Function Spaces, 1 (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en"> Luboš Pick, Charles University, Prague, Czech Republic; Alois Kufner, The Academy of Sciences of the Czech Republic, Prague, Czech Republic; Oldřich John, Charles University, Prague, Czech Republic; Svatopluk Fučík †.

Preface 5
1 Preliminaries 17
1.1 Vector space 17
1.2 Topological spaces 18
1.3 Metric, metric space 22
1.4 Norm, normed linear space 22
1.5 Modular spaces 23
1.6 Inner product, inner product space 26
1.7 Convergence, Cauchy sequences 27
1.8 Density, separability 28
1.9 Completeness 28
1.10 Subspaces 29
1.11 Products of spaces 30
1.12 Schauder bases 30
1.13 Compactness 31
1.14 Operators (mappings) 32
1.15 Isomorphism, embeddings 34
1.16 Continuous linear functionals 35
1.17 Dual space, weak convergence 36
1.18 The principle of uniform boundedness 37
1.19 Reflexivity 37
1.20 Measure spaces: general extension theory 38
1.21 The Lebesgue measure and integral 45
1.22 Modes of convergence 50
1.23 Systems of seminorms, Hahn-Saks theorem 52
2 Spaces of smooth functions 54
2.1 Multiindices and derivatives 54
2.2 Classes of continuous and smooth functions 55
2.3 Completeness 59
2.4 Separability, bases 61
2.5 Compactness 67
2.6 Continuous linear functionals 71
2.7 Extension of functions 75
3 Lebesgue spaces 78
3.1 Lp-classes 78
3.2 Lebesgue spaces 82
3.3 Mean continuity 83
3.4 Mollifiers 85
3.5 Density of smooth functions 87
3.6 Separability 87
3.7 Completeness 88
3.8 The dual space 90
3.9 Reflexivity 94
3.10 The space L8 94
3.11 Hardy inequalities 99
3.12 Sequence spaces 108
3.13 Modes of convergence 109
3.14 Compact subsets 110
3.15 Weak convergence 111
3.16 Isomorphism of Lp(O) and Lp(0, µ(O)) 112
3.17 Schauder bases 113
3.18 Weak Lebesgue spaces 117
3.19 Remarks 120
4 Orlicz spaces 124
4.1 Introduction 124
4.2 Young function, Jensen inequality 125
4.3 Complementary functions 131
4.4 The .2-condition 135
4.5 Comparison of Orlicz classes 138
4.6 Orlicz spaces 142
4.7 Hölder inequality in Orlicz spaces 147
4.8 The Luxemburg norm 150
4.9 Completeness of Orlicz spaces 153
4.10 Convergence in Orlicz spaces 154
4.11 Separability 159
4.12 The space EF(O) 161
4.13 Continuous linear functionals 167
4.14 Compact subsets of Orlicz spaces 171
4.15 Further properties of Orlicz spaces 177
4.16 Isomorphism properties, Schauder bases 179
4.17 Comparison of Orlicz spaces 182
5 Morrey and Campanato spaces 189
5.1 Introduction 189
5.2 Marcinkiewicz spaces 189
5.3 Morrey and Campanato spaces 192
5.4 Completeness 194
5.5 Relations to Lebesgue spaces 194
5.6 Some lemmas 197
5.7 Embeddings 201
5.8 The John-Nirenberg space 203
5.9 Another definition of the space JN(Q) 210
5.10 Spaces Np .(Q)
5.11 Miscellaneous remarks 215
6 Banach function spaces 219
6.1 Banach function spaces 219
6.2 Associate space 225
6.3 Absolute continuity of the norm 232
6.4 Reflexivity of Banach function spaces 239
6.5 Separability in Banach function spaces 244
7 Rearrangement-invariant spaces 253
7.1 Nonincreasing rearrangements 253
7.2 Hardy-Littlewood inequality 257
7.3 Resonant measure spaces 259
7.4 Maximal nonincreasing rearrangement 265
7.5 Hardy lemma 267
7.6 Rearrangement-invariant spaces 269
7.7 Hardy-Littlewood-Pólya principle 271
7.8 Luxemburg representation theorem 272
7.9 Fundamental function 275
7.10 Endpoint spaces 280
7.11 Almost-compact embeddings 291
7.12 Gould space 308
8 Lorentz spaces 317
8.1 Definition and basic properties 317
8.2 Embeddings between Lorentz spaces 321
8.3 The associate space 323
8.4 The fundamental function 325
8.5 Absolute continuity of norm 325
8.6 Remarks on || · ||1 8
9 Generalized Lorentz-Zygmund spaces 329
9.1 Measure-preserving transformations 329
9.2 Basic properties 330
9.3 Nontriviality 333
9.4 Fundamental function 334
9.5 Embeddings between Generalized Lorentz-Zygmund spaces 336
9.6 The associate space 348
9.7 When Generalized Lorentz-Zygmund space is Banach function space 369
9.8 Generalized Lorentz-Zygmund spaces and Orlicz spaces 372
9.9 Absolute continuity of norm 383
9.10 Lorentz-Zygmund spaces 388
9.11 Lorentz-Karamata spaces 389
10 Classical Lorentz spaces 391
10.1 Definition and basic properties 391
10.2 Functional properties 396
10.3 Embeddings 404
10.3.1 Embeddings of type . . . 408
10.3.2 Embeddings of type . . G 409
10.3.3 Embeddings of type G . . 412
10.3.4 Embeddings of type G . G 415
10.3.5 The Halperin level function 417
10.3.6 Embeddings of type Gp,8 (v) . .^ (w) 420
10.3.7 The single-weight case G 1,8(v) . .1(.) 422
10.4 Associate spaces 425
10.5 Lorentz and Orlicz spaces 427
10.6 Spaces measuring oscillation 428
10.7 The missing case 441
10.8 Embeddings 443
10.8.1 Embeddings of type S . S 445
10.8.2 Embeddings of type G . S and S . G 447
10.8.3 Embeddings of type . . S and S . . 450
11 Variable-exponent Lebesgue spaces 453
11.1 Introduction 453
11.2 Basic properties 454
11.3 Embedding relations 461
11.4 Density of smooth functions 463
11.5 Reflexivity and uniform convexity 466
11.6 Radon-Nikodým property 469
11.7 Daugavet property 471
Bibliography 475
Index 488

lt;P>"Das Buch stellt dadurch eine nützliche Informationsquelle (ergänzt durch ein umfangreiches Literaturverzeichnis)dar, in einem nicht besonders übersichtlichen Teilgebiet."
V. Losert in: Monatsh Math 186 (2018), 561–564