**Translations of Mathematical Monographs**

1983;
165 pp;
Softcover

MSC: Primary 32;

**Print ISBN: 978-0-8218-1348-5
Product Code: MMONO/56.S**

List Price: $78.00

AMS Member Price: $62.40

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4470-9
Product Code: MMONO/56.E**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

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

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*L. A. Aizenberg; Sh. A. Dautov*

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.

#### Reviews & Endorsements

“This book was originally published in Russian in 1975 as a report on the research of the authors and some other Soviet mathematicians from the early 1970s. When the English translation was proposed in 1981, the authors provided some additional chapters which discuss more recent results and which now make up about forty percent of the book. Written clearly but in the no-nonsense style of a research monograph, this book provides a rewarding look at some of the recent work of the Soviet school of complex analysis in several variables for those with some previous experience in the subject.”

-- G. B. Folland, Bulletin of the AMS

#### Table of Contents

# Table of Contents

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

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the American Edition vii8
- Preface ix10
- Introduction 112
- Integral representation of exterior differential forms and its immediate consequences 516
- Forms orthogonal to holomorphic forms 3748
- Properties of \overline{∂}-closed forms of type (𝑝,𝑛-1) 5364
- Some applications 6778
- Brief historical survey and open problems for chapters I–IV 8596
- Integral properties characterizing \overline{∂}-closed differential forms and holomorphic functions 89100
- Forms orthogonal to holomorphic forms. Weighted formula for solving the \overline{∂}-equation, and applications 125136
- Representation and multiplication of distributions in higher dimensions 137148
- Supplement to the brief historical survey 151162
- Bibliography 153164
- Subject index 163174
- Index of symbols 165176
- Back Cover Back Cover1178