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

This is the first part of the second revised and extended edition of the well established book "Function Spaces" by Alois Kufner, Oldrich John, and Svatopluk Fucík. Like the first edition this monograph is an introduction to function spaces defined in terms of differentiability and integrability classes. It provides a catalogue of various spaces and benefits as a handbook for those who use function spaces in their research or lecture courses.

This first volume is devoted to the study of function spaces, based on intrinsic properties of a function such as its size, continuity, smoothness, various forms of a control over the mean oscillation, and so on. The second volume will be dedicated to the study of function spaces of Sobolev type, in which the key notion is the weak derivative of a function of several variables.

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

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 †.