Theory of function spaces

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 $q2.10.1. Preliminaries......Page 173
2.10.2. The Main Theorem......Page 174
2.11.1. Preliminaries......Page 176
2.11.2. The Case $1 \\leq p < \\infty$......Page 178
2.11.3. The Case $O < p < 1$......Page 180
2.12.1. Characterizations of $F^s_{p,q}(R_n)$ via Lusin and Littlewood-Paley Functions......Page 182
2.12.2. Characterizations via Gauss-Weierstrass and Cauchy-Poisson Semi-Groups......Page 183
2.12.3. Characterizations via Spline Functions......Page 184
3.1.1. Motivations......Page 188
3.1.2. The Problem of Inner Descriptions......Page 189
3.1.3. The Localization Method......Page 190
3.2.1. $C^\\infty$-Domains......Page 191
3.2.2. Definitions......Page 192
3.2.3. Quasi-Banach Spaces......Page 193
3.2.4. Elementary Embedding......Page 194
3.3.1. Embedding......Page 195
3.3.2. Pointwise Multipliers......Page 197
3.3.3. Traces......Page 199
3.3.4. Extensions......Page 200
3.3.5. Equivalent Quasi-Norms......Page 202
3.3.6. Interpolation......Page 203
3.4.1. A Special Multiplication Property......Page 206
3.4.2. Inner Descriptions, Equivalent Quasi-Norms......Page 208
3.4.3. The Spaces $\\r{B}^s_{p,q}(\\Omega)$ and $\\r{F}^s_{p,q}(\\Omega)$......Page 209
4.1.1. Introduction......Page 212
4.1.2. Definitions......Page 213
4.1.3. Basic Properties of Elliptic Operators......Page 215
4.1.4. Basic Properties of Regular Elliptic Systems......Page 216
4.2.1. Introduction and the Spaces $F^{s,r}_{p,q}(R_n^+)$......Page 217
4.2.2. A Priori Estimates, Part I: $R_n^+$, constant coefficients, Dirichlet problem......Page 220
4.2.3. A Priori Estimates. Part II: $R_n^+$, constant coefficients, general boundary problem......Page 224
4.2.4. A Priori Estimates. Part III: Bounded domain, variable coefficients, general boundary problem......Page 227
4.3.1. Introduction and Hypothesis......Page 230
4.3.2. The Basic Theorem......Page 232
4.3.3. The Main Theorem......Page 233
4.3.4. Boundary Value Problems in Zygmund Spaces......Page 235
5.1.3. Definitions......Page 237
5.1.4. The Spaces $\\r{F}^s_{\\infty,q}(R_n)$......Page 239
5.2.1. Maximal Inequalities......Page 240
5.2.3. Equivalent Norms......Page 241
5.2.4. The Hardy Spaces $H_p(R_n)$, the Spaces $BMO(R_n)$......Page 243
5.2.5. Miscellaneous Properties......Page 244
6.1.2. Definitions......Page 245
6.1.3. Basic Properties......Page 247
6.1.4. Paley-Wiener-Schwartz Theorems for Ultra-Distributions......Page 248
6.2.1. Admissible Weight Functions......Page 249
6.2.2. Some Inequalities......Page 250
6.2.3. The Basic Inequality......Page 251
6.3.2. Basic Properties......Page 252
7.1.1. The Spaces $L_p^\\Omega(R_n, \\rho(x), l_q)$......Page 254
7.1.2. The Spaces $L_p(R_n, \\rho(x), l_q)$......Page 255
7.1.3. The Spaces $L_p(R_n, \\rho(x))$......Page 256
7.2.2. Basic Properties......Page 257
7.2.4. Characterizations by Approximation......Page 258
8.1.1. Introduction and Definitions......Page 260
8.1.2. Basic Properties......Page 261
8.1.3. Inner Descriptions......Page 262
8.2.2. A Mapping Property......Page 263
9.1.2. Periodic Distributions on $R_n$......Page 264
9.1.3. Function Spaces on $T_n$......Page 265
9.2.1. The Main Theorem......Page 266
9.2.4. The Problem of Strong Summability......Page 267
10.1. Anisotropic Function Spaces......Page 269
10.2. Generalizations......Page 270
10.3. Abstract Spaces and Spaces Related to Orthogonal Expansions......Page 271
References......Page 273
Index......Page 283