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. AuthorsOmar El-Fallah, Université Mohammed V-Agdal, Rabat, Morocco 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, Université de Bordeaux 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, Université Laval, Québec 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, Université Laval, Québec 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. Related Books Linear and Nonlinear Inverse Problems with Practical Applications Invariant Subspaces of Matrices with Applications Brownian Motion, Hardy Spaces and Bounded Mean Oscillation Introduction to Hp Spaces Stochastic Partial Differential Equations with Lévy Noise An Evolution Equation Approach Independent Random Variables and Rearrangement Invariant Spaces The Hardy-Littlewood Method Introduction to Circle Packing The Theory of Discrete Analytic Functions Integration and Harmonic Analysis on Compact Groups Linear and Nonlinear Inverse Problems with Practical Applications Invariant Subspaces of Matrices with Applications 123456789 PreviousNextrelated journals Proceedings of the Royal Society of Edinburgh, Section: A Mathematics A flagship publication of the Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics… Combinatorics, Probability and Computing Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability… European Journal of Applied Mathematics Since 2008 EJAM surveys have been expanded to cover Applied and Industrial Mathematics. Coverage of the journal has… Mathematika Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding… Proceedings of the Royal Society of Edinburgh, Section: A Mathematics A flagship publication of the Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics… Combinatorics, Probability and Computing Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability…