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The Dirichlet Space and Related Function Spaces

Nicola Arcozzi University of Bologna, Bologna, Italy
Richard Rochberg Washington University in Saint Louis, Saint Louis, MO
Eric T. Sawyer McMaster University, Hamilton, ON, Canada
Brett D. Wick Washington University in Saint Louis, Saint Louis, MO
Available Formats:
Hardcover ISBN: 978-1-4704-5082-3
Product Code: SURV/239
List Price: $129.00 MAA Member Price:$116.10
AMS Member Price: $103.20 Electronic ISBN: 978-1-4704-5360-2 Product Code: SURV/239.E List Price:$129.00
MAA Member Price: $116.10 AMS Member Price:$103.20
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AMS Member Price: $154.80 Click above image for expanded view The Dirichlet Space and Related Function Spaces Nicola Arcozzi University of Bologna, Bologna, Italy Richard Rochberg Washington University in Saint Louis, Saint Louis, MO Eric T. Sawyer McMaster University, Hamilton, ON, Canada Brett D. Wick Washington University in Saint Louis, Saint Louis, MO Available Formats:  Hardcover ISBN: 978-1-4704-5082-3 Product Code: SURV/239  List Price:$129.00 MAA Member Price: $116.10 AMS Member Price:$103.20
 Electronic ISBN: 978-1-4704-5360-2 Product Code: SURV/239.E
 List Price: $129.00 MAA Member Price:$116.10 AMS Member Price: $103.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$193.50 MAA Member Price: $174.15 AMS Member Price:$154.80
• Book Details

Mathematical Surveys and Monographs
Volume: 2392019; 536 pp
MSC: Primary 30; 31; 32; 39; 46; 47;

The study of the classical Dirichlet space is one of the central topics on the intersection of the theory of holomorphic functions and functional analysis. It was introduced about100 years ago and continues to be an area of active current research. The theory is related to such important themes as multipliers, reproducing kernels, and Besov spaces, among others. The authors present the theory of the Dirichlet space and related spaces starting with classical results and including some quite recent achievements like Dirichlet-type spaces of functions in several complex variables and the corona problem.

The first part of this book is an introduction to the function theory and operator theory of the classical Dirichlet space, a space of holomorphic functions on the unit disk defined by a smoothness criterion. The Dirichlet space is also a Hilbert space with a reproducing kernel, and is the model for the dyadic Dirichlet space, a sequence space defined on the dyadic tree. These various viewpoints are used to study a range of topics including the Pick property, multipliers, Carleson measures, boundary values, zero sets, interpolating sequences, the local Dirichlet integral, shift invariant subspaces, and Hankel forms. Recurring themes include analogies, sometimes weak and sometimes strong, with the classical Hardy space; and the analogy with the dyadic Dirichlet space.

The final chapters of the book focus on Besov spaces of holomorphic functions on the complex unit ball, a class of Banach spaces generalizing the Dirichlet space. Additional techniques are developed to work with the nonisotropic complex geometry, including a useful invariant definition of local oscillation and a sophisticated variation on the dyadic Dirichlet space. Descriptions are obtained of multipliers, Carleson measures, interpolating sequences, and multiplier interpolating sequences; $\overline\partial$ estimates are obtained to prove corona theorems.

Graduate students and researchers interested in classical functional analysis.

• The Dirichlet space; Foundations
• Geometry and analysis on the disk
• Hilbert spaces of holomorphic functions
• Intermezzo: Hardy spaces
• Carleson measures
• Analysis on trees
• The Pick property
• Interpolation
• The Dirichlet space; Selected topics
• Onto interpolation
• Boundary values
• Alternative norms and applications
• Shift operators and invariant subspaces
• Invariant subspaces of the Dirichlet shift
• Bilinear forms on $\mathcal {D}$
• Besov spaces on the ball
• Besov spaces on balls and trees
• Interpolating sequences
• Spaces on trees
• Corona theorems for Besov spaces in $\mathbb {C}^n$
• Some functional analysis
• Schur’s test

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Volume: 2392019; 536 pp
MSC: Primary 30; 31; 32; 39; 46; 47;

The study of the classical Dirichlet space is one of the central topics on the intersection of the theory of holomorphic functions and functional analysis. It was introduced about100 years ago and continues to be an area of active current research. The theory is related to such important themes as multipliers, reproducing kernels, and Besov spaces, among others. The authors present the theory of the Dirichlet space and related spaces starting with classical results and including some quite recent achievements like Dirichlet-type spaces of functions in several complex variables and the corona problem.

The first part of this book is an introduction to the function theory and operator theory of the classical Dirichlet space, a space of holomorphic functions on the unit disk defined by a smoothness criterion. The Dirichlet space is also a Hilbert space with a reproducing kernel, and is the model for the dyadic Dirichlet space, a sequence space defined on the dyadic tree. These various viewpoints are used to study a range of topics including the Pick property, multipliers, Carleson measures, boundary values, zero sets, interpolating sequences, the local Dirichlet integral, shift invariant subspaces, and Hankel forms. Recurring themes include analogies, sometimes weak and sometimes strong, with the classical Hardy space; and the analogy with the dyadic Dirichlet space.

The final chapters of the book focus on Besov spaces of holomorphic functions on the complex unit ball, a class of Banach spaces generalizing the Dirichlet space. Additional techniques are developed to work with the nonisotropic complex geometry, including a useful invariant definition of local oscillation and a sophisticated variation on the dyadic Dirichlet space. Descriptions are obtained of multipliers, Carleson measures, interpolating sequences, and multiplier interpolating sequences; $\overline\partial$ estimates are obtained to prove corona theorems.

Graduate students and researchers interested in classical functional analysis.

• The Dirichlet space; Foundations
• Geometry and analysis on the disk
• Hilbert spaces of holomorphic functions
• Intermezzo: Hardy spaces
• Carleson measures
• Analysis on trees
• The Pick property
• Interpolation
• The Dirichlet space; Selected topics
• Onto interpolation
• Boundary values
• Alternative norms and applications
• Shift operators and invariant subspaces
• Invariant subspaces of the Dirichlet shift
• Bilinear forms on $\mathcal {D}$
• Besov spaces on the ball
• Besov spaces on balls and trees
• Interpolating sequences
• Spaces on trees
• Corona theorems for Besov spaces in $\mathbb {C}^n$
• Some functional analysis
• Schur’s test
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