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E-raamat: Primer on the Dirichlet Space

(Université de Bordeaux), (Université Laval, Québec), (Université Laval, Québec),
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Primer on the Dirichlet SpaceSuurem pilt
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The Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.

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The first systematic account of the Dirichlet space, one of the most fundamental Hilbert spaces of analytic functions.
Preface xi
1 Basic notions
1(14)
1.1 The Dirichlet space
1(3)
1.2 Reproducing kernels
4(2)
1.3 Multiplication
6(1)
1.4 Composition
7(1)
1.5 Douglas' formula
8(3)
1.6 Weighted Dirichlet spaces
11(4)
Notes on
Chapter 1
14(1)
2 Capacity
15(14)
2.1 Potentials, energy and capacity
15(4)
2.2 Equilibrium measures
19(2)
2.3 Cantor sets
21(3)
2.4 Logarithmic capacity
24(5)
Notes on
Chapter 2
27(2)
3 Boundary behavior
29(21)
3.1 The Cauchy transform
29(2)
3.2 Beurling's theorem
31(4)
3.3 Weak-type and strong-type inequalities
35(4)
3.4 Sharpness results
39(6)
3.5 Exponentially tangential approach regions
45(5)
Notes on
Chapter 3
48(2)
4 Zero sets
50(21)
4.1 Zero sets and uniqueness sets
50(4)
4.2 Moduli of zero sets
54(7)
4.3 Boundary zeros I: sets of capacity zero
61(3)
4.4 Boundary zeros II: Carleson sets
64(3)
4.5 Arguments of zero sets
67(4)
Notes on
Chapter 4
69(2)
5 Multipliers
71(22)
5.1 Definition and elementary properties
71(5)
5.2 Carleson measures
76(8)
5.3 Pick interpolation
84(5)
5.4 Zeros of multipliers
89(4)
Notes on
Chapter 5
91(2)
6 Conformal invariance
93(15)
6.1 Mobius invariance
93(3)
6.2 Composition operators
96(6)
6.3 Compactness criteria
102(6)
Notes on
Chapter 6
106(2)
7 Harmonically weighted Dirichlet spaces
108(24)
7.1 Dμ-spaces and the local Dirichlet integral
108(2)
7.2 The local Douglas formula
110(5)
7.3 Approximation in Dμ
115(2)
7.4 Outer functions
117(5)
7.5 Lattice operations in Dμ
122(3)
7.6 Inner functions
125(7)
Notes on
Chapter 7
130(2)
8 Invariant subspaces
132(14)
8.1 The shift operator on Dμ
132(3)
8.2 Characterization of the shift operator
135(5)
8.3 Invariant subspaces of Dμ
140(6)
Notes on
Chapter 8
145(1)
9 Cyclicity
146(35)
9.1 Cyclicity in Dμ
146(5)
9.2 Cyclicity in D and boundary zero sets
151(3)
9.3 The Brown--Shields conjecture
154(5)
9.4 Measure conditions and distance functions
159(7)
9.5 Cyclicity via duality
166(5)
9.6 Bergman--Smirnov exceptional sets
171(10)
Notes on
Chapter 9
179(2)
Appendix A Hardy spaces
181(6)
A.1 Hardy spaces
181(2)
A.2 Inner and outer functions
183(2)
A.3 The Smirnov class
185(2)
Appendix B The Hardy--Littlewood maximal function
187(2)
B.1 Weak-type inequality for the maximal function
187(2)
Appendix C Positive definite matrices
189(4)
C.1 Basic facts about positive definite matrices
189(1)
C.2 Hadamard products
190(3)
Appendix D Regularization and the rising-sun lemma
193(4)
D.1 Increasing regularization
193(2)
D.2 Proof of the regularization lemma
195(2)
References 197(8)
Index of notation 205(2)
Index 207
Omar El-Fallah is professor at Université Mohammed V-Agdal in Rabat, Morocco. He has published more than twenty research articles and has supervised eight doctoral students. Karim Kellay is professor at Université Bordeaux 1, France. He is the author of 24 research articles and has supervised three doctoral students. Javad Mashreghi is Professor of Mathematics at Université Laval in Québec. His main fields of interest are complex analysis, operator theory and harmonic analysis. He has given numerous graduate and undergraduate courses in different institutions in English, French and Persian. Mashreghi has published several research articles, three conference proceedings, two undergraduate textbooks in French and one graduate textbook, entitled Representation Theorems for Hardy Spaces (Cambridge University Press, 2009). He was awarded the prestigious G. de B. Robinson Award of CMS (Canadian Mathematical Society), a publication award, for two long research articles in the Canadian Journal of Mathematics. Thomas Ransford is holder of a senior-level Canada Research Chair at Université Laval in Québec. His main research interests are in complex analysis, functional analysis and potential theory. He is the author of Potential Theory in the Complex Plane (Cambridge University Press, 1995) and of more than 70 research articles. He has supervised nearly 40 graduate students and postdoctoral fellows.
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