On the Bergmann kernel function in hyperholomorphic analysis
type: Item
instance of: scholarly article
title: On the Bergmann kernel function in hyperholomorphic analysis
zbMATH Open document ID: 0877.30025
DOI: 10.1023/A:1017916828448
published in: Acta Applicandae Mathematicae
publication date: 1997-08-17T00:00:00Z
review text: The hyperholomorphic Bergman kernel function \({}_\psi{\mathcal B}\) for a domain \(\Omega\subset\mathbf R ^4\) is introduced as a special quaternionic ``derivative'' of the Green function for \(\Omega\). It is shown that \({}_\psi{\mathcal B}\) is hyperholomorphic, Hermitian symmetric and reproduces hyperholomorphic functions. An integral representation of \({}_\psi{\mathcal B} \) is further obtained which expresses \({}_\psi{\mathcal B}\) as a sum of a function continuous on \(\overline\Omega\times\overline\Omega\) and an integral over \(\partial\Omega\) of a differential form on \(\mathbf R ^4\) which does not depend on \(\Omega\). As a consequence, the equality \({}^\psi B =I-\bar S\cdot S + C\) is proved for the Bergman projector \({}^\psi B \), where \(\bar S,S\) are certain singular integral operators and \(C\) is a compact operator. In more detail, let \(\psi=\{\psi^k\}_{k=0}^3\) be an orthonormal basis for the quaternions \(\mathbf H(\simeq\mathbf R ^4)\) over \({\mathbf R}\) and \(g\) the Green function for the domain \(\Omega\). A function \(f:\Omega\to\mathbf H\) is called (left-) \(\psi\)-hyperholomorphic if \({}^\psi D [f]:=\sum_{k=0}^3 \psi^k \frac {\partial f}{\partial x_k}=0\), and the hyperholomorphic Bergman kernel is defined by \({}_\psi{\mathcal B}(x,\xi)={}^{\bar\psi} D_x \cdot{}^\psi D _\xi[g](x,\xi)\). Let further \({}^\psi B [f](x)=\int_\Omega {}_\psi{\mathcal B} (x,\xi)f(\xi) d\xi\) (the Bergman operator), \({}^\psi T [f](x)=-\int_\Omega K_\psi(\tau-x) f(\tau) d\tau\), where \(K_\psi(x)=(2\pi^2|x|^4)^{-1} \sum_{k=0}^3 \bar\psi^k x_k\) is the fundamental solution of the operator \({}^\psi D \), and for \(\phi=\{\phi^k\}_{k=0}^3\) another orthonormal basis of \(\mathbf H \) such that \(\sum_{k=0}^3 \phi^k \cdot \psi^k =0\) introduce the operator \({}^{\phi,\psi}S [f](x)=2\pi^{-2}\int_\Omega (\sum_{k=0}^3 (\tau_k-x_k)\bar\psi^k) (\sum_{k=0}^3 (\tau_k-x_k)\bar\phi^k) |\tau-x|^{-6}f(\tau) d\tau\). Then \({}^\psi B \), \({}^\psi T \) and \({}^{\p…
reviewed by: Miroslav Engliš
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 986858
zbMATH Keywords: Bergman kernel function
zbMATH Keywords: hyperholomorphic functions
zbMATH Keywords: integral representation
author: M. Shapiro
author: Nikolai Vasilevski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 986858