A primer on the Dirichlet Spaces

Omar El-Fallah (1) , Karim Kellay (2) , Javad Mashreghi (3) , Thomas Ransford (3)

1 Département de Mathématiques
2 IMB - Institut de Mathématiques de Bordeaux
3 Département de Mathématiques et de Statistiques

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.

Domaines: Variables complexes [math.CV]; Analyse fonctionnelle [math.FA]; Analyse classique [math.CA]

Omar El-Fallah, Karim Kellay, Javad Mashreghi, Thomas Ransford. A primer on the Dirichlet Spaces. Cambridge University Press, pp.230, 2014, Cambridge Tracts in Mathematics (No. 203), 9781107239425. ⟨10.1017/CBO9781107239425⟩. ⟨hal-00835054⟩