The Cauchy-Riemann Complex

Integrale Darstellungen von Funktionen und Differentialformen sind ein wichtiges Werkzeug, um quantitative Resultate für holomorphe Funktionen auf komplexen Mannigfaltigkeiten zu entwickeln. Dieses Buch ist eine Monographie zu einem Spezialgebiet der komplexen Analysis.

Advanced Methods of Complex Analysis Applied to Classical and New Problems

Autorentext
Prof. Dr. Ingo Lieb ist Professor für Mathematik an der Universität Bonn. Er ist Autor der beiden Bücher "Funktionentheorie" und "Ausgewählte Kapitel aus der Funktionentheorie" in der Reihe vieweg studium/Aufbaukurs Mathematik.
Prof. Dr. Joachim Michel ist Professor für Mathematik am "Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville" (L.M.P.A.) in Calais, Frankreich.

Klappentext
The method of integral representations is developed in order to establish 1. classical fundamental results of complex analysis both elementary and advanced, 2. subtle existence and regularity theorems for the Cauchy-Riemann equations on complex manifolds. These results are then applied to important function theoretic questions. The book can be used for advanced courses and seminars at the graduate level; it contains to a large extent material which has not yet been covered in text books.

Inhalt
I The Bochner-Martinelli-Koppelman Formula.- §1 Forms on Product Manifolds.- §2 The Complex Laplacian.- §3 The Fundamental Solution.- §4 The Bochner-Martinelli-Koppelman Formula.- §5 Types of Kernels and Regularity Properties.- §6 Derivatives of the BMK Transform.- §7 Applications of the BMK Formula.- §8 Cauchy-Riemann Functions.- §9 The Bochner-Martinelli Transform for Currents.- §10 Regularity Properties of Isotropic Operators.- §11 Notes.- II Cauchy-Fantappiè Forms.- §1 The Koppelman Formula.- §2 A Generalisation of the Bochner-Martinelli-Koppelman Formula.- §3 Notes.- III Strictly Pseudoconvex Domains in ?n.- §1 Strict Pseudoconvexity.- §2 The Levi Polynomial and Holomorphic Support Functions.- §3 The Basic Homotopy Formula for the Ball.- §4 The Basic Integral Representation.- §5 Admissible Kernels and Lp-Estimates.- §6 Levi's Problem and Vanishing of Cohomology.- §7 The Henkin-Ramírez Formula.- §8 Convex Domains of Finite Type.- §9 Notes.- IV Strictly Pseudoconvex Manifolds.- §1 The Real Laplacian.- §2 Generalised Isotropic Operators.- §3 The Parametrix.- §4 Harmonic Forms and Finiteness Theorems on Compact Manifolds.- §5 Basic Integral Representation on Hermitian Manifolds.- §6 The Levi Problem on Strictly Pseudoconvex Manifolds.- §7 Vanishing of Dolbeault Cohomology Groups.- §8 Notes.- V The a-Neumann Problem.- §1 Operators on Hilbert Spaces.- §2 Hilbert Spaces of Differential Forms.- §3 The Generalised Cauchy Condition.- §4 The Friedrichs-Hörmander Lemma.- §5 The Self-adjointness of the Complex Laplacian and Hörmander's Density Theorem.- §6 The $$ \overline \partial $$-Neumann Problem.- §7 Notes.- VI Integral Representations for the $$ \overline \partial $$-Neumann Problem.- §1 The Basic IntegralRepresentation.- §2 Cancellation of Singularities.- §3 The Bergman Projection.- §4 Z-operators.- §5 The Structure of the Kernels Tq.- §6 Asymptotic Development of the Neumann Operator.- §7 Notes.- VII Regularity Properties of Admissible Operators.- §1 Spaces of Functions and Differential Forms.- §2 Behaviour of Ao-operators on Lp-spaces.- §3 Regularity Properties of A1-operators.- §4 Regularity Properties of E1?2n-operators.- §5 Notes.- VIII Regularity of the $$ \overline \partial $$-Neumann Problem and Applications.- §1 The Basic Hölder Estimate.- §2 The Basic Sobolev Estimate.- §3 The Basic Ck-Estimate.- §4 Dolbeault Cohomology Spaces.- §5 Regularity of the Bergman Projection.- §6 The L1-theory of the $$ \overline \partial $$-Neumann Problem.- §7 Gleason's Problem for Ck-functions.- §8 Stability of Estimates for the $$ \overline \partial $$-Neumann Problem.- §9 Mergelyan's Approximation Theorem with Ck Boundary Values on Hermitian Manifolds.- §10 Notes.- Notations.