Differential Forms Orthogonal to Holomorphic Functions or Forms, and Their Properties

✍ Scribed by Lev Abramovich Aĭzenberg, Shamilʹ Abdullovich Dautov

• Publisher: American Mathematical Society
• Year: 1983
• Tongue: English
• Leaves: 173
• Series: Translation of Mathematical Monographs 56
• Edition: 1
• Category: Library

✦ Synopsis

The authors consider the problem of characterizing the exterior differential forms which are orthogonal to holomorphic functions (or forms) in a domain $D\subset {\mathbf C}^n$ with respect to integration over the boundary, and some related questions. They give a detailed account of the derivation of the Bochner-Martinelli-Koppelman integral representation of exterior differential forms, which was obtained in 1967 and has already found many important applications. They study the properties of $\overline \partial$-closed forms of type $(p, n - 1), 0\leq p\leq n - 1$, which turn out to be the duals (with respect to the orthogonality mentioned above) to holomorphic functions (or forms) in several complex variables, and resemble holomorphic functions of one complex variable in their properties.

✦ Table of Contents

Cover......Page 1
TABLE OF CONTENTS......Page 3
PREFACE TO THE AMERICAN EDITION......Page 5
PREFACE......Page 7
INTRODUCTION......Page 9
§1. The Martinelli-Bochner-Koppelman formula......Page 13
§2. Theorems on the saltus of forms......Page 26
§3.. Characterization of the trace of a hol©merphic form on the boundary of a domain......Page 36
§4. Some cases of solvability of the problem......Page 39
§5. Polynomials orthogonal to holomorphic functions......Page 45
§6. Forms orthogonal to holomorphic forms: the case of strictly pseudoconvex domains......Page 53
§7. The general case......Page 55
§8. Converse theorems......Page 57
§9. The theorems of Runge and Morera......Page 61
§10. The f ilst Cousin problem, separation of singularities and domains of existence......Page 65
§11. Theorems of approximation on compact sets......Page 68
§12. Generalization of the theorems of Hartogs and F. and M. Riesz......Page 75
§13. On the general form of integral representations of holomorphic functions......Page 79
§14. Representation of distributions in 6l)' (R^(2n-1)) by a-closed exterior differential forms of type (n, n - 1)......Page 84
§ 15. Multiplication of distributions in '1 , (R^2n -1))......Page 90
BRIEF HISTORICAL SURVEY AND OPEN PROBLEMS FOR CHAPTERS I-IV......Page 93
§16. A characteristic property of a-closed forms and forms of class B......Page 97
§17. Holomorphy of continuous functions representable by the Martinelli-Bochner integral; criteria for the holomorphy of integrals of the Martinelli-Bochner type......Page 102
§18. The traces of holomorphic functions on the Shilov boundary of a circular domain......Page 115
§19. Computation of an integral of Martinell i-Bochner type for the case of the ball......Page 119
§20. Differential boundary conditions for the holomorphy of functions......Page 124
§21. Forms orthogonal to holomorphic forms......Page 133
§22. Generalization of Theorem 8.1......Page 138
§23. Weighted formula for solving the a-problem in strictly convex domains and zeros of functions of the Nevanlinna-Dzhrbashyan class......Page 140
§24. Harmonic representation of distributions......Page 145
§25. Multiplication of distributions and its properties......Page 149
§26. Examples of products of distributions......Page 152
SUPPLEMENT TO THE BRIEF HISTORICAL SURVEY......Page 159
BIBLIOGRAPHY......Page 161
SUBJECT INDEX......Page 171
INDEX OF SYMBOLS......Page 173