The Cauchy-Riemann Complex Integral Formulae and Neumann Problem

This book presents complex analysis of several variables from the point of view of the Cauchy-Riemann equations and integral representations. A more detailed description of our methods and main results can be found in the introduction. Here we only make some remarks on our aims and on the required background knowledge. Integral representation methods serve a twofold purpose: 1° they yield regularity results not easily obtained by other methods and 2°, along the way, they lead to a fairly simple development of parts of the classical theory of several complex variables. We try to reach both aims. Thus, the first three to four chapters, if complemented by an elementary chapter on holomorphic functions, can be used by a lecturer as an introductory course to com­ plex analysis. They contain standard applications of the Bochner-Martinelli-Koppelman integral representation, a complete presentation of Cauchy-Fantappie forms giving also the numerical constants of the theory, and a direct study of the Cauchy-Riemann com­ plex on strictly pseudoconvex domains leading, among other things, to a rather elementary solution of Levi's problem in complex number space en. Chapter IV carries the theory from domains in en to strictly pseudoconvex subdomains of arbitrary - not necessarily Stein - manifolds. We develop this theory taking as a model classical Hodge theory on compact Riemannian manifolds; the relation between a parametrix for the real Laplacian and the generalised Bochner-Martinelli-Koppelman formula is crucial for the success of the method.

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.

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.