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**Function theory of several complex variables.
Reprint of the 1992 2nd ed. with corrections.**
*(English)*
Zbl 1087.32001

Providence, RI: American Mathematical Society (AMS), AMS Chelsea Publishing (ISBN 0-8218-2724-3/hbk). xvi, 564 p. (2001).

The present book is the reprint of the 1992 2nd edition (Zbl 0471.32008, Zbl 0776.32001). Now one can say with certainty that the book of Steven Krantz has stood the test by the time. Apparently, the time came when on studying complex analysis should be acquainted with bases of function theory not only of one, but also of several complex variables. This book makes available a comprehensive, detailed and carefully organized treatment of the foundations for multidimensional complex analysis. It includes the matter which one looks for in an undergraduate textbook, but it also provides all sorts of topics and exercises which stretch the imagination and show how useful the subject really is. It is to be hoped that many new students will be drawn to function theory of several complex variables by this valuable text.

From the Contents: An introduction to the subject. What is a holomorphic function? Comparison of \(\mathbb C^1\) and \(\mathbb C^n\). The Bochner-Martinelli formula. Applications of Cauchy theory and the \(\bar\partial\) equation. Basic properties of harmonic functions. The Bergman kernel. The Szegö and Poisson-Szegö kernels. Subharmonicity and its applications. Subharmonic functions. Pluriharmonic and plurisubharmonic functions. Power series. Hartogs’s theorem on separate analyticity. Convexity and pseudoconvexity, and analytic discs. Domains of holomorphy. Examples of domains of holomorphy and the edge-of-the-wedge theorem. Hörmander’s solution of the \(\bar\partial\) equation. Unbounded operators on a Hilbert space. An existence and regularity theorems for the \(\bar\partial\) operator. Solution of the Levi problem and other applications of \(\bar\partial\) techniques. An extension problem. Singular functions on strongly pseudoconvex domains. Hefer’s lemma and Henkin’s integral representation. Approximation problems. Cousin problems, cohomology, and sheaves. Dolbeault isomorphism. Algebraic properties of the ring of germs of holomorphic functions. Sheaf of divisors, Chern classes, and the obstruction to solving Cousin II. Coherent analytic sheaves. Applications of the Cartan and Oka theorems: the structure of ideals. Zeros of one holomorphic function. Zero sets for different \(H^p\) spaces are different. Zero sets of \(H^p\) functions do not have a simple mass distribution characterization. Some harmonic analysis. Review of the classical theory of \(H^p\) spaces on the disc. Three propositions about the Poisson kernel. Subharmonicity, harmonic majorants, and boundary values. Pointwise convergence for harmonic functions on domains in \(\mathbb R^N\). Boundary values of holomorphic functions in \(\mathbb C^n\). Admissible convergence. The Lindelöf principle. Additional tangential phenomena: Lipschitz spaces. Constructive methods. Story of the inner functions problem. The Hakim/Løw/Sibony construction of inner functions. Further results obtained with constructive methods. Integral formulas for solutions to the \(\bar\partial\) problem and norm estimates. The Henkin integral formula. Estimates for the Henkin solution on domains in \(\mathbb C^2\). Smoothness of the Henkin solution and the case of arbitrary \(\bar\partial\)-closed forms. First Cousin problem with bounds and uniform approximation on strongly pseudoconvex domains in \(\mathbb C^2\). Holomorphic mappings and invariant metrics. Classical theory of holomorphic mappings. Invariant metrics. The theorem of Bun Wong and Rosay. Smoothness to the boundary of biholomorphic mappings. Concept of finite type. Complex analytic dynamics. Manifolds. Area measures. Exterior algebra. Vectors, covectors, and differential forms. List of notation. Bibliography. Index.

From the Contents: An introduction to the subject. What is a holomorphic function? Comparison of \(\mathbb C^1\) and \(\mathbb C^n\). The Bochner-Martinelli formula. Applications of Cauchy theory and the \(\bar\partial\) equation. Basic properties of harmonic functions. The Bergman kernel. The Szegö and Poisson-Szegö kernels. Subharmonicity and its applications. Subharmonic functions. Pluriharmonic and plurisubharmonic functions. Power series. Hartogs’s theorem on separate analyticity. Convexity and pseudoconvexity, and analytic discs. Domains of holomorphy. Examples of domains of holomorphy and the edge-of-the-wedge theorem. Hörmander’s solution of the \(\bar\partial\) equation. Unbounded operators on a Hilbert space. An existence and regularity theorems for the \(\bar\partial\) operator. Solution of the Levi problem and other applications of \(\bar\partial\) techniques. An extension problem. Singular functions on strongly pseudoconvex domains. Hefer’s lemma and Henkin’s integral representation. Approximation problems. Cousin problems, cohomology, and sheaves. Dolbeault isomorphism. Algebraic properties of the ring of germs of holomorphic functions. Sheaf of divisors, Chern classes, and the obstruction to solving Cousin II. Coherent analytic sheaves. Applications of the Cartan and Oka theorems: the structure of ideals. Zeros of one holomorphic function. Zero sets for different \(H^p\) spaces are different. Zero sets of \(H^p\) functions do not have a simple mass distribution characterization. Some harmonic analysis. Review of the classical theory of \(H^p\) spaces on the disc. Three propositions about the Poisson kernel. Subharmonicity, harmonic majorants, and boundary values. Pointwise convergence for harmonic functions on domains in \(\mathbb R^N\). Boundary values of holomorphic functions in \(\mathbb C^n\). Admissible convergence. The Lindelöf principle. Additional tangential phenomena: Lipschitz spaces. Constructive methods. Story of the inner functions problem. The Hakim/Løw/Sibony construction of inner functions. Further results obtained with constructive methods. Integral formulas for solutions to the \(\bar\partial\) problem and norm estimates. The Henkin integral formula. Estimates for the Henkin solution on domains in \(\mathbb C^2\). Smoothness of the Henkin solution and the case of arbitrary \(\bar\partial\)-closed forms. First Cousin problem with bounds and uniform approximation on strongly pseudoconvex domains in \(\mathbb C^2\). Holomorphic mappings and invariant metrics. Classical theory of holomorphic mappings. Invariant metrics. The theorem of Bun Wong and Rosay. Smoothness to the boundary of biholomorphic mappings. Concept of finite type. Complex analytic dynamics. Manifolds. Area measures. Exterior algebra. Vectors, covectors, and differential forms. List of notation. Bibliography. Index.

Reviewer: Vasily A. Chernecky (Odessa)

### MSC:

32-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces |

37F99 | Dynamical systems over complex numbers |

32F45 | Invariant metrics and pseudodistances in several complex variables |