On Bergmann kernel functions in quaternion analysis
The authors deal with the problem of generalizing the Bergman kernel function to four dimensions with the help of quaternion analysis. A necessary tool is to generalize the notion of holomorphic functions in quaternion analysis to hyperholomorphic ones. That means the normal basic elements in the classical operator \((D)\) are substituted by an orthonormal set \((\psi = \{\psi^{0}, \psi^{1}, \psi^{2}, \psi^{3}\})\) of quaternions, which leads to operators \((^{\psi}D)\) and \((D^{\psi})\), whose kernels are the right and left hyperholomorphic functions. \((K_{\psi}:= ^{\overline{\psi}} D(\Theta))\) is the fundamental solution of \((^{\psi}D)\), where \((\Theta)\) is the fundamental solution of the Laplace operator in \((R^4)\). Then the quaternion Bergman kernel can be defined using Green's function \((g(x,\xi))\) of the domain \((\Omega)\) by \[ _{\psi}B:= D^{\psi}_{\xi}\:^{\overline{\psi}} D_{x}[g] = ^{\overline{\psi}} D_{x}\:^{\psi}D_{\xi}[g]. \] For this kernel holomorphicity and integral representations -- using \((K_{\psi})\) -- are shown. Here it is necessary to use another orthonormal set \((\phi)\) with \((\sum{\psi^{k} \phi^{k}} = 0)\) and the related operators. A theorem about the necessary function spaces for the Bergman operator follows.
scientific article; zbMATH DE number 1309946