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

by Rocha-Chavez; Reynaldo

• ISBN13: 9781584882466
• ISBN10: 1584882468
• Edition: 1st
• Format: Nonspecific Binding
• Copyright: 2001-08-03
• Publisher: Chapman & Hall/

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.

Table of Contents

Introduction	1	(8)
Differential forms	9	(10)
Usual notation	9	(1)
Complex differential forms	10	(1)
Operations on complex differential forms	11	(3)
Integration with respect to a part of variables	14	(1)
The differential form |F|	15	(1)
More spaces of differential forms	16	(3)
Differential forms with coefficients in 2 x 2-matrices	19	(42)
Classes Gp(Ω), Gp(Omega;)	19	(1)
Matrix-valued differential forms	19	(2)
The hyperholomorphic Cauchy-Riemann operators on B1 and B1	21	(3)
Formula for d (F * G)	24	(1)
Differential matrix forms of the unit normal	24	(4)
Formula for d (F * σ * G)	28	(4)
Exterior differentiation and the hyperholomorphic Cauchy-Riemann operators	32	(1)
Stokes formula compatible with the hyperholomorphic Cauchy-Riemann operators	32	(2)
The Cauchy kernel for the null-sets of the hyperholomorphic Cauchy-Riemann operators	34	(1)
Structure of the product KD * σ	35	(4)
Borel-Pompeiu (or Cauchy-Green) formula for smooth differential matrix-forms	39	(22)
Structure of the Borel-Pompeiu formula	44	(3)
The case m = 1	47	(1)
The case m = 2	48	(3)
Notations for some integrals in C2	51	(3)
Formulas of the Borel-Pompeiu type in C2	54	(1)
Complements to the Borel-Pompeiu-type formulas in C2	55	(1)
The case m > 2	55	(2)
Notations for some integrals in Cm	57	(1)
Formulas of the Borel-Pompeiu type in Cm	58	(1)
Complements to the Borel-Pompeiu-type formulas in Cm	58	(3)
Hyperholomorphic functions and differential forms in Cm	61	(14)
Hyperholomorphy in Cm	61	(1)
Hyperholomorphy in one variable	62	(1)
Hyperholomorphy in two variables	63	(2)
Hyperholomorphy in three variables	65	(5)
Hyperholomorphy for any number of variables	70	(3)
Observation about right-hand-side hyperholomorphy	73	(2)
Hyperholomorphic Cauchy's integral theorems	75	(6)
The Cauchy integral theorem for left-hyperholo-morphic matrix-valued differential forms	75	(1)
The Cauchy integral theorem for right-G-hyper-holomorphic m.v.d.f.	75	(1)
Some auxiliary computations	76	(1)
More auxiliary computations	77	(1)
The Cauchy integral theorem for holomorphic functions of several complex variables	78	(1)
The Cauchy integral theorem for antiholomorphic functions of several complex variables	78	(1)
The Cauchy integral theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables	79	(1)
Concluding remarks	80	(1)
Hyperholomorphic Morera's theorems	81	(8)
Left-hyperholomorphic Morera theorem	81	(1)
Version of a right-hyperholomorphic Morera theorem	82	(2)
Morera's theorem for holomorphic functions of several complex variables	84	(1)
Morera's theorem for antiholomorphic functions of several complex variables	85	(1)
The Morera theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables	86	(3)
Hyperholomorphic Cauchy's integral representations	89	(6)
Cauchy's integral representation for left-hyperholomorphic matrix-valued differential forms	89	(1)
A consequence for holomorphic functions	90	(1)
A consequence for antiholomorphic functions	90	(1)
A consequence for holomorphic-like functions	91	(1)
Bochner-Martinelli integral representation for holomorphic functions of several complex variables, and hyperholomorphic function theory	92	(1)
Bochner-Martinelli integral representation for antiholomorphic functions of several complex variables, and hyperholomorphic function theory	92	(1)
Bochner-Martinelli integral representation for functions holomorphic in some variables and antiholomorphic in the rest, and hyperholomorphic function theory	93	(2)
Hyperholomorphic D-problem	95	(22)
Some reasonings from one variable theory	95	(2)
Right inverse operators to the hyperholomorphic Cauchy-Riemann operators	97	(13)
Structure of the formula of Theorem 7.2	99	(2)
Case m = 1	101	(1)
Case m = 2	102	(4)
Case m > 2	106	(3)
Analogs of (7.17)	109	(1)
Commutativity relations for T-type operators	110	(1)
Solution of the hyperholomorphic D-problem	110	(1)
Structure of the general solution of the hyperholomorphic D-problem	111	(3)
D-type problem for the Hodge-Dirac operator	114	(3)
Complex Hodge-Dolbeault system, the ∂-problem and the Koppelman formula	117	(50)
Definition of the complex Hodge-Dolbeault system	117	(1)
Relation with hyperholomorphic case	118	(1)
The Cauchy integral theorem for solutions of degree p for the complex Hodge-Dolbeault system	119	(2)
The Cauchy integral theorem for arbitrary solutions of the complex Hodge-Dolbeault system	121	(1)
Morera's theorem for solutions of degree p for the complex Hodge-Dolbeault system	122	(1)
Morera's theorem for arbitrary solutions of the complex Hodge-Dolbeault system	123	(1)
Solutions of a fixed degree	124	(1)
Arbitrary solutions	124	(1)
Bochner-Martinelli-type integral representation for solutions of degree s of the complex Hodge-Dolbeault system	125	(1)
Bochner-Martinelli-type integral representation for arbitrary solutions of the complex Hodge-Dolbeault system	126	(1)
Solution of the ∂-type problem for the complex Hodge-Dolbeault system in a bounded domain in Cm	127	(1)
Complex ∂-problem and the ∂-type problem for the complex Hodge-Dolbeault system	128	(2)
∂-problem for differential forms	130	(1)
∂-problem for functions of several complex variables	131	(1)
General situation of the Borel-Pompeiu representation	132	(6)
Partial derivatives of integrals with a weak singularity	138	(2)
Theorem 8.15 in C2	140	(1)
Formula (8.14.3) in C2	141	(2)
Integral representation (8.14.3) for a (0, 1)-differential form in C2, in terms of its coefficients	143	(1)
Koppelman's formula in C2	143	(1)
Koppelman's formula in C2 for a (0, 1) - differential form, in terms of its coefficients	144	(1)
Comparison of Propositions 8.18 and 8.20	145	(2)
Koppelman's formula in C2 and hyperholomorphic theory	147	(1)
Definition of H,K	147	(1)
A reformulation of the Borel-Pompeiu formula	148	(3)
Identity (8.14.4) for a d.f. of a fixed degree	151	(2)
About the Koppelman formula	153	(6)
Auxiliary computations	159	(3)
The Koppelman formula for solutions of the complex Hodge-Dolbeault system	162	(1)
Appendix: properties of H,K	163	(4)
Hyperholomorphic theory and Clifford analysis	167	(28)
One way to introduce a complex Clifford algebra	167	(3)
Classical definition of a complex Clifford algebra	168	(2)
Some differential operators on Wm-valued functions	170	(3)
Factorization of the Laplace operator	171	(2)
Relation of the operators ∂ and ∂ with the Dirac operator of Clifford analysis	173	(1)
Matrix algebra with entries from Wm	174	(1)
The matrix Dirac operators	175	(2)
Factorization of the Laplace operator on Wm- valued functions	176	(1)
The fundamental solution of the matrix Dirac operators	177	(2)
Borel-Pompeiu formulas for Wm-valued functions	179	(1)
Monogenic Wm-valued functions	180	(1)
Cauchy's integral representations for monogenic Wm-valued functions	180	(1)
Clifford algebra with the Witt basis and differential forms	181	(2)
Relation between the two matrix algebras	183	(6)
Operators D and D	185	(4)
Cauchy's integral representation for left-hyperholomorphic matrix-valued differential forms	189	(1)
Hyperholomorphic theory and Clifford analysis	190	(5)
Bibliography	195	(6)
Index	201