The Cauchy Transform, Potential Theory and Conformal Mapping

Steven R. Bell, PhD, professor, Department of Mathematics, Purdue University, West Lafayette, Indiana, USA, and Fellow of the AMS

Introduction. The Improved Cauchy Integral Formula. The Cauchy Transform. The Hardy Space, the Szegö Projection, and the Kerzman-Stein Formula. The Kerzman-Stein Operator and Kernel. The Classical Definition of the Hardy Space. The Szegö Kernel Function. The Riemann Mapping Function. A Density Lemma and Consequences. Solution of the Dirichlet Problem in Simply Connected Domains. The Case of Real Analytic Boundary. The Transformation Law for the Szegö Kernel under Conformal Mappings. The Ahlfors Map of a Multiply Connected Domain. The Dirichlet Problem in Multiply Connected Domains. The Bergman Space. Proper Holomorphic Mappings and the Bergman Projection.The Solid Cauchy Transform. The Classical Neumann Problem. Harmonic Measure and the Szegö Kernel. The Neumann Problem in Multiply Connected Domains. The Dirichlet Problem Again. Area Quadrature Domains. Arc Length Quadrature Domains. The Hilbert Transform. The Bergman Kernel and the Szegö Kernel. Pseudo-Local Property of the Cauchy Transform and Consequences. Zeroes of the Szegö Kernel. The Kerzman-Stein Integral Equation. Local Boundary Behavior of Holomorphic Mappings. The Dual Space of A∞(Ω). The Green’s Function and the Bergman Kernel. Zeroes of the Bergman Kernel. Complexity in Complex Analysis. Area Quadrature Domains and the Double. The Cauchy-Kovalevski Theorem for the Cauchy-Riemann Operator.