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| eBook ISBN: | 978-1-4704-4470-9 |
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Book DetailsTranslations of Mathematical MonographsVolume: 56; 1983; 165 ppMSC: Primary 32
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.
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Table of Contents
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Chapters
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Introduction
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Integral representation of exterior differential forms and its immediate consequences
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Forms orthogonal to holomorphic forms
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Properties of $\overline \{\partial \}$-closed forms of type $(p,n-1) $
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Some applications
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Brief historical survey and open problems for chapters I–IV
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Integral properties characterizing $\overline \{\partial \}$-closed differential forms and holomorphic functions
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Forms orthogonal to holomorphic forms. Weighted formula for solving the $\overline \{\partial \}$-equation, and applications
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Representation and multiplication of distributions in higher dimensions
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Reviews
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“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
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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.
-
Chapters
-
Introduction
-
Integral representation of exterior differential forms and its immediate consequences
-
Forms orthogonal to holomorphic forms
-
Properties of $\overline \{\partial \}$-closed forms of type $(p,n-1) $
-
Some applications
-
Brief historical survey and open problems for chapters I–IV
-
Integral properties characterizing $\overline \{\partial \}$-closed differential forms and holomorphic functions
-
Forms orthogonal to holomorphic forms. Weighted formula for solving the $\overline \{\partial \}$-equation, and applications
-
Representation and multiplication of distributions in higher dimensions
-
“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
