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Author Lieb, Ingo. author The Cauchy-Riemann Complex [electronic resource] : Integral Formulae and Neumann Problem / by Ingo Lieb, Joachim Michel Wiesbaden : Vieweg+Teubner Verlag, 2002 http://dx.doi.org/10.1007/978-3-322-91608-2 X, 362 p. online resource

SUMMARY

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

CONTENT

I The Bochner-Martinelli-Koppelman Formula -- ยง1 Forms on Product Manifolds -- ยง2 The Complex Laplacian -- ยง3 The Fundamental Solution -- ยง4 The Bochner-Martinelli-Koppelman Formula -- ยง5 Types of Kernels and Regularity Properties -- ยง6 Derivatives of the BMK Transform -- ยง7 Applications of the BMK Formula -- ยง8 Cauchy-Riemann Functions -- ยง9 The Bochner-Martinelli Transform for Currents -- ยง10 Regularity Properties of Isotropic Operators -- ยง11 Notes -- II Cauchy-Fantappiรจ Forms -- ยง1 The Koppelman Formula -- ยง2 A Generalisation of the Bochner-Martinelli-Koppelman Formula -- ยง3 Notes -- III Strictly Pseudoconvex Domains in ?n -- ยง1 Strict Pseudoconvexity -- ยง2 The Levi Polynomial and Holomorphic Support Functions -- ยง3 The Basic Homotopy Formula for the Ball -- ยง4 The Basic Integral Representation -- ยง5 Admissible Kernels and Lp-Estimates -- ยง6 Leviโ{128}{153}s Problem and Vanishing of Cohomology -- ยง7 The Henkin-Ramรญrez Formula -- ยง8 Convex Domains of Finite Type -- ยง9 Notes -- IV Strictly Pseudoconvex Manifolds -- ยง1 The Real Laplacian -- ยง2 Generalised Isotropic Operators -- ยง3 The Parametrix -- ยง4 Harmonic Forms and Finiteness Theorems on Compact Manifolds -- ยง5 Basic Integral Representation on Hermitian Manifolds -- ยง6 The Levi Problem on Strictly Pseudoconvex Manifolds -- ยง7 Vanishing of Dolbeault Cohomology Groups -- ยง8 Notes -- V The a-Neumann Problem -- ยง1 Operators on Hilbert Spaces -- ยง2 Hilbert Spaces of Differential Forms -- ยง3 The Generalised Cauchy Condition -- ยง4 The Friedrichs-Hรถrmander Lemma -- ยง5 The Self-adjointness of the Complex Laplacian and Hรถrmanderโ{128}{153}s Density Theorem -- ยง6 The $$\overline \partial$$-Neumann Problem -- ยง7 Notes -- VI Integral Representations for the $$\overline \partial$$-Neumann Problem -- ยง1 The Basic Integral Representation -- ยง2 Cancellation of Singularities -- ยง3 The Bergman Projection -- ยง4 Z-operators -- ยง5 The Structure of the Kernels Tq -- ยง6 Asymptotic Development of the Neumann Operator -- ยง7 Notes -- VII Regularity Properties of Admissible Operators -- ยง1 Spaces of Functions and Differential Forms -- ยง2 Behaviour of Ao-operators on Lp-spaces -- ยง3 Regularity Properties of A1-operators -- ยง4 Regularity Properties of E1?2n-operators -- ยง5 Notes -- VIII Regularity of the $$\overline \partial$$-Neumann Problem and Applications -- ยง1 The Basic Hรถlder Estimate -- ยง2 The Basic Sobolev Estimate -- ยง3 The Basic Ck-Estimate -- ยง4 Dolbeault Cohomology Spaces -- ยง5 Regularity of the Bergman Projection -- ยง6 The L1-theory of the $$\overline \partial$$-Neumann Problem -- ยง7 Gleasonโ{128}{153}s Problem for Ck-functions -- ยง8 Stability of Estimates for the $$\overline \partial$$-Neumann Problem -- ยง9 Mergelyanโ{128}{153}s Approximation Theorem with Ck Boundary Values on Hermitian Manifolds -- ยง10 Notes -- Notations

Mathematics Mathematical analysis Analysis (Mathematics) Partial differential equations Mathematics Partial Differential Equations Analysis

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