Szegő projections for Hardy spaces of monogenic functions and applications
S. Bell and later Y.-B. Chung studied the Szegő projection for a smooth, bounded and simply connected domain in the complex plane. Connections to further kernel functions (Cauchy kernel, Garabedian kernel, Bergman kernel) are lined out. Bell obtained a representation of the solution of Dirichlet's problem in terms of the Szegő projection. The authors consider a corresponding problem in higher dimensions. For this reason left and right Hardy spaces are introduced. Furthermore, left and right Hilbert transforms and Cauchy integral operators are considered. Defined by Plemelj-Sokhotzkij formulae the Cauchy-transforms (right and left) can be expressed in terms of the Hilbert transforms (left and right). The following orthogonal decomposition of the Hilbert space \(L^2(b\Omega)\), \(\Omega\) Lipschitzean domain in \(\mathbb{R}^n\), is given, where \(v\in H^2_l(b\Omega)^+\) iff \(v=\overline{Hn}\) for some \(H\in H^2_r(b\Omega)\) with \(n\) being the unit vector of the outer normal on \(b\Omega\). The projections onto the corresponding subspaces are called Szegő projections which are applied for an explicit and elegant description of the solution of the Dirichlet problem. Further applications are mentioned.
scientific article; zbMATH DE number 1780381