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

Reynaldo Rocha-Chavez, Michael Shapiro, Frank Sommen

August 3, 2001 by Chapman and Hall/CRC
Reference - 216 Pages
ISBN 9781584882466 - CAT# C2468
Series: Chapman & Hall/CRC Research Notes in Mathematics Series

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  • Offers a completely new approach to studying classical problems in several complex variables
  • Gives the solution of the ?-problem for functions and differential forms in a domain of an arbitrary shape
  • Constructs a universal Cauchy kernel in that clarifies the nature of the Koppelman formula
  • Extends the concept of holomorphic functions to the setting of differential forms
  • Provides a solid starting point for exploring other difficult problems in several complex variables
  • Summary

    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.