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