Author | Rudin, Walter, 1921-, author |
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Title | Function Theory in the Unit Ball of โn [electronic resource] / by Walter Rudin |
Imprint | New York, NY : Springer New York, 1980 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8098-6 |
Descript | XIII, 438 p. online resource |
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 -- 2 The Automorphisms of B -- 2.1 Cartanโs Uniqueness Theorem -- 2.2 The Automorphisms -- 2.3 The Cayley Transform -- 2.4 Fixed Points and Affine Sets -- 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 -- 4 The Invariant Laplacian -- 4.1 The Operator $$ \tilde \Delta $$ -- 4.2 Eigenfunctions of $$ \tilde \Delta $$ -- 4.3 ?-Harmonic Functions -- 4.4 Pluriharmonic Functions -- 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 Calderรณn, Privalov, Plessner -- 5.6 The Spaces N(B) and Hp(B) -- 5.7 Appendix: Marcinkiewicz Interpolation -- 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 Lipschitz Functions -- 6.5 Toeplitz Operators -- 6.6 Gleasonโs Problem -- 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 Majorants -- 7.5 The Isometries of Hp(B) -- 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 Lindelรถf-?irka Theorem -- 8.5 The Julia-Carathรฉodory Theorem -- 9 Measures Related to the Ball Algebra -- 9.1 Introduction -- 9.2 Valskiiโs Decomposition -- 9.3 Henkinโs 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) -- 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-รksendal Theorem -- 10.5 Smooth Interpolation Sets -- 10.6 Determining Sets -- 10.7 Peak Sets for Smooth Functions -- 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 -- 12 Unitarily Invariant Function Spaces -- 12.1 Spherical Harmonics -- 12.2 The Spaces H(p, q) -- 12.3 U-Invariant Spaces on S -- 12.4 U-Invariant Subalgebras of C(S) -- 12.5 The Case n = 2 -- 13 Moebius-Invariant Function Spaces -- 13.1.?-Invariant Spaces on S -- 13.2.?-Invariant Subalgebras of C0(B) -- 13.3.?-Invariant Subspaces of C(B) -- 13.4 Some Applications -- 14 Analytic Varieties -- 14.1 The Weierstrass Preparation Theorem -- 14.2 Projections of Varieties -- 14.3 Compact Varieties in ?n -- 14.4 Hausdorff Measures -- 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 -- 16 The $$ {\bar \partial } $$ -Problem -- 16.1 Differential Forms -- 16.2 Differential Forms in ?n -- 16.3 The $$ {\bar \partial } $$ -Problem with Compact Support -- 16.4 Some Computations -- 16.5 Koppelmanโs Cauchy Formula -- 16.6 The $$ {\bar \partial } $$ -Problem in Convex Regions -- 16.7 An Explicit Solution in B -- 17 The Zeros of Nevanlinna Functions -- 17.1 The Henkin-Skoda Theorem -- 17.2 Plurisubharmonic Functions -- 17.3 Areas of Zero-Varieties -- 18 Tangential Cauchy-Riemann Operators -- 18.1 Extensions from the Boundary -- 18.2 Unsolvable Differential Equations -- 18.3 Boundary Values of Pluriharmonic Functions -- 19 Open Problems -- 19.1 The Inner Function Conjecture -- 19.2 RP-Measures -- 19.3 Miscellaneous Problems