# The Dirichlet Space and Related Function Spaces

###### Nicola Arcozzi, Richard Rochberg, Eric T. Sawyer, and Brett D. Wick
Publisher:
AMS
Publication Date:
2019
Number of Pages:
536
Format:
Hardcover
Series:
Mathematical Surveys and Monographs
Price:
129.00
ISBN:
978-1-4704-5082-3
Category:
Monograph
[Reviewed by
Mihaela Poplicher
, on
12/22/2019
]
This is a wonderful monograph published by AMS, a culmination of many years of work, research and collaboration of the four authors. The book will be very useful for mathematicians doing research on Function Spaces, and Dirichlet Space $D$ in particular, but it can also be used for graduate-level courses (for students who have already taken complex analysis, measure theory, functional analysis), as well as for seminars.

The book has 17 (seventeen) chapters grouped in three parts:
• Part 1. The Dirichlet Space: Foundations (seven chapters);
• Part 2. The Dirichlet Space: Selected topics (six chapters);
• Part 3. Besov Spaces on the Ball (four chapters).
Also, there are two appendices:
• Some Functional Analysis;
• Schur’s Test.
In the first part of the book, the focus is on the Dirichlet Space as a reproducing kernel Hilbert space, investigating multipliers, Carleson measures, zero sets, interpolating sequences for both the space and for its multiplier algebra.

The second part of the book considers more specialized topics related to the Dirichlet Space, including interpolation, boundary behavior, alternative norms, the local Dirichlet integral, shift-invariant subspaces, and Hankel forms.

The third part of the book extends the discussion to Besov Spaces on balls and trees, interpolating sequences, spaces on trees, Corona Theorems for Besov Spaces in $C^{n}$. Therefore, in the final part of the book, the authors expand their viewpoint in two fundamental ways: the discussions is moved beyond functions of a single variable towards considerations of spaces of holomorphic functions on the ball of in $C^{n}$ and the treatment of Banach spaces of functions beyond Hilbert spaces is introduced. With these extensions, new techniques need to be introduced. The final chapters contain only a few theorems, some with elaborate proofs. The authors consider that the main importance of these chapters lies beyond the specific results,  and is in the techniques and tools that are developed.

The authors emphasize the two themes that run as subtexts throughout the monograph:
• The background role of the Hardy space $H^{2}$. The study of this space is the historical and conceptual basis for much of modern function theoretic operator theory and understanding how classical $H^{2}$ results do or do not generalize is a continuing major theme. Also, there is the more recent insight that both $H^{2}$ and $D$ are Hilbert spaces whose reproducing kernels have the complete Pick property and, as such, they share some basic properties;
• The study of discrete ”tree” models of the disk and ball, and function spaces on those trees. Spaces on trees are very valuable tools, both conceptually and technically.

In the first part of the book there are some exercises, inteded "to encourage the reader to keep the pencil and paper nearby."

The book has an extensive bibliography of more than 350 papers, books, and articles cited.  This shows the meticulous work of the authors and can help the reader gain a deeper understanding of the material.

In summary, this monograph is very informative, wonderful to read, a great achievement of the authors. Mathematicians young and old will be enriched from studying it, pencil in hand.

Mihaela Poplicher earned her PhD in Functional Analysis at the University of New Hampshire, under the supervision of Professor Eric Nordgren. After a couple of years at Southern Illinois University in Carbondale, she got a position at the University of Cincinnati, where she is now Associate Professor in the Department of Mathematics.
He email is:   mihaela.poplicher@uc.edu