Sobolev Besov and Triebel Lizorkin Spaces on Quantum Tori 1st Edition by Xiao Xiong, Quanhua Xu, Zhi Yin 1470443759 9781470443757

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ISBN 10: 1470443759 
ISBN 13: 9781470443757
Author: Xiao Xiong; Quanhua Xu; Zhi Yin

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 $dtimes d$-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 Poincare type inequality for Sobolev spaces.

Sobolev Besov and Triebel Lizorkin Spaces on Quantum Tori 1st Table of contents:

Chapter 0. Introduction
Basic properties
Embedding
Characterizations
Interpolation
Multipliers

Chapter 1. Preliminaries
1.1. Noncommutative LpL^pLp-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 KKK-functional of (Lp,Wkp)(L^p, W^p_k)(Lp,Wkp​)
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 multipliers
7.1. Fourier multipliers on Sobolev spaces
7.2. Fourier multipliers on Besov spaces
7.3. Fourier multipliers on Triebel-Lizorkin spaces

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Xiao Xiong,Quanhua Xu,Zhi Yin,Sobolev Besov