Instead of cover......Page 1
Series......Page 2
Title page......Page 3
Copyright page......Page 4
Contents......Page 5
Preface......Page 9
List of Symbols......Page 12
0.1 Vector space......Page 21
0.3 Norm. Normed linear space......Page 23
0.4 Inner product. Inner product space......Page 24
0.5 Convergence. Fundamental sequences......Page 25
0.8 Subspaces......Page 26
0.10 Schauder bases......Page 27
0.11 Compactness......Page 28
0.12 Operators (mappings)......Page 29
0.13 Isomorphism. Imbedding......Page 31
0.14 Continuous linear functionals......Page 33
0.15 Dual space. Weak convergence......Page 34
0.16 Reflexivity......Page 35
Part I Smooth functions......Page 37
1.1 Multi-indices, derivatives......Page 39
1.2 Basic function classes......Page 41
1.3 Completeness......Page 45
1.4 Separability, bases......Page 47
1.5 Compactness......Page 54
1.6 Abstract functions......Page 59
1.7 Continuous linear functional......Page 63
1.8 Extension of functions......Page 67
1.9 Various remarks......Page 72
Part II Integrable functions......Page 77
2.1 Preliminaries—Lebesgue integral......Page 79
2.2 $\mathcal{L}_p$-classes and some integral inequalities......Page 83
2.3 Definition of the Lebesgue spaces......Page 88
2.4 Mean continuity......Page 89
2.5 Mollifiers......Page 91
2.6 Density theorems......Page 93
2.8 Completeness......Page 94
2.9 Dual space of $L_p(\Omega)$ $(1
2.10 Reflexivity......Page 101
2.11 The space $L_\infty(\Omega)$......Page 102
2.12 Relations between various types of convergence......Page 107
2.13 Relatively compact subsets of $L_p(\Omega)$......Page 108
2.14 Weak convergence......Page 110
2.15 Isomorphism between $L_p(\Omega)$ and $L_p(0,\mu(\Omega))$......Page 111
2.16 Schauder bases in $L_p(\Omega)$......Page 112
2.17 Remarks......Page 116
2.18 Weak Lebesgue spaces $L_p^w(\Omega)$ and their properties......Page 123
2.19 Definition and basic properties of the Bochner integral......Page 126
2.20 The spaces $L_p(\Omega;X)$......Page 133
2.21 Abstract functions with finite variation......Page 138
2.22 Dual space of $L_p(0,1;X)$......Page 140
3.1 Introduction......Page 146
3.2 Young's function. Jensen's inequality......Page 148
3.3 Complementary functions. Young's inequality......Page 154
3.4 The $\Delta_2$-condition......Page 157
3.5 Further properties of Orlicz classes......Page 160
3.6 Orlicz spaces......Page 165
3.7 An extension of Hoelder's inequality......Page 169
3.8 The Luxemburg norm......Page 173
3.9 Completeness of Orlicz spaces......Page 176
3.10 Convergence in Orlicz spaces......Page 177
3.11 Separability......Page 181
3.12 The space $E_\Phi(\Omega)$......Page 184
3.13 Continuous linear functional......Page 187
3.14 Compact subsets of Orlicz spaces......Page 192
3.15 Further properties of Orlicz spaces......Page 198
3.16 Isomorphism properties. Schauder bases......Page 202
3.17 Comparison of Orlicz spaces......Page 205
3.18 Further results. Remarks. Generalizations......Page 212
4.2 Marcinkiewicz spaces and their connection with the spaces $L_p^w(\Omega)$......Page 229
4.3 Morrey and Campanato spaces. Definitions and basic properties......Page 233
4.5 Some lemmas......Page 237
4.6 Relations between the spaces $L_{p,\lambda}^C(\Omega)$ and the spaces $L_{p,\lambda}^M(\Omega)$, $C^{0,\alpha}(\bar{\Omega})$......Page 242
4.7 $L_{p,N}^C(\Omega)$ and the John-Nirenberg space......Page 244
4.8 Another definition of the space $JN(Q)$. Remarks......Page 250
4.9 Spaces $N_{p,\lambda}(Q)$ and their relation to the spaces $L_{p,\lambda}^C(Q)$......Page 254
4.10 Miscellaneous remarks......Page 256
4.11 Connections with partial differential equations......Page 261
Part III Differentiabie functions......Page 267
5 Classical Sobolev spaces......Page 269
5.1 Abstract construction of Sobolev spaces......Page 270
5.2 Spaces $W^{k,p}(\Omega)$ and $W_0^{k,p}(\Omega)$......Page 275
5.3 A short excursion in the theory of distributions......Page 280
5.4 Spaces $H^{k,p}(\Omega)$......Page 284
5.5 Relations between $W^{k,p}(\Omega)$ and $H^{k,p}(\Omega)$. Special domains......Page 287
5.6 Beppo Levi spaces......Page 293
5.7 Imbedding theorems......Page 297
5.8 Compact imbeddings......Page 311
5.9 Dual spaces of $H^{k,p}(\Omega)$ and $H_0^{k,p}(\Omega)$......Page 313
5.11 Equivalent norms. Poincare's inequality......Page 316
5.12 Various remarks......Page 320
6.2 Preliminaries. Description of the boundary. Notation......Page 324
6.3 Spaces $L_p(\partial\Omega)$. Surface integral......Page 330
6.4 Imbedding of Sobolev spaces in the spaces $L_q(\partial\Omega)$......Page 336
6.5 Behavior of traces in the case of variable boundary......Page 340
6.6 Density theorem. Characterization of $W_0^{1,p}(\Omega)$......Page 344
6.7 Spaces $W^{k,p}(\partial\Omega)$ and the traces......Page 347
6.8 Spaces $W^{k,p}(\Omega)$ with $k$ noninteger and their traces......Page 350
6.9 Inverse theorem......Page 358
6.10 Remarks......Page 362
7 Sobolev-Orlicz spaces......Page 366
7.1 Definitions. Density results......Page 367
7.2 Imbedding theorems......Page 372
7.3 Traces......Page 383
7.4 Compact imbeddings......Page 387
7.5 Various remarks......Page 389
8 Some generalizations......Page 396
8.1 A brief survey of Sobolev spaces......Page 397
8.2 Nikol'skii spaces......Page 401
8.3 Slobodeckii spaces......Page 405
8.4 Besov spaces......Page 408
8.5 Anisotropic spaces. The problem of the domain......Page 413
8.6 Spaces of Bessel potentials (Lizorkin spaces)......Page 417
8.7 Spaces with "negative derivatives". Triebel spaces......Page 422
8.8 Anisotropic spaces revisited......Page 426
8.9 Still other spaces......Page 429
8.10 Sobolev weight spaces......Page 437
References......Page 444
Subject index......Page 463
Author index......Page 470
