Theory of function spaces

Hans Triebel9783764313814, 3764313811

Table of contents :
Cover……Page 1
Series……Page 2
Title page……Page 3
Date-line……Page 4
Preface……Page 5
Contents……Page 7
1.1. Introduction……Page 11
1.2.1. Distributions……Page 12
1.2.2. $L_p$-Spaces and Quasi-Banach Spaces……Page 13
1.2.3. Maximal Inequalities……Page 14
1.3.1. A Maximal Inequality……Page 15
1.3.2. Inequalities for the Lebesgue Measure……Page 17
1.3.3. Inequalities for Atomic Measures……Page 19
1.3.5. A Representation Formula……Page 20
1.4.1. Definition and Main Inequalities……Page 22
1.4.2. Basic Properties……Page 23
1.4.3. Further Properties……Page 24
1.5.1. Definition and Criterion……Page 25
1.5.2. A Multiplier Theorem……Page 26
1.5.4. Further Multiplier Assertions……Page 28
1.6.2. Maximal Inequalities……Page 29
1.6.3. A Multiplier Theorem……Page 31
1.6.4. Further Multiplier Assertions……Page 32
2.2.1. On the History of Function Spaces……Page 33
2.2.2. The Constructive Spaces……Page 35
2.2.3. The Criterion……Page 38
2.2.4. Decomposition Method, the Principle……Page 40
2.2.5. Approximation Procedures……Page 43
2.3.1. Definition……Page 45
2.3.2. Equivalent Quasi-Norms and Elementary Embeddings……Page 46
2.3.3. Basic Properties……Page 47
2.3.4. The Spaces $F^s_{infty,q}(R_n)$……Page 50
2.3.5. An Orientation and Some Historical Remarks……Page 51
2.3.6. Maximal Inequalities……Page 52
2.3.7. A Fourier Multiplier Theorem……Page 57
2.3.8. Lifting Property and Related Equivalent Quasi-Forms……Page 58
2.3.9. Diversity of the Spaces $B^s_{p,q}(R_n)$ and $F^s_{p,q}(R_n)$……Page 61
2.4.1. Preliminaries……Page 62
2.4.2. Real Interpolation for the Spaces $B^s_{p,q}(R_n)$ and $B^s_{p,q}(R_n)$ with Fixed $p$……Page 64
2.4.4. Complex Interpolation: Definitions……Page 66
2.4.6. Some Preparations……Page 68
2.4.7. Complex Interpolation for the Spaces $B^s_{p,q}(R_n)$ and $F^s_{p,q}(R_n)$……Page 69
2.4.8. Fourier Multipliers for the Spaces $F^s_{p,q}(R_n)$……Page 73
2.4.9. The Spaces $L^Omega_p(R_n,l_q)$: Complex Interpolation and Fourier Multipliers……Page 77
2.5.1. An Orientation……Page 78
2.5.2. Nikol’skij Representations……Page 79
2.5.3. Characterizations by Approximation……Page 80
2.5.4. Lizorkin Representations……Page 85
2.5.5. Discrete Representations and Schauder Bases for $B^s_{p,q}(R_n)$……Page 86
2.5.6. The Bessel-Potential Spaces $H^s_p(R_n)$ and the Soboiev Spaces $W^m_p(R_n)……Page 87
2.5.7. The Besov Spaces $A^s_{p,q}(R_n)$ and the Zygmund Spaces $mathcal{C}^s(R_n)$……Page 89
2.5.8. The Local Hardy Spaces $h_p(R_n)$, the Space $bmo(R_n)$……Page 91
2.5.9. Characterizations by Maximal Functions of Differences……Page 94
2.5.10. Characterizations of the Spaces $F^s_{p,q}(R_n)$ by Differences……Page 101
2.5.11. Characterizations of the Spaces $F^s_{p,q}(R_n)$ by Ball Means of Differences……Page 105
2.5.12. Characterizations of the Spaces $B^s_{p,q}(R_n)$ by Differences; the Spaces $A^s_{p,q}(R_n)$ and $mathcal{C}^s(R_n)$……Page 109
2.5.13. Fubini Type Theorems……Page 114
2.6.1. Definitions and Preliminaries……Page 117
2.6.3. Properties of the Classes $mathfrak{M}_p$……Page 120
2.6.4. Properties of the Classes $mathfrak{M}_p^H$……Page 124
2.6.5. Convolution Algebras……Page 127
2.6.6. The Classes $mathfrak{M}_{p,q}$……Page 128
2.7.1. Embedding Theorems for Different Metrics……Page 129
2.7.2. Traces……Page 131
2.8.2. General Multipliers……Page 140
2.8.3. Multiplication Algebras……Page 145
2.8.4. The Classes $P_{p,alpha}(R_n)$……Page 146
2.8.5. Special Multipliers for $B^s_{p,q}(R_n)$……Page 149
2.8.6. Two Propositions……Page 154
2.8.7. Characteristic Functions as Multipliers……Page 158
2.8.8. Further Multipliers……Page 165
2.9.1. The Spaces $B^s_{p,q}(R_n^+)$ and $F^s_{p,q}(R_n^+)$……Page 166
2.9.2. The Case $min(p,q)>1$……Page 167
2.9.3. The Case $0 < p leq q < infty$ and $n = 1$……Page 170
2.9.4. The Extension Theorem……Page 171
2.9.5. The Case $q