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.

Steven G. Krantz, Washington University, St. Louis, MO.

Chapters
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 equation
Chapter 5. Solution of the Levi problem and other applications of 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 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