A Primer on the Dirichlet Space

The first systematic account of the Dirichlet space, one of the most fundamental Hilbert spaces of analytic functions.

Omar El-Fallah (Author), Karim Kellay (Author), Javad Mashreghi (Author), Thomas Ransford (Author)

9781107047525, Cambridge University Press

Hardback, published 16 January 2014

226 pages, 5 b/w illus. 110 exercises
23.1 x 15.2 x 2.3 cm, 0.47 kg

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.

Preface
1. Basic notions
2. Capacity
3. Boundary behavior
4. Zero sets
5. Multipliers
6. Conformal invariance
7. Harmonically weighted Dirichlet spaces
8. Invariant subspaces
9. Cyclicity
Appendix A. Hardy spaces
Appendix B. The Hardy–Littlewood maximal function
Appendix C. Positive definite matrices
Appendix D. Regularization and the rising-sun lemma
References
Index of notation
Index.

Subject Areas: Functional analysis & transforms [PBKF], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB], Mathematics [PB]