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Table of Contents
Preface; Contents; 1 Introduction and Review; 1.1 Harmonic Analysis on the Disc; 1.1.1 The Boundary Behavior of Holomorphic Functions; Exercises; 2 Boundary Behavior; 2.1 The Modern Era; 2.1.1 Spaces of Homogeneous Type; 2.2 Estimates for the Poisson Kernel; 2.3 Subharmonicity and Boundary Values; 2.4 Pointwise Convergence for Harmonic Functions; 2.5 Boundary Values of Holomorphic Functions; 2.6 Admissible Convergence; Exercises; 3 The Heisenberg Group; 3.1 Prolegomena; 3.2 The Upper Half Plane in C; 3.3 The Significance of the Heisenberg Group; 3.4 The Heisenberg Group Action on U
3.5 The Nature of ∂U3.6 The Heisenberg Group as a Lie Group; 3.7 Classical Analysis; 3.7.1 The Folland-Stein Theorem; 3.8 Calderón-Zygmund Theory; Exercises; 4 Analysis on the Heisenberg Group; 4.1 A Deeper Look at the Heisenberg Group; 4.2 L2 Boundedness of Calderón-Zygmund Integrals; 4.3 The Cotlar-Knapp-Stein Lemma; 4.4 Lp Boundedness of Calderón-Zygmund Integrals; 4.5 Calderón-Zygmund Applications; 4.6 The Szegő Integral on the Heisenberg Group; 4.7 The Poisson-Szegő Integral; 4.8 Applications of the Paley-Wiener Theorem; Exercises; 5 Reproducing Kernels; 5.1 Reproducing Kernels
6.6 The Behavior of the Singularity6.7 A Real Bergman Space; 6.8 Relation Between Bergman and Szegő; 6.8.1 Introduction; 6.8.2 The Case of the Disc; 6.8.3 The Unit Ball in Cn; 6.8.4 Strongly Pseudoconvex Domains; 6.9 The Annulus; 6.10 Multiply Connected Domains; 6.11 The Sobolev Bergman Kernel; 6.12 The Theorem of Ramadanov; 6.13 More on the Szegő Kernel; 6.14 Boundary Localization; 6.14.1 Definitions and Notation; 6.14.2 A Representative Result; 6.14.3 The More General Result in the Plane; 6.14.4 Domains in Higher-Dimensional Complex Space; Exercises; 7 The Bergman Metric
7.1 Smoothness of Biholomorphic Mappings7.2 The Bergman Metric at the Boundary; 7.3 Inequivalence of the Ball and the Polydisc; Exercises; 8 Geometric and Analytic Ideas; 8.1 Bergman Representative Coordinates; 8.2 The Berezin Transform; 8.2.1 Preliminary Remarks; 8.2.2 Introduction to the Poisson-Bergman Kernel; 8.2.3 Boundary Behavior; 8.3 Ideas of Fefferman; 8.4 The Invariant Laplacian; 8.5 The Dirichlet Problem for the Invariant Laplacian; 8.6 Concluding Remarks; Exercises; 9 Additional Analytic Topics; 9.1 The Worm Domain; 9.2 Additional Worm Ideas
3.5 The Nature of ∂U3.6 The Heisenberg Group as a Lie Group; 3.7 Classical Analysis; 3.7.1 The Folland-Stein Theorem; 3.8 Calderón-Zygmund Theory; Exercises; 4 Analysis on the Heisenberg Group; 4.1 A Deeper Look at the Heisenberg Group; 4.2 L2 Boundedness of Calderón-Zygmund Integrals; 4.3 The Cotlar-Knapp-Stein Lemma; 4.4 Lp Boundedness of Calderón-Zygmund Integrals; 4.5 Calderón-Zygmund Applications; 4.6 The Szegő Integral on the Heisenberg Group; 4.7 The Poisson-Szegő Integral; 4.8 Applications of the Paley-Wiener Theorem; Exercises; 5 Reproducing Kernels; 5.1 Reproducing Kernels
6.6 The Behavior of the Singularity6.7 A Real Bergman Space; 6.8 Relation Between Bergman and Szegő; 6.8.1 Introduction; 6.8.2 The Case of the Disc; 6.8.3 The Unit Ball in Cn; 6.8.4 Strongly Pseudoconvex Domains; 6.9 The Annulus; 6.10 Multiply Connected Domains; 6.11 The Sobolev Bergman Kernel; 6.12 The Theorem of Ramadanov; 6.13 More on the Szegő Kernel; 6.14 Boundary Localization; 6.14.1 Definitions and Notation; 6.14.2 A Representative Result; 6.14.3 The More General Result in the Plane; 6.14.4 Domains in Higher-Dimensional Complex Space; Exercises; 7 The Bergman Metric
7.1 Smoothness of Biholomorphic Mappings7.2 The Bergman Metric at the Boundary; 7.3 Inequivalence of the Ball and the Polydisc; Exercises; 8 Geometric and Analytic Ideas; 8.1 Bergman Representative Coordinates; 8.2 The Berezin Transform; 8.2.1 Preliminary Remarks; 8.2.2 Introduction to the Poisson-Bergman Kernel; 8.2.3 Boundary Behavior; 8.3 Ideas of Fefferman; 8.4 The Invariant Laplacian; 8.5 The Dirichlet Problem for the Invariant Laplacian; 8.6 Concluding Remarks; Exercises; 9 Additional Analytic Topics; 9.1 The Worm Domain; 9.2 Additional Worm Ideas