Holomorphic Functions and Integral Representations in Several Complex Variables

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Springer Science & Business Media, Jun 26, 1998 - Mathematics - 392 pages
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.
 

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Contents

III
1
IV
18
V
31
VI
40
VII
42
VIII
43
IX
48
X
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XXX
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XXXI
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XXXII
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XXXIII
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XXXIV
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XXXV
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XXXVI
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XXXVII
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XI
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XIII
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XIV
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XV
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XVI
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XVII
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XVIII
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XIX
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XX
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XXI
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XXIII
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XXIV
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XXV
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XXVI
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XXVII
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XXVIII
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XXIX
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XXXVIII
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XXXIX
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XL
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XLI
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XLII
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XLIII
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XLIV
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XLV
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XLVI
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XLVII
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XLVIII
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XLIX
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L
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LI
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LII
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LIII
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LIV
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Page 366 - Solutions with bounds for the equations of H. Lewy and PoincareLelong. Construction of functions of Nevanlinna class with given zeroes in a strongly pseudoconvex domain. Dokl. Akad. Nauk SSSR 224(1975), 771-774; Engl.
Page 367 - Soc. 73(1967), 373-377. 2. The Cauchy integral for differential forms. Bull. Amer. Math. Soc. 73(1967), 554-556. [Krai Krantz, SG: 1. Optimal Lipschitz and L' regularity for the equation flu = /on strongly pseudoconvex domains.
Page 365 - Grauert, H.: On Levi's problem and the embedding of real-analytic manifolds. Ann. of Math.
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