Hua operators on bounded homogeneous domains in \(\mathbb{C}^ n\) and alternative reproducing kernels for holomorphic functions
type: Item
instance of: scholarly article
title: Hua operators on bounded homogeneous domains in \(\mathbb{C}^ n\) and alternative reproducing kernels for holomorphic functions
zbMATH Open document ID: 0887.43005
published in: Journal of Functional Analysis
publication date: 1998-04-19T00:00:00Z
review text: A canonical system of differential operators for bounded homogeneous domains (more general than the Hermite symmetric domains) in \(\mathbb{C}^n\) is defined by contracting the operator \(\partial\overline\partial\) with the holomorphic curvature tensor, which generalizes the classical Hua operators in the Cartan domains. Functions annihilated by this system are called Hua-harmonic. The main result proved here is the possibility of representing them as ``Poisson integrals'', more precisely, every bounded Hua-harmonic function is a Poisson integral over the Bergman-Shilov boundary of a unique \(L^\infty\) function with respect to the Poisson kernel of the Laplace-Beltrami operator. The authors also consider harmonic functions defined as the null space of a single real differential operator invariant under a particular solvable Lie group which acts transitively on the bounded homogeneous domain. A result in this respect is that there exists an operator satisfying the following three conditions: annihilating holomorphic functions; satisfying the Hörmander condition; having the Bergman-Shilov boundary as its maximal boundary. It follows that the corresponding bounded harmonic functions are in one-to-one correspondence with the \(L^\infty\) functions on the Bergman-Shilov boundary under the Poisson integral.
reviewed by: Na Jisheng
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1106590
zbMATH Keywords: Hua operator
zbMATH Keywords: bounded homogeneous domain
zbMATH Keywords: Poisson kernel
zbMATH Keywords: Bergman-Shilov boundary
zbMATH Keywords: differential operators
author: Ewa Damek
author: Andrzej Hulanicki
author: Richard Penney
MaRDI profile type: Publication
full work available at URL: jfan.1997.3117
OpenAlex ID: W2094217795
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DOI: 10.1006/JFAN.1997.3117
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description: scientific article; zbMATH DE number 1106590