C单位球上的函数理论 影印版PDF|Epub|txt|kindle电子书版本网盘下载

C单位球上的函数理论 影印版PDF格式电子书版下载

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Chapter 1 Preliminaries1

1.1 Some Terminology1

1.2 The Cauchy Formula in Polydiscs3

1.3 Differentiation7

1.4 Integrals over Spheres12

1.5 Homogeneous Expansions19

Chapter 2 The Automorphisms of B23

2.1 Cartan's Uniqueness Theorem23

2.2 The Automorphisms25

2.3 The Cayley Transform31

2.4 Fixed Points and Affine Sets32

Chapter 3 Integral Representations36

3.1 The Bergman Integral in B36

3.2 The Cauchy Integral in B38

3.3 The Invariant Poisson Integral in B50

Chapter 4 The Invariant Laplacian47

4.1 The Operator？47

4.2 Eigenfunctions of ？49

4.3 M-Harmonic Functions55

4.4 Pluriharmonic Functions59

Chapter 5 Boundary Behavior of Poisson Integrals65

5.1 A Nonisotropic Metric on S65

5.2 The Maximal Function of a Measure on S67

5.3 Differentiation of Measures on S70

5.4 K-Limits of Poisson Integrals72

5.5 Theorems of Calderón,Privalov,Plessner79

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

5.7 Appendix:Marcinkiewicz Interpolation88

Chapter 6 Boundary Behavior of Cauchy Integrals91

6.1 An Inequality92

6.2 Cauchy Integrals of Measures94

6.3 Cauchy Integrals of Lp-Functions99

6.4 Cauchy Integrals of Lipschitz Functions101

6.5 Toeplitz Operators110

6.6 Gleason's Problem114

Chapter 7 Some Lp-Topics120

7.1 Projections of Bergman Type120

7.2 Relations between Hp and Lp？H126

7.3 Zero-Varieties133

7.4 Pluriharmonic Majorants145

7.5 The Isometries of Hp(B)152

Chapter 8 Consequences of the Schwarz Lemma161

8.1 The Schwarz Lemma in B161

8.2 Fixed-Point Sets in B165

8.3 An Extension Problem166

8.4 The Lindel？f-？irka Theorem168

8.5 The Julia-Carathéodory Theorem174

Chapter 9 Measures Related to the Ball Algebra185

9.1 Introduction185

9.2 Valskii's Decomposition187

9.3 Henkin's Theorem189

9.4 A General Lebesgue Decomposition191

9.5 A General E.and M.Riesz Theorem195

9.6 The Cole-Range Theorem198

9.7 Pluriharmonic Majorants198

9.8 The Dual Space of A(B)202

Chapter 10 Interpolation Sets for the Ball Algebra204

10.1 Some Equivalences204

10.2 A Theorem of Varopoulos207

10.3 A Theorem of Bishop209

10.4 The Davie-？ksendal Theorem211

10.5 Smooth Interpolation Sets214

10.6 Determining Sets222

10.7 Peak Sets for Smooth Functions229

Chapter 11 Boundary Behavior of H∞-Functions234

11.1 A Fatou Theorem in One Variable234

11.2 Boundary Values on Curves in S237

11.3 Weak-Convergence244

11.4 A Problem on Extreme Values247

Chapter 12 Unitarily Invariant Function Spaces253

12.1 Spherical Harmonics253

12.2 The Spaces H(p,q)255

12.3 U-Invariant Spaces on S259

12.4 U-Invariant Subalgebras of C(S)264

12.5 The Case n=2270

Chapter 13 Moebius-Invariant Function Spaces278

13.1 .U-Invariant Spaces on S278

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

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

13.4 Some Applications285

Chapter 14 Analytic Varieties288

14.1 The Weierstrass Preparation Theorem288

14.2 Projections of Varieties291

14.3 Compact Varieties in ？n294

14.4 Hausdorff Measures295

Chapter 15 Proper Holomorphic Maps300

15.1 The Structure of Proper Maps300

15.2 Balls vs.Polydiscs305

15.3 Local Theorems309

15.4 Proper Maps from B to B314

15.5 A Characterization of B319

Chapter 16 The ？-Problem330

16.1 Differential Forms330

16.2 Differential Forms in Cn335

16.3 The ？-Problem with Compact Support338

16.4 Some Computations341

16.5 Koppelman's Cauchy Formula346

16.6 The ？-Problem in Convex Regions350

16.7 An Explicit Solution in B357

Chapter 17 The Zeros of Nevanlinna Functions364

17.1 The Henkin-Skoda Theorem364

17.2 Plurisubharmonic Functions366

17.3 Areas of Zero-Varieties381

Chapter 18 Tangential Cauchy-Riemann Operators387

18.1 Extensions from the Boundary387

18.2 Unsolvable Differential Equations395

18.3 Boundary Values of Pluriharmonic Functions397

Chapter 19 Open Problems403

19.1 The Inner Function Conjecture403

19.2 RP-Measures409

19.3 Miscellaneous Problems413

Bibliography419

Index431