Integral Theorems for Functions and Differential Forms in C(m) (Chapman & Hall/CRC research notes in mathematics series ; 428)

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. c2468fm.pdf 1 Integral theorems for functions and differential forms in Cm 1 Contents 3 Introduction 9 C2468ch01.pdf 17 Integral theorems for functions and differential forms in Cm 17 Table of Contents -1 Chapter 1: Differential forms 17 1.1 Usual notation 17 1.2 Complex differential forms 18 1.3 Operations on complex differential forms 19 1.4 Integration with respect to a part of variables 22 1.5 The differential form |F| 23 1.6 More spaces of differential forms 24 C2468ch02.pdf 26 Integral theorems for functions and differential forms in Cm 26 Table of Contents -1 Chapter 2: Differential forms with coefficients in 2 x 2-matrices 26 2.1 Classes Gp ( ), Gp ( ) 26 2.2 Matrix-valued differential forms 26 2.3 The hyperholomorphic Cauchy-Riemann operators on G1 and G1 28 2.4 Formula for d (F ^* G) 31 2.5 Differential matrix forms of the unit normal 31 2.6 Formula for d ( F ^* o ^* G) 35 2.7 Exterior differentiation and the hyperholomorphic Cauchy-Riemann operators 39 2.8 Stokes formula compatible with the hyperholo morphic Cauchy-Riemann operators 39 2.9 The Cauchy kernel for the null-sets of the hyperholomorphic Cauchy-Riemann operators 41 2.10 Structure of the product KD ^* o 42 2.11 Borel-Pompeiu (or Cauchy-Green) formula for smooth differential matrix-forms 46 2.11.1 Structure of the Borel-Pompeiu formula 51 2.11.2 The case m=1 54 2.11.3 The case m=2 55 2.11.4 Notations for some integrals in C 2 58 2.11.5 Formulas of the Borel-Pompeiu type in C 2 61 2.11.6 Complements to the Borel-Pompeiu-type formulas in C 2 62 2.11.7 The case m >2 62 2.11.8 Notations for some integrals in C m 64 2.11.9 Formulas of the Borel-Pompeiu type in C m 65 2.11.10 Complements to the Borel-Pompeiu-type formulas in C m 65 C2468ch03.pdf 67 Integral theorems for functions and differential forms in Cm 67 Table of Contents -1 Chapter 3: Hyperholomorphic functions and differential forms in Cm 67 3.1 Hyperholomorphy in Cm 67 3.2 Hyperholomorphy in one variable 68 3.3 Hyperholomorphy in two variables 69 3.4 Hyperholomorphy in three variables 71 3.5 Hyperholomorphy for any number of variables 76 3.6 Observation about right-hand-side hyperholomorphy 79 C2468ch04.pdf 81 Integral theorems for functions and differential forms in Cm 81 Table of Contents -1 Chapter 4: Hyperholomorphic Cauchy’s integral theorems 81 4.1 The Cauchy integral theorem for left-hyperholomorphicmatrix- valued differential forms 81 4.2 The Cauchy integral theorem for right-G-hyperholomorphic m.v.d.f. 81 4.3 Some auxiliary computations 82 4.4 More auxiliary computations 83 4.5 The Cauchy integral theorem for holomorphic functions of several complex variables 84 4.6 The Cauchy integral theoremfor antiholomorphic functions of several complex variables 84 4.7 The Cauchy integral theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables 85 4.8 Concluding remarks 86 C2468ch05.pdf 87 Integral theorems for functions and differential forms in Cm 87 Table of Contents -1 Chapter 5: Hyperholomorphic Morera’s theorems 87 5.1 Left-hyperholomorphic Morera theorem 87 5.2 Version of a right-hyperholomorphic Morera theorem 88 5.3 Morera’s theorem for holomorphic functions of several complex variables 90 5.4 Morera’s theorem for antiholomorphic functions of several complex variables 91 5.5 The Morera theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables 92 C2468ch06.pdf 94 Integral theorems for functions and differential forms in Cm 94 Table of Contents -1 Chapter 6: Hyperholomorphic Cauchy’s integral representations 94 6.1 Cauchy’s integral representation for lefthyperholomorphicmatrix- valued differential forms 94 6.2 A consequence for holomorphic functions 95 6.3 Aconsequence for antiholomorphic functions 95 6.4 Aconsequence for holomorphic-like functions 96 6.5 Bochner-Martinelli integral representation for holomorphic functions of several complex variables, and hyperholomorphic function theory 97 6.6 Bochner-Martinelli integral representation for antiholomorphic functions of several complex variables, and hyperholomorphic function theory 97 6.7 Bochner-Martinelli integral representation for functions holomorphic in some variables and antiholomorphic in the rest, and hyperholomorphic function theory 98 C2468ch07.pdf 100 Integral theorems for functions and differential forms in Cm 100 Table of Contents -1 Chapter 7: Hyperholomorphic D-problem 100 7.1 Some reasonings from one variable theory 100 7.2 Right inverse operators to the hyperholomorphic Cauchy-Riemann operators 102 7.2.1 Structure of the formula of Theorem 7.2 104 7.2.2 Case m = 1 106 7.2.3 Case m = 2 107 7.2.4 Case m > 2 111 7.2.5 Analogs of (7.1.7) 114 7.2.6 Commutativity relations for T-type operators 115 7.3 Solution of the hyperholomorphic D-problem 115 7.4 Structure of the general solution of the hyperholomorphic D-problem 116 7.5 D-type problem for the Hodge-Dirac operator 119 C2468ch08.pdf 121 Integral theorems for functions and differential forms in Cm 121 Table of Contents -1 Chapter 8: Complex Hodge-Dolbeault system, the @-problem and the Koppelman formula 121 8.1 Definition of the complex Hodge-Dolbeault system 121 8.2 Relation with hyperholomorphic case 122 8.3 The Cauchy integral theorem for solutions of degree p for the complex Hodge-Dolbeault system 123 8.4 The Cauchy integral theorem for arbitrary solutions of the complex Hodge-Dolbeault system 125 8.5 Morera’s theorem for solutions of degree p for the complex Hodge-Dolbeault system 126 8.6 Morera’s theorem for arbitrary solutions of the complex Hodge-Dolbeault system 127 8.7 Solutions of a fixed degree 128 8.8 Arbitrary solutions 128 8.9 Bochner-Martinelli-type integral representation for solutions of degree s of the complex Hodge-Dolbeault system 129 8.10 Bochner-Martinelli-type integral representation for arbitrary solutions of the complex Hodge-Dolbeault system 130 8.11 Solution of the a-type problem for the complex Hodge-Dolbeault system in a bounded domain in Cm 131 8.12 Complex a-problem and the a-type problem for the complex Hodge-Dolbeault system 132 8.13 a-problem for differential forms 134 8.13.1 a-problem for functions of several complex variables 135 8.14 General situation of the Borel-Pompeiu representation 136 8.15 Partial derivatives of integrals with a weak singularity 142 8.16 Theorem 8.15 in C2 144 8.17 Formula (8.14.3) in C2 145 8.18 Integral representation (8.14.3) for a (0,1)-differential form in C2, in terms of its coefficients 147 8.19 Koppelman’s formula in C2 147 8.20 Koppelman’s formula in C2 for a (0,1) - differential form, in terms of its coefficients 148 8.21 Comparison of Propositions 8.18 and 8.20 149 8.22 Koppelman’s formula in C2 and hyperholomorphic theory 151 8.23 Definition of pH;K 151 8.24 A reformulation of the Borel-Pompeiu formula 152 8.25 Identity (8.14.4) for a d.f. of a fixed degree 155 8.26 About the Koppelman formula 157 8.27 Auxiliary computations 163 8.28 TheKoppelman formula for solutions of the complex Hodge-Dolbeault system 166 8.29 Appendix: properties of pH;K 167 C2468ch09.pdf 170 Integral theorems for functions and differential forms in Cm 170 Table of Contents -1 Chapter 9: Hyperholomorphic theory and Clifford analysis 170 9.1 One way to introduce a complex Clifford algebra 170 9.1.1 Classical definition of a complex Clifford algebra 171 9.2 Some differential operators on Wm-valued functions 173 9.2.1 Factorization of the Laplace operator 174 9.3 Relation of the operators @ and @ with the Dirac operator of Clifford analysis 176 9.4 Matrix algebra with entries from Wm 177 9.5 The matrix Dirac operators 178 9.5.1 Factorization of the Laplace operator on Wm-valued functions 179 9.6 The fundamental solution of the matrix Dirac operators 180 9.7 Borel-Pompeiu formulas for Wm-valued functions 182 9.8 Monogenic Wm-valued functions 183 9.9 Cauchy’s integral representations for monogenic Wm-valued functions 183 9.10 Clifford algebra with the Witt basis and differential forms 184 9.11 Relation between the two matrix algebras 186 9.11.1 Operators D and D 188 9.12 Cauchy’s integral representation for left-hyperholomorphic matrix-valued differential forms 192 9.13 Hyperholomorphic theory and Clifford analysis 193 C2468bib.pdf 197 Integral theorems for functions and differential forms in Cm 197 Table of Contents -1 Bibliography 197 Integral theorems for functions and differential forms in Cm......Page 1 Contents......Page 3 Introduction......Page 9 1.1 Usual notation......Page 17 Table of Contents......Page 0 1.2 Complex differential forms......Page 18 1.3 Operations on complex differential forms......Page 19 1.4 Integration with respect to a part of variables......Page 22 1.5 The differential form |F|......Page 23 1.6 More spaces of differential forms......Page 24 2.2 Matrix-valued differential forms......Page 26 2.3 The hyperholomorphic Cauchy-Riemann operators on G1 and G1......Page 28 2.5 Differential matrix forms of the unit normal......Page 31 2.6 Formula for d ( F ^* o ^* G)......Page 35 2.8 Stokes formula compatible with the hyperholo morphic Cauchy-Riemann operators......Page 39 2.9 The Cauchy kernel for the null-sets of the hyperholomorphic Cauchy-Riemann operators......Page 41 2.10 Structure of the product KD ^* o......Page 42 2.11 Borel-Pompeiu (or Cauchy-Green) formula for smooth differential matrix-forms......Page 46 2.11.1 Structure of the Borel-Pompeiu formula......Page 51 2.11.2 The case m=1......Page 54 2.11.3 The case m=2......Page 55 2.11.4 Notations for some integrals in C 2......Page 58 2.11.5 Formulas of the Borel-Pompeiu type in C 2......Page 61 2.11.7 The case m >2......Page 62 2.11.8 Notations for some integrals in C m......Page 64 2.11.10 Complements to the Borel-Pompeiu-type formulas in C m......Page 65 3.1 Hyperholomorphy in Cm......Page 67 3.2 Hyperholomorphy in one variable......Page 68 3.3 Hyperholomorphy in two variables......Page 69 3.4 Hyperholomorphy in three variables......Page 71 3.5 Hyperholomorphy for any number of variables......Page 76 3.6 Observation about right-hand-side hyperholomorphy......Page 79 4.2 The Cauchy integral theorem for right-G-hyperholomorphic m.v.d.f.......Page 81 4.3 Some auxiliary computations......Page 82 4.4 More auxiliary computations......Page 83 4.6 The Cauchy integral theoremfor antiholomorphic functions of several complex variables......Page 84 4.7 The Cauchy integral theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables......Page 85 4.8 Concluding remarks......Page 86 5.1 Left-hyperholomorphic Morera theorem......Page 87 5.2 Version of a right-hyperholomorphic Morera theorem......Page 88 5.3 Morera’s theorem for holomorphic functions of several complex variables......Page 90 5.4 Morera’s theorem for antiholomorphic functions of several complex variables......Page 91 5.5 The Morera theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables......Page 92 6.1 Cauchy’s integral representation for lefthyperholomorphicmatrix- valued differential forms......Page 94 6.3 Aconsequence for antiholomorphic functions......Page 95 6.4 Aconsequence for holomorphic-like functions......Page 96 6.6 Bochner-Martinelli integral representation for antiholomorphic functions of several complex variables, and hyperholomorphic function theory......Page 97 6.7 Bochner-Martinelli integral representation for functions holomorphic in some variables and antiholomorphic in the rest, and hyperholomorphic function theory......Page 98 7.1 Some reasonings from one variable theory......Page 100 7.2 Right inverse operators to the hyperholomorphic Cauchy-Riemann operators......Page 102 7.2.1 Structure of the formula of Theorem 7.2......Page 104 7.2.2 Case m = 1......Page 106 7.2.3 Case m = 2......Page 107 7.2.4 Case m > 2......Page 111 7.2.5 Analogs of (7.1.7)......Page 114 7.3 Solution of the hyperholomorphic D-problem......Page 115 7.4 Structure of the general solution of the hyperholomorphic D-problem......Page 116 7.5 D-type problem for the Hodge-Dirac operator......Page 119 8.1 Definition of the complex Hodge-Dolbeault system......Page 121 8.2 Relation with hyperholomorphic case......Page 122 8.3 The Cauchy integral theorem for solutions of degree p for the complex Hodge-Dolbeault system......Page 123 8.4 The Cauchy integral theorem for arbitrary solutions of the complex Hodge-Dolbeault system......Page 125 8.5 Morera’s theorem for solutions of degree p for the complex Hodge-Dolbeault system......Page 126 8.6 Morera’s theorem for arbitrary solutions of the complex Hodge-Dolbeault system......Page 127 8.8 Arbitrary solutions......Page 128 8.9 Bochner-Martinelli-type integral representation for solutions of degree s of the complex Hodge-Dolbeault system......Page 129 8.10 Bochner-Martinelli-type integral representation for arbitrary solutions of the complex Hodge-Dolbeault system......Page 130 8.11 Solution of the a-type problem for the complex Hodge-Dolbeault system in a bounded domain in Cm......Page 131 8.12 Complex a-problem and the a-type problem for the complex Hodge-Dolbeault system......Page 132 8.13 a-problem for differential forms......Page 134 8.13.1 a-problem for functions of several complex variables......Page 135 8.14 General situation of the Borel-Pompeiu representation......Page 136 8.15 Partial derivatives of integrals with a weak singularity......Page 142 8.16 Theorem 8.15 in C2......Page 144 8.17 Formula (8.14.3) in C2......Page 145 8.19 Koppelman’s formula in C2......Page 147 8.20 Koppelman’s formula in C2 for a (0,1) - differential form, in terms of its coefficients......Page 148 8.21 Comparison of Propositions 8.18 and 8.20......Page 149 8.23 Definition of pH;K......Page 151 8.24 A reformulation of the Borel-Pompeiu formula......Page 152 8.25 Identity (8.14.4) for a d.f. of a fixed degree......Page 155 8.26 About the Koppelman formula......Page 157 8.27 Auxiliary computations......Page 163 8.28 TheKoppelman formula for solutions of the complex Hodge-Dolbeault system......Page 166 8.29 Appendix: properties of pH;K......Page 167 9.1 One way to introduce a complex Clifford algebra......Page 170 9.1.1 Classical definition of a complex Clifford algebra......Page 171 9.2 Some differential operators on Wm-valued functions......Page 173 9.2.1 Factorization of the Laplace operator......Page 174 9.3 Relation of the operators @ and @ with the Dirac operator of Clifford analysis......Page 176 9.4 Matrix algebra with entries from Wm......Page 177 9.5 The matrix Dirac operators......Page 178 9.5.1 Factorization of the Laplace operator on Wm-valued functions......Page 179 9.6 The fundamental solution of the matrix Dirac operators......Page 180 9.7 Borel-Pompeiu formulas for Wm-valued functions......Page 182 9.9 Cauchy’s integral representations for monogenic Wm-valued functions......Page 183 9.10 Clifford algebra with the Witt basis and differential forms......Page 184 9.11 Relation between the two matrix algebras......Page 186 9.11.1 Operators D and D......Page 188 9.12 Cauchy’s integral representation for left-hyperholomorphic matrix-valued differential forms......Page 192 9.13 Hyperholomorphic theory and Clifford analysis......Page 193 Bibliography......Page 197