Ingo Lieb, Joachim Michel

The Cauchy-Riemann Complex (eBook, PDF)

Integral Formulae and Neumann Problem

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Vieweg+Teubner Verlag
• Seitenzahl: 362
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9783322916082
• Artikelnr.: 53136734

Autorenporträt

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.

Inhaltsangabe

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.