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A Primer on the Dirichlet Space

Book Description

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Series
  4. Title
  5. Copyright
  6. Dedication
  7. Contents
  8. Preface
  9. 1 Basic notions
    1. 1.1 The Dirichlet space
    2. 1.2 Reproducing kernels
    3. 1.3 Multiplication
    4. 1.4 Composition
    5. 1.5 Douglas’ formula
    6. 1.6 Weighted Dirichlet spaces
    7. Notes on Chapter 1
  10. 2 Capacity
    1. 2.1 Potentials, energy and capacity
    2. 2.2 Equilibrium measures
    3. 2.3 Cantor sets
    4. 2.4 Logarithmic capacity
    5. Notes on Chapter 2
  11. 3 Boundary behavior
    1. 3.1 The Cauchy transform
    2. 3.2 Beurling’s theorem
    3. 3.3 Weak-type and strong-type inequalities
    4. 3.4 Sharpness results
    5. 3.5 Exponentially tangential approach regions
    6. Notes on Chapter 3
  12. 4 Zero sets
    1. 4.1 Zero sets and uniqueness sets
    2. 4.2 Moduli of zero sets
    3. 4.3 Boundary zeros I: sets of capacity zero
    4. 4.4 Boundary zeros II: Carleson sets
    5. 4.5 Arguments of zero sets
    6. Notes on Chapter 4
  13. 5 Multipliers
    1. 5.1 Definition and elementary properties
    2. 5.2 Carleson measures
    3. 5.3 Pick interpolation
    4. 5.4 Zeros of multipliers
    5. Notes on Chapter 5
  14. 6 Conformal invariance
    1. 6.1 Möbius invariance
    2. 6.2 Composition operators
    3. 6.3 Compactness criteria
    4. Notes on Chapter 6
  15. 7 Harmonically weighted Dirichlet spaces
    1. 7.1 D[sub(μ)]-spaces and the local Dirichlet integral
    2. 7.2 The local Douglas formula
    3. 7.3 Approximation in D[sub(μ)]
    4. 7.4 Outer functions
    5. 7.5 Lattice operations in D[sub(μ)]
    6. 7.6 Inner functions
    7. Notes on Chapter 7
  16. 8 Invariant subspaces
    1. 8.1 The shift operator on D[sub(μ)]
    2. 8.2 Characterization of the shift operator
    3. 8.3 Invariant subspaces of D[sub(μ)]
    4. Notes on Chapter 8
  17. 9 Cyclicity
    1. 9.1 Cyclicity in D[sub(μ)]
    2. 9.2 Cyclicity in D and boundary zero sets
    3. 9.3 The Brown–Shields conjecture
    4. 9.4 Measure conditions and distance functions
    5. 9.5 Cyclicity via duality
    6. 9.6 Bergman–Smirnov exceptional sets
    7. Notes on Chapter 9
  18. Appendix A Hardy spaces
    1. A.1 Hardy spaces
    2. A.2 Inner and outer functions
    3. A.3 The Smirnov class
  19. Appendix B The Hardy–Littlewood maximal function
    1. B.1 Weak-type inequality for the maximal function
  20. Appendix C Positive definite matrices
    1. C.1 Basic facts about positive definite matrices
    2. C.2 Hadamard products
  21. Appendix D Regularization and the rising-sun lemma
    1. D.1 Increasing regularization
    2. D.2 Proof of the regularization lemma
  22. References
  23. Index of notation
  24. Index