Contents Introduction and Review Boundary Behavior The Heisenberg Group Analysis on the Heisenberg Group

Transcription

1 Contents 1 Introduction and Review Harmonic Analysis on the Disc The Boundary Behavior of Holomorphic Functions... 4 Exercises Boundary Behavior The Modern Era Spaces of Homogeneous Type Estimates for the Poisson Kernel Subharmonicity and Boundary Values Pointwise Convergence for Harmonic Functions Boundary Values of Holomorphic Functions Admissible Convergence Exercises The Heisenberg Group Prolegomena The Upper Half Plane in C The Significance of the Heisenberg Group The Heisenberg Group Action on U The Nature The Heisenberg Group as a Lie Group Classical Analysis The Folland Stein Theorem Calderón Zygmund Theory Exercises Analysis on the Heisenberg Group A Deeper Look at the Heisenberg Group L 2 Boundedness of Calderón Zygmund Integrals The Cotlar Knapp Stein Lemma L p Boundedness of Calderón Zygmund Integrals ix

2 x Contents 4.5 Calderón Zygmund Applications The Szegő Integral on the Heisenberg Group The Poisson SzegőIntegral Applications of the Paley Wiener Theorem Exercises Reproducing Kernels Reproducing Kernels Canonical Integral Formulas Formulas with Holomorphic Kernel Asymptotic Expansion for the Kernel Constructive Kernels vs. Canonical Kernels Exercises Moreonthe Kernels The Bergman Kernel Smoothness to the Boundary of K Calculating the Bergman Kernel The Poincaré Bergman Distance on the Disc Construction of the Bergman Kernel by Way of Differential Equations Construction of the Bergman Kernel by Way of Conformal Invariance The Szegő and Poisson SzegőKernels Aronszajn Theory A New Basis Additional Examples The Behavior of the Singularity A Real Bergman Space Relation Between Bergman and Szegő Introduction The Case of the Disc The Unit Ball in C n Strongly Pseudoconvex Domains The Annulus Multiply Connected Domains The Sobolev Bergman Kernel The Theorem of Ramadanov More on the SzegőKernel Boundary Localization Definitions and Notation A Representative Result The More General Result in the Plane Domains in Higher-Dimensional Complex Space Exercises

3 Contents xi 7 The Bergman Metric Smoothness of Biholomorphic Mappings The Bergman Metric at the Boundary Inequivalence of the Ball and the Polydisc Exercises Geometric and Analytic Ideas Bergman Representative Coordinates The Berezin Transform Preliminary Remarks Introduction to the Poisson Bergman Kernel Boundary Behavior Ideas of Fefferman The Invariant Laplacian The Dirichlet Problem for the Invariant Laplacian Concluding Remarks Exercises Additional Analytic Topics The Worm Domain Additional Worm Ideas Alternative Versions of the Worm Domain Pathologies of the Bergman Projection Pathologies of the Bergman Kernel Kohn s Projection Formula Boundary Behavior of the Kernel Hörmander s Result on Boundary Behavior Fefferman s Asymptotic Expansion Regularity for the Dirichlet Problem Plurisubharmonic Defining Functions Proof of Theorem Uses of the Monge Ampère Equation An Example of Barrett A Hilbert Integral Exercises Cauchy Riemann Equations Solution The Inhomogeneous Cauchy Riemann Equations Some Notation Statement of Problem The Main Estimate Special Boundary Charts and Technical Matters Beginning of the Proof of the Main Theorem Estimates in the Sobolev 1=2 Norm Proof of the Main Theorem

4 xii Contents 10.9 Solution of Problem Appendix to Section Exercises A Few Miscellaneous Topics Ideas of Christ/Geller Square Functions Ideas of Nagel/Stein and Di Biase H 1 and BMO Factorization of Hardy Space Functions The Atomic Theory of Hardy Spaces Concluding Remarks Exercises Bibliography Index

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