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

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Transcription:

Contents 1 Introduction and Review... 1 1.1 Harmonic Analysis on the Disc... 1 1.1.1 The Boundary Behavior of Holomorphic Functions... 4 Exercises... 15 2 Boundary Behavior... 19 2.1 The Modern Era... 19 2.1.1 Spaces of Homogeneous Type... 23 2.2 Estimates for the Poisson Kernel... 25 2.3 Subharmonicity and Boundary Values... 28 2.4 Pointwise Convergence for Harmonic Functions... 35 2.5 Boundary Values of Holomorphic Functions... 37 2.6 Admissible Convergence... 43 Exercises... 53 3 The Heisenberg Group... 59 3.1 Prolegomena... 60 3.2 The Upper Half Plane in C... 60 3.3 The Significance of the Heisenberg Group... 62 3.4 The Heisenberg Group Action on U... 66 3.5 The Nature of @U... 68 3.6 The Heisenberg Group as a Lie Group... 69 3.7 Classical Analysis... 73 3.7.1 The Folland Stein Theorem... 74 3.8 Calderón Zygmund Theory... 79 Exercises... 88 4 Analysis on the Heisenberg Group... 89 4.1 A Deeper Look at the Heisenberg Group... 89 4.2 L 2 Boundedness of Calderón Zygmund Integrals... 91 4.3 The Cotlar Knapp Stein Lemma... 92 4.4 L p Boundedness of Calderón Zygmund Integrals... 94 ix

x Contents 4.5 Calderón Zygmund Applications... 95 4.6 The Szegő Integral on the Heisenberg Group... 96 4.7 The Poisson SzegőIntegral... 97 4.8 Applications of the Paley Wiener Theorem... 98 Exercises... 112 5 Reproducing Kernels... 115 5.1 Reproducing Kernels... 115 5.2 Canonical Integral Formulas... 116 5.3 Formulas with Holomorphic Kernel... 119 5.4 Asymptotic Expansion for the Kernel... 124 5.5 Constructive Kernels vs. Canonical Kernels... 124 Exercises... 126 6 Moreonthe Kernels... 131 6.1 The Bergman Kernel... 131 6.1.1 Smoothness to the Boundary of K... 143 6.1.2 Calculating the Bergman Kernel... 144 6.1.3 The Poincaré Bergman Distance on the Disc... 149 6.1.4 Construction of the Bergman Kernel by Way of Differential Equations... 150 6.1.5 Construction of the Bergman Kernel by Way of Conformal Invariance... 153 6.2 The Szegő and Poisson SzegőKernels... 155 6.3 Aronszajn Theory... 161 6.4 A New Basis... 162 6.5 Additional Examples... 165 6.6 The Behavior of the Singularity... 166 6.7 A Real Bergman Space... 167 6.8 Relation Between Bergman and Szegő... 168 6.8.1 Introduction... 168 6.8.2 The Case of the Disc... 169 6.8.3 The Unit Ball in C n... 173 6.8.4 Strongly Pseudoconvex Domains... 175 6.9 The Annulus... 177 6.10 Multiply Connected Domains... 179 6.11 The Sobolev Bergman Kernel... 179 6.12 The Theorem of Ramadanov... 182 6.13 More on the SzegőKernel... 184 6.14 Boundary Localization... 184 6.14.1 Definitions and Notation... 185 6.14.2 A Representative Result... 186 6.14.3 The More General Result in the Plane... 188 6.14.4 Domains in Higher-Dimensional Complex Space... 188 Exercises... 190

Contents xi 7 The Bergman Metric... 195 7.1 Smoothness of Biholomorphic Mappings... 195 7.2 The Bergman Metric at the Boundary... 206 7.3 Inequivalence of the Ball and the Polydisc... 208 Exercises... 209 8 Geometric and Analytic Ideas... 213 8.1 Bergman Representative Coordinates... 213 8.2 The Berezin Transform... 216 8.2.1 Preliminary Remarks... 216 8.2.2 Introduction to the Poisson Bergman Kernel... 217 8.2.3 Boundary Behavior... 220 8.3 Ideas of Fefferman... 224 8.4 The Invariant Laplacian... 226 8.5 The Dirichlet Problem for the Invariant Laplacian... 236 8.6 Concluding Remarks... 241 Exercises... 242 9 Additional Analytic Topics... 245 9.1 The Worm Domain... 245 9.2 Additional Worm Ideas... 251 9.3 Alternative Versions of the Worm Domain... 259 9.4 Pathologies of the Bergman Projection... 260 9.5 Pathologies of the Bergman Kernel... 265 9.6 Kohn s Projection Formula... 267 9.7 Boundary Behavior of the Kernel... 268 9.7.1 Hörmander s Result on Boundary Behavior... 269 9.7.2 Fefferman s Asymptotic Expansion... 275 9.8 Regularity for the Dirichlet Problem... 282 9.9 Plurisubharmonic Defining Functions... 286 9.10 Proof of Theorem 9.9.1... 288 9.11 Uses of the Monge Ampère Equation... 291 9.12 An Example of Barrett... 294 9.13 A Hilbert Integral... 304 Exercises... 308 10 Cauchy Riemann Equations Solution... 309 10.1 The Inhomogeneous Cauchy Riemann Equations... 309 10.2 Some Notation... 313 10.3 Statement of the @-Neumann Problem... 326 10.4 The Main Estimate... 331 10.5 Special Boundary Charts and Technical Matters... 338 10.6 Beginning of the Proof of the Main Theorem... 348 10.7 Estimates in the Sobolev 1=2 Norm... 353 10.8 Proof of the Main Theorem... 363

xii Contents 10.9 Solution of the @-Neumann Problem... 371 Appendix to Section 10.8... 376 Exercises... 379 11 A Few Miscellaneous Topics... 395 11.1 Ideas of Christ/Geller... 395 11.2 Square Functions... 396 11.3 Ideas of Nagel/Stein and Di Biase... 399 11.4 H 1 and BMO... 399 11.5 Factorization of Hardy Space Functions... 401 11.6 The Atomic Theory of Hardy Spaces... 401 11.7 Concluding Remarks... 402 Exercises... 402 Bibliography... 405 Index... 419

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