Offers a completely new approach to studying classical problems in several complex variablesGives the solution of the ?-problem for functions and differential forms in a domain of an arbitrary shapeConstructs a universal Cauchy kernel in that clarifies the nature of the Koppelman formulaExtends the concept of holomorphic functions to the setting of differential formsProvides a solid starting point for exploring other difficult problems in several complex variables
The theory of holomorphic functions of several complex variables emerged from the attempt to generalize the theory in one variable to the multidimensional situation. Research in this area has led to the discovery of many sophisticated facts, structures, ideas, relations, and applications. This deepening of knowledge, however, has also revealed more and more paradoxical differences between the structures of the two theories.
The authors of this Research Note were driven by the quest to construct a theory in several complex variables that has the same structure as the one-variable theory. That is, they sought a reproducing kernel for the whole class that is universal and from same class. Integral Theorems for Functions and Differential Forms in Cm documents their success. Their highly original approach allowed them to obtain new results and refine some well-known results from the classical theory of several complex variables. The 'hyperholomorphic" theory they developed proved to be a kind of direct sum of function theories for two Dirac-type operators of Clifford analysis considered in the same domain.
In addition to new results and methods, this work presents a first-look at a brand new setting, based upon the natural language of differential forms, for complex analysis. Integral Theorems for Functions and Differential Forms in Cm reveals a deep link between the fields of several complex variables theory and Clifford analysis. It will have a strong influence on researchers in both areas, and undoubtedly will change the general viewpoint on the methods and ideas of several complex variables theory.
Table of Contents
Differential Forms with Coefficients in 2 x 2 Matrices
Hyperholomorphic Functions and Differential Forms in Cm
Hyperholomorphic Cauchy's Integral Theorems
Hyperholomorphic Morera's Theorems
Hyperholomorphic Cauchy's Intergral Representations
Complex Hodge-Dolbeault System, the ?-Problem and the Koppelman Formula
Relation Between Hyperholomorphic Theory and Clifford Analysis
"…the book will be interesting to specialists in complex analysis and its applications".
- Mathematical Reviews, 2003a
"This well-written book is a valuable contribution to the broad field of interactions between complex analysis and partial differential equations...Moreover, the book can be used for individual studies, because fundamental concepts and important theorems are explained in detail."
-Mathematical Reviews, Issue 94a