Sobolev, Besov, and Triebel-Lizorkin spaces on quantum tori / Xiao Xiong, Quanhua Xu, Zhi Yin.

Format:

Book

Author/Creator:

Xiong, Xiao, 1989- author.

Xu, Quanhua, author.

Yin, Zhi, 1984- author.

Series:

Memoirs of the American Mathematical Society ; Volume 252, Number 1203.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 252, Number 1203

Language:

English

Subjects (All):

Function spaces.

Sobolev spaces.

Lipschitz spaces.

Torus (Geometry).

Physical Description:

1 online resource (130 pages).

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2018]

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.

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.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-4375-9