Function Theory of Several Complex Variables (AMS Chelsea Publishing)

This work departs from earlier treatments of the subject by emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, the boundary behavior of holomorphic functions, inner functions, invariant metrics, and mapping theory. While due homage is paid to the more traditional algebraic theory (sheaves, Cousin problems, etc.), the student with a background in real and complex variable theory, harmonic analysis, and differential equations will be most comfortable with this treatment. It is currently the only book on the subject with exercises and a large number of examples. The theory of several complex variables can be studied from several different perspectives. In this book, Steven Krantz approaches the subject from the point of view of a classical analyst, emphasizing its function-theoretic aspects. He has taken particular care to write the book with the student in mind, with uniformly extensive and helpful explanations, numerous examples, and plentiful exercises of varying difficulty. In the spirit of a student-oriented text, Krantz begins with an introduction to the subject, including an insightful comparison of analysis of several complex variables with the more familiar theory of one complex variable. The main topics in the book include integral formulas, convexity and pseudoconvexity, methods from harmonic analysis, and several aspects of the $\overline{\partial}$ problem. Some further topics are zero sets of holomorphic functions, estimates, partial differential equations, approximation theory, the boundary behavior of holomorphic functions, inner functions, invariant metrics, and holomorphic mappings. While due attention is paid to algebraic aspects of several complex variables (sheaves, Cousin problems, etc.), the student with a background in real and complex variable theory, harmonic analysis, and differential equations will be most comfortable with this treatment. This book is suitable for a first graduate course in several complex variables. Chapter 0. An Introduction To The Subject Chapter 1. Some Integral Formulas Chapter 2. Subharmonicity And Its Applications Chapter 3. Convexity Chapter 4. Hörmander's Solution Of The $\bar \partial $ Equation Chapter 5. Solution Of The Levi Problem And Other Applications Of $\bar \partial $ Techniques Chapter 6. Cousin Problems, Cohomology, And Sheaves Chapter 7. The Zero Set Of A Holomorphic Function Chapter 8. Some Harmonic Analysis Chapter 9. Constructive Methods Chapter 10. Integral Formulas For Solutions To The $\bar \partial $ Problem And Norm Estimates Chapter 11. Holomorphic Mappings And Invariant Metrics Appendix I. Manifolds Appendix Ii. Area Measures Appendix Iii. Exterior Algebra Appendix Iv. Vectors, Covectors, And Differential Forms Steven G. Krantz. Includes Bibliographical References (p. 515-552) And Index. This edition has been revised and updated to include two new chapters on recent developments in the field. Steven G. Krantz. Includes Bibliographical References (p. 507-544) And Index.