C单位球上的函数理论 影印版PDF电子书下载

Chapter 1 Preliminaries 1

1.1 Some Terminology 1

1.2 The Cauchy Formula in Polydiscs 3

1.3 Differentiation 7

1.4 Integrals over Spheres 12

1.5 Homogeneous Expansions 19

Chapter 2 The Automorphisms of B 23

2.1 Cartan's Uniqueness Theorem 23

2.2 The Automorphisms 25

2.3 The Cayley Transform 31

2.4 Fixed Points and Affine Sets 32

Chapter 3 Integral Representations 36

3.1 The Bergman Integral in B 36

3.2 The Cauchy Integral in B 38

3.3 The Invariant Poisson Integral in B 50

Chapter 4 The Invariant Laplacian 47

4.1 The Operator？ 47

4.2 Eigenfunctions of ？ 49

4.3 M-Harmonic Functions 55

4.4 Pluriharmonic Functions 59

Chapter 5 Boundary Behavior of Poisson Integrals 65

5.1 A Nonisotropic Metric on S 65

5.2 The Maximal Function of a Measure on S 67

5.3 Differentiation of Measures on S 70

5.4 K-Limits of Poisson Integrals 72

5.5 Theorems of Calderón,Privalov,Plessner 79

5.6 The Spaces N(B)and Hp(B) 83

5.7 Appendix:Marcinkiewicz Interpolation 88

Chapter 6 Boundary Behavior of Cauchy Integrals 91

6.1 An Inequality 92

6.2 Cauchy Integrals of Measures 94

6.3 Cauchy Integrals of Lp-Functions 99

6.4 Cauchy Integrals of Lipschitz Functions 101

6.5 Toeplitz Operators 110

6.6 Gleason's Problem 114

Chapter 7 Some Lp-Topics 120

7.1 Projections of Bergman Type 120

7.2 Relations between Hp and Lp？H 126

7.3 Zero-Varieties 133

7.4 Pluriharmonic Majorants 145

7.5 The Isometries of Hp(B) 152

Chapter 8 Consequences of the Schwarz Lemma 161

8.1 The Schwarz Lemma in B 161

8.2 Fixed-Point Sets in B 165

8.3 An Extension Problem 166

8.4 The Lindel？f-？irka Theorem 168

8.5 The Julia-Carathéodory Theorem 174

Chapter 9 Measures Related to the Ball Algebra 185

9.1 Introduction 185

9.2 Valskii's Decomposition 187

9.3 Henkin's Theorem 189

9.4 A General Lebesgue Decomposition 191

9.5 A General E.and M.Riesz Theorem 195

9.6 The Cole-Range Theorem 198

9.7 Pluriharmonic Majorants 198

9.8 The Dual Space of A(B) 202

Chapter 10 Interpolation Sets for the Ball Algebra 204

10.1 Some Equivalences 204

10.2 A Theorem of Varopoulos 207

10.3 A Theorem of Bishop 209

10.4 The Davie-？ksendal Theorem 211

10.5 Smooth Interpolation Sets 214

10.6 Determining Sets 222

10.7 Peak Sets for Smooth Functions 229

Chapter 11 Boundary Behavior of H∞-Functions 234

11.1 A Fatou Theorem in One Variable 234

11.2 Boundary Values on Curves in S 237

11.3 Weak-Convergence 244

11.4 A Problem on Extreme Values 247

Chapter 12 Unitarily Invariant Function Spaces 253

12.1 Spherical Harmonics 253

12.2 The Spaces H(p,q) 255

12.3 U-Invariant Spaces on S 259

12.4 U-Invariant Subalgebras of C(S) 264

12.5 The Case n=2 270

Chapter 13 Moebius-Invariant Function Spaces 278

13.1 .U-Invariant Spaces on S 278

13.2 .U-Invariant Subalgebras of Co(B) 280

13.3 .U-Invariant Subspaces of C(？) 283

13.4 Some Applications 285

Chapter 14 Analytic Varieties 288

14.1 The Weierstrass Preparation Theorem 288

14.2 Projections of Varieties 291

14.3 Compact Varieties in ？n 294

14.4 Hausdorff Measures 295

Chapter 15 Proper Holomorphic Maps 300

15.1 The Structure of Proper Maps 300

15.2 Balls vs.Polydiscs 305

15.3 Local Theorems 309

15.4 Proper Maps from B to B 314

15.5 A Characterization of B 319

Chapter 16 The ？-Problem 330

16.1 Differential Forms 330

16.2 Differential Forms in Cn 335

16.3 The ？-Problem with Compact Support 338

16.4 Some Computations 341

16.5 Koppelman's Cauchy Formula 346

16.6 The ？-Problem in Convex Regions 350

16.7 An Explicit Solution in B 357

Chapter 17 The Zeros of Nevanlinna Functions 364

17.1 The Henkin-Skoda Theorem 364

17.2 Plurisubharmonic Functions 366

17.3 Areas of Zero-Varieties 381

Chapter 18 Tangential Cauchy-Riemann Operators 387

18.1 Extensions from the Boundary 387

18.2 Unsolvable Differential Equations 395

18.3 Boundary Values of Pluriharmonic Functions 397

Chapter 19 Open Problems 403

19.1 The Inner Function Conjecture 403

19.2 RP-Measures 409

19.3 Miscellaneous Problems 413

Bibliography 419

Index 431