Higher Laplace-Beltrami operators on bounded symmetric domains
type: Item
instance of: scholarly article
title: Higher Laplace-Beltrami operators on bounded symmetric domains
zbMATH Open document ID: 1409.32013
author: Miroslav Engliš
author: Hao Xu
publication date: 2018-09-18T00:00:00Z
review text: Let \(\mathcal D(\Omega)\) denote the algebra of all biholomorphically invariant differential operators on a bounded symmetric domain \(\Omega\) of rank \(r\). The higher Laplace-Beltrami operators are defined by \[ \bar{L}_mf=g^{a_1\bar{b}_1}\dots g^{a_m\bar{b}_m}f_{/\bar{b}_1\dots\bar{b}_ma_1\dots a_m}, \] where \(/\) in the subscript denotes covariant differentiation and \(g^{a_i\bar{b}_i}\) are contravariant metric tensors. The first author conjectured that on any Hermitian symmetric space \(\Omega\) of rank \(r\geq1\), the operators \(L_{2m-1}\), \(1\leq m\leq r\) generate \(\mathcal D(\Omega)\). The above-mentioned conjecture were partially proved (for \(r\leq3\) and for simple Hermitian symmetric spaces of rank \(r\geq2\)). In the paper under review the authors prove the conjecture for any bounded symmetric domain of rank \(r\leq 6\) and present an idea of a possible general proof by the method they used.
reviewed by: Paweł Zapałowski
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 6938662
zbMATH Keywords: higher Laplace-Beltrami operators
zbMATH Keywords: bounded symmetric domains
zbMATH Keywords: Bergman kernel
MaRDI profile type: Publication
full work available at URL: s10114-018-8162-y
OpenAlex ID: W2854784509
cites work: \(Q_p\)-spaces on bounded symmetric domains
cites work: QUANTIZATION
cites work: On the asymptotic expansion of Bergman kernel
cites work: Berezin Quantization and Reproducing Kernels on Complex Domains
cites work: Invariant Operators and the Berezin Transform on Cartan Domains
cites work: The asymptotics of a Laplace integral on a Kähler manifold
cites work: A Forelli-Rudin construction and asymptotics of weighted Bergman kernels
cites work: Covariant Cauchy-Riemann operators and higher Laplacians on Kähler manifolds.
cites work: Q4896581
cites work: Function spaces and reproducing kernels on bounded symmetric domains
cites work: Balanced metrics on some Hartogs type domains over bounded symmetric domains
cites work: Identification of Berezin-Toeplitz deformation quantization
cites work: THE TIAN–YAU–ZELDITCH ASYMPTOTIC EXPANSION FOR REAL ANALYTIC KÄHLER METRICS
cites work: On the coefficients of TYZ expansion of locally Hermitian symmetric spaces
cites work: Harmonic analysis on the quantized Riemann sphere
cites work: Higher Laplacians on pseudo-Hermitian symmetric spaces
cites work: The Berezin transform and invariant differential operators
cites work: A closed formula for the asymptotic expansion of the Bergman kernel
cites work: Q3803786
cites work: Invariant differential operators on Hermitian symmetric spaces and their eigenvalues
DOI: 10.1007/S10114-018-8162-Y
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6938662