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 Poincare type inequality for Sobolev spaces.
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1 | (8) |
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3 | (1) |
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4 | (1) |
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4 | (2) |
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6 | (1) |
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6 | (3) |
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9 | (10) |
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1.1 Noncommutative Lp-spaces |
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9 | (1) |
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10 | (2) |
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12 | (2) |
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14 | (5) |
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19 | (16) |
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2.1 Distributions on quantum tori |
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19 | (2) |
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2.2 Definitions and basic properties |
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21 | (4) |
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2.3 A Poincare-type inequality |
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25 | (3) |
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28 | (3) |
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2.5 The link with the classical Sobolev spaces |
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31 | (4) |
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35 | (24) |
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3.1 Definitions and basic properties |
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35 | (7) |
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3.2 A general characterization |
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42 | (6) |
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3.3 The characterizations by Poisson and heat semigroups |
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48 | (3) |
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3.4 The characterization by differences |
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51 | (3) |
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3.5 Limits of Besov norms |
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54 | (1) |
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3.6 The link with the classical Besov spaces |
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55 | (4) |
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Chapter 4 Triebel-Lizorkin spaces |
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59 | (24) |
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59 | (9) |
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4.2 Definitions and basic properties |
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68 | (4) |
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4.3 A general characterization |
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72 | (4) |
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4.4 Concrete characterizations |
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76 | (4) |
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4.5 Operator-valued Triebel-Lizorkin spaces |
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80 | (3) |
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83 | (10) |
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5.1 Interpolation of Besov and Sobolev spaces |
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83 | (5) |
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5.2 The K-functional of (Lp, Wkp) |
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88 | (3) |
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5.3 Interpolation of Triebel-Lizorkin spaces |
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91 | (2) |
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93 | (10) |
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6.1 Embedding of Besov spaces |
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93 | (2) |
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6.2 Embedding of Sobolev spaces |
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95 | (5) |
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100 | (3) |
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Chapter 7 Fourier multiplier |
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103 | (10) |
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7.1 Fourier multipliers on Sobolev spaces |
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103 | (4) |
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7.2 Fourier multipliers on Besov spaces |
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107 | (3) |
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7.3 Fourier multipliers on Triebel-Lizorkin spaces |
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110 | (3) |
| Acknowledgements |
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113 | (2) |
| Bibliography |
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115 | |
Xiao Xiong, Wuhan University, China, and Universite de Franche-Comte, Besancon, France.
Quanhua Xu, Wuhan University, China, and Universite de Franche-Comte, Besancon, France.
Zhi Yin, Wuhan University, China.
