Pick / Kufner / John Function Spaces, 1

Volume 1
2. revidierte and ext. ed
ISBN: 978-3-11-025042-8
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 494 Seiten

Reihe: ISSN

Zielgruppe

Graduate Students, Lecturers, and Researchers in Mathematics; Academic Libraries

Autoren/Hrsg.

Pick, Luboš

Pick, Lubos

Kufner, Alois

John, Oldrich

Fucík, Svatopluk

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

1;Preface;5
2;1 Preliminaries;17
2.1;1.1 Vector space;17
2.2;1.2 Topological spaces;18
2.3;1.3 Metric, metric space;22
2.4;1.4 Norm, normed linear space;22
2.5;1.5 Modular spaces;23
2.6;1.6 Inner product, inner product space;26
2.7;1.7 Convergence, Cauchy sequences;27
2.8;1.8 Density, separability;28
2.9;1.9 Completeness;28
2.10;1.10 Subspaces;29
2.11;1.11 Products of spaces;30
2.12;1.12 Schauder bases;30
2.13;1.13 Compactness;31
2.14;1.14 Operators (mappings);32
2.15;1.15 Isomorphism, embeddings;34
2.16;1.16 Continuous linear functionals;35
2.17;1.17 Dual space, weak convergence;36
2.18;1.18 The principle of uniform boundedness;37
2.19;1.19 Reflexivity;37
2.20;1.20 Measure spaces: general extension theory;38
2.21;1.21 The Lebesgue measure and integral;45
2.22;1.22 Modes of convergence;50
2.23;1.23 Systems of seminorms, Hahn-Saks theorem;52
3;2 Spaces of smooth functions;54
3.1;2.1 Multiindices and derivatives;54
3.2;2.2 Classes of continuous and smooth functions;55
3.3;2.3 Completeness;59
3.4;2.4 Separability, bases;61
3.5;2.5 Compactness;67
3.6;2.6 Continuous linear functionals;71
3.7;2.7 Extension of functions;75
4;3 Lebesgue spaces;78
4.1;3.1 Lp-classes;78
4.2;3.2 Lebesgue spaces;82
4.3;3.3 Mean continuity;83
4.4;3.4 Mollifiers;85
4.5;3.5 Density of smooth functions;87
4.6;3.6 Separability;87
4.7;3.7 Completeness;88
4.8;3.8 The dual space;90
4.9;3.9 Reflexivity;94
4.10;3.10 The space L8;94
4.11;3.11 Hardy inequalities;99
4.12;3.12 Sequence spaces;108
4.13;3.13 Modes of convergence;109
4.14;3.14 Compact subsets;110
4.15;3.15 Weak convergence;111
4.16;3.16 Isomorphism of Lp(O) and Lp(0, µ(O));112
4.17;3.17 Schauder bases;113
4.18;3.18 Weak Lebesgue spaces;117
4.19;3.19 Remarks;120
5;4 Orlicz spaces;124
5.1;4.1 Introduction;124
5.2;4.2 Young function, Jensen inequality;125
5.3;4.3 Complementary functions;131
5.4;4.4 The .2-condition;135
5.5;4.5 Comparison of Orlicz classes;138
5.6;4.6 Orlicz spaces;142
5.7;4.7 Hölder inequality in Orlicz spaces;147
5.8;4.8 The Luxemburg norm;150
5.9;4.9 Completeness of Orlicz spaces;153
5.10;4.10 Convergence in Orlicz spaces;154
5.11;4.11 Separability;159
5.12;4.12 The space EF(O);161
5.13;4.13 Continuous linear functionals;167
5.14;4.14 Compact subsets of Orlicz spaces;171
5.15;4.15 Further properties of Orlicz spaces;177
5.16;4.16 Isomorphism properties, Schauder bases;179
5.17;4.17 Comparison of Orlicz spaces;182
6;5 Morrey and Campanato spaces;189
6.1;5.1 Introduction;189
6.2;5.2 Marcinkiewicz spaces;189
6.3;5.3 Morrey and Campanato spaces;192
6.4;5.4 Completeness;194
6.5;5.5 Relations to Lebesgue spaces;194
6.6;5.6 Some lemmas;197
6.7;5.7 Embeddings;201
6.8;5.8 The John-Nirenberg space;203
6.9;5.9 Another definition of the space JN(Q);210
6.10;5.10 Spaces Np;.(Q);213
6.11;5.11 Miscellaneous remarks;215
7;6 Banach function spaces;219
7.1;6.1 Banach function spaces;219
7.2;6.2 Associate space;225
7.3;6.3 Absolute continuity of the norm;232
7.4;6.4 Reflexivity of Banach function spaces;239
7.5;6.5 Separability in Banach function spaces;244
8;7 Rearrangement-invariant spaces;253
8.1;7.1 Nonincreasing rearrangements;253
8.2;7.2 Hardy-Littlewood inequality;257
8.3;7.3 Resonant measure spaces;259
8.4;7.4 Maximal nonincreasing rearrangement;265
8.5;7.5 Hardy lemma;267
8.6;7.6 Rearrangement-invariant spaces;269
8.7;7.7 Hardy-Littlewood-Pólya principle;271
8.8;7.8 Luxemburg representation theorem;272
8.9;7.9 Fundamental function;275
8.10;7.10 Endpoint spaces;280
8.11;7.11 Almost-compact embeddings;291
8.12;7.12 Gould space;308
9;8 Lorentz spaces;317
9.1;8.1 Definition and basic properties;317
9.2;8.2 Embeddings between Lorentz spaces;321
9.3;8.3 The associate space;323
9.4;8.4 The fundamental function;325
9.5;8.5 Absolute continuity of norm;325
9.6;8.6 Remarks on || · ||1;8;327
10;9 Generalized Lorentz-Zygmund spaces;329
10.1;9.1 Measure-preserving transformations;329
10.2;9.2 Basic properties;330
10.3;9.3 Nontriviality;333
10.4;9.4 Fundamental function;334
10.5;9.5 Embeddings between Generalized Lorentz-Zygmund spaces;336
10.6;9.6 The associate space;348
10.7;9.7 When Generalized Lorentz-Zygmund space is Banach function space;369
10.8;9.8 Generalized Lorentz-Zygmund spaces and Orlicz spaces;372
10.9;9.9 Absolute continuity of norm;383
10.10;9.10 Lorentz-Zygmund spaces;388
10.11;9.11 Lorentz-Karamata spaces;389
11;10 Classical Lorentz spaces;391
11.1;10.1 Definition and basic properties;391
11.2;10.2 Functional properties;396
11.3;10.3 Embeddings;404
11.3.1;10.3.1 Embeddings of type . . .;408
11.3.2;10.3.2 Embeddings of type . . G;409
11.3.3;10.3.3 Embeddings of type G . .;412
11.3.4;10.3.4 Embeddings of type G . G;415
11.3.5;10.3.5 The Halperin level function;417
11.3.6;10.3.6 Embeddings of type Gp,8 (v) . .^ (w);420
11.3.7;10.3.7 The single-weight case G 1,8(v) . .1(.);422
11.4;10.4 Associate spaces;425
11.5;10.5 Lorentz and Orlicz spaces;427
11.6;10.6 Spaces measuring oscillation;428
11.7;10.7 The missing case;441
11.8;10.8 Embeddings;443
11.8.1;10.8.1 Embeddings of type S . S;445
11.8.2;10.8.2 Embeddings of type G . S and S . G;447
11.8.3;10.8.3 Embeddings of type . . S and S . .;450
12;11 Variable-exponent Lebesgue spaces;453
12.1;11.1 Introduction;453
12.2;11.2 Basic properties;454
12.3;11.3 Embedding relations;461
12.4;11.4 Density of smooth functions;463
12.5;11.5 Reflexivity and uniform convexity;466
12.6;11.6 Radon-Nikodým property;469
12.7;11.7 Daugavet property;471
13;Bibliography;475
14;Index;488

Über Autor(innen)

Luboš Pick, Charles University, Prague, Czech Republic; Alois Kufner, The Academy of Sciences of the Czech Republic, Prague, Czech Republic; Oldrich John, Charles University, Prague, Czech Republic; Svatopluk Fucík †.