Holomorphic Functions and Integral Representations in Several Complex Variables
Holomorphic Functions and Integral Representations in Several Complex Variables
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DOI：
10.1007/978-1-4757-1918-5
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发表时间：
1998-06
期刊：
影响因子：
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通讯作者：
R. Range
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文献类型：
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作者：
R. Range
本书的主题是多变量的复分析。本文从标准局部结果开始，深入讨论与全纯函数在多个变量中的显着可拓性质相关的“复凸性”的各种基本概念。然后，它继续全面介绍积分表示，并以 C 中全纯域和严格伪凸域上的实质性全局结果的完整证明作为结论，包括例如 C. Fefferman 著名的映射定理。本书最重要的新特点是系统地包含了过去 20 年中围绕积分表示和柯西-黎曼方程估计的许多发展。特别是，积分表示是与许多早期涉及交换代数和层理论和/或偏微分方程的方法的书籍相比，它是用于发展全局理论的主要工具，我认为这种方法具有几个优点：（1）它在一个复杂变量中使用了分析师熟悉的多变量版本的工具，因此有助于弥合一个和多个变量的复杂分析之间经常存在的差距；（2）它可以直接得出深入的全局结果，而无需引入大量新机制；具体的积分表示有助于估计，因此为早期方法无法访问的应用程序打开了大门。
The subject of this book is Complex Analysis in Several Variables. This text begins at an elementary level with standard local results, followed by a thorough discussion of the various fundamental concepts of" complex convexity" related to the remarkable extension properties of holomorphic functions in more than one variable. It then continues with a comprehensive introduction to integral representations, and concludes with complete proofs of substantial global results on domains of holomorphy and on strictly pseudoconvex domains inC", including, for example, C. Fefferman's famous Mapping Theorem. The most important new feature of this book is the systematic inclusion of many of the developments of the last 20 years which centered around integral representations and estimates for the Cauchy-Riemann equations. In particu lar, integral representations are the principal tool used to develop the global theory, in contrast to many earlier books on the subject which involved methods from commutative algebra and sheaf theory, and/or partial differ ential equations. I believe that this approach offers several advantages:(1) it uses the several variable version of tools familiar to the analyst in one complex variable, and therefore helps to bridge the often perceived gap between com plex analysis in one and in several variables;(2) it leads quite directly to deep global results without introducing a lot of new machinery; and (3) concrete integral representations lend themselves to estimations, therefore opening the door to applications not accessible by the earlier methods.