《Cn单位球上的函数理论》(作者鲁丁)是springer数学经典教材系列之一，表述清晰易懂，自然流畅，用很少的实分析、复分析和泛函分析基本知识做铺垫，全面介绍了球上基本原理。既是一本很好的参考书，又是一本高年级教程。

List of Symbols and Notations

Chapter 1

Preliminaries

1.1 Some Terminology

1.2 The Cauchy Formula in Polydiscs

1.3 Differentiation

1.4 Integrals over Spheres

1.5 Homogeneous Expansions

Chapter 2

The Automorphisms of B

2.1 Cartan's Uniqueness Theorem

2.2 The Automorphisms

2.3 The Cayley Transform

2.4 Fixed Points and Afline Sets

Chapter 3

Integral Representations

3.1 The Bergman Integral in B

3.2 The Cauchy Integral in B

3.3 The Invariant Poisson Integral in B

Chapter 4

The lnvariant Laplacian

4.1 The Operator

4.2 Eigenfunctions of □

4.3 □-Harmonie Functions

4.4 Pluriharmonic Functions

Chapter 5

Boundary Behavior of Poisson Integrals

5.1 A Nonisotropic Metric on S

5.2 The Maximal Function of a Measure on S

5.3 Differentiation of Measures on S

5.4 K-Limits of Poisson Integrals

5.5 Theorems of Calder6n. Privalov, Plessner

5.6 The Spaces N(B) and H□(B)

5.7 Appendix: Marcinkiewicz Interpolation

Chapter 6

Boundary Behavior of Cauchy Integrals

6,1 An Inequality

6.2 Cauchy Integrals of Measures

6.3 Cauchy Integrals of LP-Functions

6.4 Cauchy Integrals of Lipschltz Functions

6.5 Toeplitz Operators

6.6 Gleason's Problem

Chapter 7

Some LP-Topics

7.1 Projections of Bergman Type

7.2 Relations between Hp and Lp□H

7.3 Zero-Varieties

7.4 Pluriharmonic Majoranls

7.5 The Isometties of HP(B)

Chapter 8

Consequences of the Schwarz Lemma

8.1 The Schwarz Lemma in B

8.2 Fixed-Point Sets in B

8.3 An Extension Problem

8.4 The Liodel6f-□irka Theorem

8,5 The Julia-Carath6odory Theorem

Chapter 9

Measures Related to the Ball Algebra

9.1 Introduction

9.2 Valskii's Decomposition

9.3 Henkin's Theorem

9.4 A General Lebesgue Decomposition

9.5 A General F. and M. Riesz Theorem

9.6 The Cole-Range Theorem

9.7 Pluriharmonic Majorants

9.8 The Dual Space of A(B)

Chapter 10

Interpolation Sets for the Ball Algebra

10.1 Some Equivalences

10.2 A Theorem of Varopoulos

10.3 A Theorem of Bishop

10.4 The Davie-□ksendal Theorem

10.5 Smooth Interpolation Sets

10.6 Determining Sets

10.7 Peak Sets for Smooth Functions

Chapter 11

Boundary Behavior of H□-Functions

11.1 A Fatou Theorem in One Variable

11.2 Boundary Values on Curves in S

11.3 Weak*-Convergence

11.4 A Problem on Extreme Values

Chapter 12

Unitarily Invariant Function Spaces

12.1 Spherical Harmonics

12.2 The Spaces H~, q)

12.3 □-Invariant Spaceson S

12.4 □-lnvariant Subalgebras of C(S)

12.5 The Case n = 2

Chapter 13

Moebius-lnvariant Function Spaces

13.1 □-Invariant Spaces on S

13.2 □-Invariant Subalgebras of Co(B)

13.3 □-lnvariant Subspaces of C(□)

13.4 Some Applications

Chapter 14

Analytic Varieties

14.1 The Weierstrass Preparation Theorem

14.2 Projections of Varieties

14.3 Compact Varieties in C"

14.4 Hausdorff Measures

Chapter 15

Proper Holomorphic Maps

15.1 The Structure of Proper Maps

15.2 Balls vs. Polydiscs

15.3 Local Theorems

15.4 Proper Maps from B to B

15.5 A Characterization of B

Chapter 16

The □-problem

16.1 Differential Forms

16.2 Differential Forms in C"

16.3 The □-problem with Compact Support

16.4 Some Computations

16.5 Koppelman's Cauchy Formula

16.6 The g-problem in Convex Regions

16.7 An Explicit Solution in B

Chapter 17

The Zeros of Nevanlinna Functions

17.1 The Henkin-Skoda Theorem

17.2 Plurisubharmonic Functions

17.3 Areas of Zero-Varieties

Chapter 18

Tangential Cauchy-Riemann Operators

18.1 Extensions from the Boundary

18.2 Unsolvable Differential Equations

18.3 Boundary Values of Pluriharmonic Functions

Chapter 19

Open Problems

19.1 The Inner Function Conjecture

19.2 RP-Measures

19.3 Miscellaneous Problems

Bibliography

Index