theory for the Heisenberg group
cites work: A duality for symmetric spaces with applications to group representations
cites work: \({\mathcal M}\)-subspaces of \(X_ \lambda\)
cites work: Weighted Plancherel formula. Irreducible unitary representations and eigenspace representations.
cites work: Die Differentialgleichungen in der Theorie der elliptischen Modulfunktionen. (The differential equations in the theory of elliptic modular functions)
cites work: Q3692234
cites work: Q4083962
cites work: Q3958740
cites work: The Plancherel formula for spherical functions with a one-dimensional \(K\)-type on a simply connected simple Lie group of Hermitian type
cites work: Harmonic analysis as spectral theory of Laplacians
cites work: A weighted Plancherel formula II. The case of the ball
DOI: 10.1006/JFAN.1998.3318
description: scientific article; zbMATH DE number 1246379