Integral Theorems for Functions and Differential Forms in C(m)

This Research Note provides a deep link between the active fields of Several Complex Variables Theory and Clifford Analysis. The authors draw on their extensive research in the field to establish integral theorems for Several Complex Variables Theory, which forms a bridge between it and Function Theory in one complex variable. There has recently been a strong period of activity in the field, and Integral Theorems for Functions and Differential Forms in Cm provides a good "summing up" by presenting a complete and up-to-date survey, as well as new results and methods.

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.

IntroductionDifferential FormsDifferential Forms with Coefficients in 2 x 2 MatricesHyperholomorphic Functions and Differential Forms in CmHyperholomorphic Cauchy's Integral TheoremsHyperholomorphic Morera's TheoremsHyperholomorphic Cauchy's Intergral RepresentationsHyperholomorphic D-ProblemComplex Hodge-Dolbeault System, the ?-Problem and the Koppelman FormulaRelation Between Hyperholomorphic Theory and Clifford Analysis