The Dirichlet space: A Survey

Abstract

In this paper we survey many results on the Dirichlet space of analytic functions. Our focus is more on the classical Dirichlet space on the disc and not the potential generalizations to other domains or several variables. Additionally, we focus mainly on certain function theoretic properties of the Dirichlet space and omit covering the interesting connections between this space and operator theory. The results discussed in this survey show what is known about the Dirichlet space and compares it with the related results for the Hardy space.

35 pages, typoes corrected, some open problems added

References in corpus (3)

• Function Spaces Related to the Dirichlet Space
• Onto Interpolating Sequences for the Dirichlet Space
• Potential Theory on Trees, Graphs and Ahlfors Regular Metric Spaces

Cited by in corpus (11)

• Cyclicity in Dirichlet-type spaces and extremal polynomials
• Winding of simple walks on the square lattice
• Weak product spaces of Dirichlet series
• Weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order $m$: Explicit formulae for reproducing kernels and asymptotic
• Approximation numbers of composition operators on the Dirichlet space
• Multi-parameter Carleson embeddings for $p\ne 2$ on $T^2$ or for $p=2$ on $T^4$, and why the proofs fail
• Operator analogues of Mahler's measure
• Differences between the potential theories on a tree and on a bi-tree
• Weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order $m$ in high dimensions
• Inner functions in weighted Hardy spaces
• Bargmann transfoms associated with reproducing kernel Hilbert space and application to Dirichlet spaces