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Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2018Copyright date: ©2018Edition: 1st edDescription: 1 online resource (130 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470443757
Subject(s): Genre/Form: Additional physical formats: Print version:: Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum ToriDDC classification:
  • 515/.73
LOC classification:
  • QA323 .X566 2018
Online resources:
Contents:
Cover -- Title page -- Chapter 0. Introduction -- Basic properties -- Embedding -- Characterizations -- Interpolation -- Multipliers -- Chapter 1. Preliminaries -- 1.1. Noncommutative _{ }-spaces -- 1.2. Quantum tori -- 1.3. Fourier multipliers -- 1.4. Hardy spaces -- Chapter 2. Sobolev spaces -- 2.1. Distributions on quantum tori -- 2.2. Definitions and basic properties -- 2.3. A Poincaré-type inequality -- 2.4. Lipschitz classes -- 2.5. The link with the classical Sobolev spaces -- Chapter 3. Besov spaces -- 3.1. Definitions and basic properties -- 3.2. A general characterization -- 3.3. The characterizations by Poisson and heat semigroups -- 3.4. The characterization by differences -- 3.5. Limits of Besov norms -- 3.6. The link with the classical Besov spaces -- Chapter 4. Triebel-Lizorkin spaces -- 4.1. A multiplier theorem -- 4.2. Definitions and basic properties -- 4.3. A general characterization -- 4.4. Concrete characterizations -- 4.5. Operator-valued Triebel-Lizorkin spaces -- Chapter 5. Interpolation -- 5.1. Interpolation of Besov and Sobolev spaces -- 5.2. The K-functional of ( _{ }, _{ }^{ }) -- 5.3. Interpolation of Triebel-Lizorkin spaces -- Chapter 6. Embedding -- 6.1. Embedding of Besov spaces -- 6.2. Embedding of Sobolev spaces -- 6.3. Compact embedding -- Chapter 7. Fourier multiplier -- 7.1. Fourier multipliers on Sobolev spaces -- 7.2. Fourier multipliers on Besov spaces -- 7.3. Fourier multipliers on Triebel-Lizorkin spaces -- Acknowledgements -- Bibliography -- Back Cover.
Summary: This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus \mathbb{T}^d_\theta (with \theta a skew symmetric real d\times d-matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincar� type inequality for Sobolev spaces.
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Cover -- Title page -- Chapter 0. Introduction -- Basic properties -- Embedding -- Characterizations -- Interpolation -- Multipliers -- Chapter 1. Preliminaries -- 1.1. Noncommutative _{ }-spaces -- 1.2. Quantum tori -- 1.3. Fourier multipliers -- 1.4. Hardy spaces -- Chapter 2. Sobolev spaces -- 2.1. Distributions on quantum tori -- 2.2. Definitions and basic properties -- 2.3. A Poincaré-type inequality -- 2.4. Lipschitz classes -- 2.5. The link with the classical Sobolev spaces -- Chapter 3. Besov spaces -- 3.1. Definitions and basic properties -- 3.2. A general characterization -- 3.3. The characterizations by Poisson and heat semigroups -- 3.4. The characterization by differences -- 3.5. Limits of Besov norms -- 3.6. The link with the classical Besov spaces -- Chapter 4. Triebel-Lizorkin spaces -- 4.1. A multiplier theorem -- 4.2. Definitions and basic properties -- 4.3. A general characterization -- 4.4. Concrete characterizations -- 4.5. Operator-valued Triebel-Lizorkin spaces -- Chapter 5. Interpolation -- 5.1. Interpolation of Besov and Sobolev spaces -- 5.2. The K-functional of ( _{ }, _{ }^{ }) -- 5.3. Interpolation of Triebel-Lizorkin spaces -- Chapter 6. Embedding -- 6.1. Embedding of Besov spaces -- 6.2. Embedding of Sobolev spaces -- 6.3. Compact embedding -- Chapter 7. Fourier multiplier -- 7.1. Fourier multipliers on Sobolev spaces -- 7.2. Fourier multipliers on Besov spaces -- 7.3. Fourier multipliers on Triebel-Lizorkin spaces -- Acknowledgements -- Bibliography -- Back Cover.

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus \mathbb{T}^d_\theta (with \theta a skew symmetric real d\times d-matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincar� type inequality for Sobolev spaces.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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