Inequalities: A Journey into Linear Analysis

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Introduction......Page 13
1.1 Measure......Page 16
1.2 Measurable functions......Page 19
1.3 Integration......Page 21
1.4 Notes and remarks......Page 24
2.1 Cauchy's inequality......Page 25
2.2 Inner-product spaces......Page 26
2.3 The Cauchy--Schwarz inequality......Page 27
Exercises......Page 29
3.1 The arithmetic mean–geometric mean inequality......Page 31
3.2 Applications......Page 33
Exercises......Page 35
4.1 Convex sets and convex functions......Page 36
4.2 Convex functions on an interval......Page 38
4.3 Directional derivatives and sublinear functionals......Page 41
4.4 The Hahn--Banach theorem......Page 43
4.5 Normed spaces, Banach spaces and Hilbert space......Page 46
4.6 The Hahn--Banach theorem for normed spaces......Page 48
4.7 Barycentres and weak integrals......Page 51
4.8 Notes and remarks......Page 52
Exercises......Page 53
5.1 Lp spaces, and Minkowski's inequality......Page 57
5.2 The Lebesgue decomposition theorem......Page 59
5.3  The reverse Minkowski inequality......Page 61
5.4 Hölder's inequality......Page 62
5.5 The inequalities of Liapounov and Littlewood......Page 66
5.6 Duality......Page 67
5.7 The Loomis--Whitney inequality......Page 69
5.8 A Sobolev inequality......Page 72
5.9 Schur's theorem and Schur's test......Page 74
5.10 Hilbert's absolute inequality......Page 77
Exercises......Page 79
6.1 Banach function spaces......Page 82
6.2 Function space duality......Page 84
6.3 Orlicz spaces......Page 85
Exercises......Page 88
7.1 Decreasing rearrangements......Page 90
7.2 Rearrangement-invariant Banach function spaces......Page 92
7.3 Muirhead's maximal function......Page 93
7.4 Majorization......Page 96
7.5 Calderón's interpolation theorem and its converse......Page 100
7.6 Symmetric Banach sequence spaces......Page 103
7.7 The method of transference......Page 105
7.8 Finite doubly stochastic matrices......Page 109
7.9 Schur convexity......Page 110
7.10 Notes and remarks......Page 112
Exercises......Page 113
8.1 The Hardy--Riesz inequality…......Page 115
8.2 The Hardy--Riesz inequality (p=1)......Page 117
8.3 Related inequalities......Page 118
8.4 Strong type and weak type......Page 120
8.5 Riesz weak type......Page 123
8.6 Hardy, Littlewood, and a batsman's averages......Page 124
8.7 Riesz's sunrise lemma......Page 126
8.8 Differentiation almost everywhere......Page 129
8.9 Maximal operators in higher dimensions......Page 130
8.11 Convolution kernels......Page 133
8.12 Hedberg's inequality......Page 137
8.13 Martingales......Page 139
8.15 The martingale convergence theorem......Page 142
Exercises......Page 145
9.1 Hadamard's three lines inequality......Page 147
9.2 Compatible couples and intermediate spaces......Page 148
9.3 The Riesz--Thorin interpolation theorem......Page 150
9.4 Young's inequality......Page 152
9.5 The Hausdorff--Young inequality......Page 153
9.6 Fourier type......Page 155
9.7 The generalized Clarkson inequalities......Page 157
9.8 Uniform convexity......Page 159
Exercises......Page 162
10.1 The Marcinkiewicz interpolation theorem: I......Page 166
10.2 Lorentz spaces......Page 168
10.3 Hardy's inequality......Page 170
10.4 The scale of Lorentz spaces......Page 171
10.5 The Marcinkiewicz interpolation theorem: II......Page 174
Exercises......Page 177
11.1 The conjugate Poisson kernel......Page 179
11.2 The Hilbert transform on L2(R)......Page 180
11.3 The Hilbert transform on Lp(R) for…......Page 182
11.4 Hilbert's inequality for sequences......Page 186
11.5 The Hilbert transform on T......Page 187
11.6 Multipliers......Page 191
11.7 Singular integral operators......Page 192
11.8 Singular integral operators on Lp(Rd) for…......Page 195
Exercises......Page 197
12.1 The contraction principle......Page 199
12.2 The reflection principle, and Lévy's inequalities......Page 201
12.3 Khintchine's inequality......Page 204
12.4 The law of the iterated logarithm......Page 206
12.5 Strongly embedded subspaces......Page 208
12.6 Stable random variables......Page 210
12.7 Sub-Gaussian random variables......Page 211
12.8 Kahane's theorem and Kahane's inequality......Page 213
Exercises......Page 216
13.1 Bonami's inequality......Page 218
13.2 Kahane's inequality revisited......Page 222
13.3 The theorem of Latala and Oleszkiewicz......Page 223
13.4 The logarithmic Sobolev inequality on…......Page 225
13.5 Gaussian measure and the Hermite polynomials......Page 228
13.6 The central limit theorem......Page 231
13.7 The Gaussian hypercontractive inequality......Page 233
13.8 Correlated Gaussian random variables......Page 235
13.9 The Gaussian logarithmic Sobolev inequality......Page 237
13.10 The logarithmic Sobolev inequality in higher dimensions......Page 239
13.11 Beckner's inequality......Page 241
13.12 The Babenko--Beckner inequality......Page 242
Exercises......Page 244
14.1 Hadamard's inequality......Page 245
14.2 Hadamard numbers......Page 246
14.3 Error-correcting codes......Page 249
14.4 Note and remark......Page 250
15.1 Jordan normal form......Page 251
15.2 Riesz operators......Page 252
15.3 Related operators......Page 253
15.4 Compact operators......Page 254
15.5 Positive compact operators......Page 255
15.6 Compact operators between Hilbert spaces......Page 257
15.7 Singular numbers, and the Rayleigh--Ritz minimax formula......Page 258
15.8 Weyl's inequality and Horn's inequality......Page 259
15.9 Ky Fan's inequality......Page 262
15.10 Operator ideals......Page 263
15.11 The Hilbert--Schmidt class......Page 265
15.12 The trace class......Page 268
15.13 Lidskii's trace formula......Page 269
15.14 Operator ideal duality......Page 272
Exercises......Page 273
16.1 Unconditional convergence......Page 275
16.2 Absolutely summing operators......Page 277
16.3 (p,q)-summing operators......Page 278
16.4 Examples of p-summing operators......Page 281
16.5 (p,2)-summing operators between Hilbert spaces......Page 283
16.6 Positive operators on L1......Page 285
16.7 Mercer's theorem......Page 286
16.8 p-summing operators between Hilbert spaces…......Page 288
16.9 Pietsch's domination theorem......Page 289
16.10 Pietsch's factorization theorem......Page 291
16.11 p-summing operators between Hilbert spaces…......Page 293
16.12 The Dvoretzky--Rogers theorem......Page 294
16.13 Operators that factor through a Hilbert space......Page 296
16.14 Notes and remarks......Page 299
Exercises......Page 300
17.1 The approximation, Gelfand and Weyl numbers......Page 301
17.2 Subadditive and submultiplicative properties......Page 303
17.3 Pietsch's inequality......Page 306
17.4 Eigenvalues of p-summing and (p,2)-summing endomorphisms......Page 308
Exercises......Page 313
18.1 Littlewood's 4/3 inequality......Page 314
18.2 Grothendieck's inequality......Page 316
18.3 Grothendieck's theorem......Page 318
18.4 Another proof, using Paley's inequality......Page 319
18.5 The little Grothendieck theorem......Page 322
18.6 Type and cotype......Page 324
18.7 Gaussian type and cotype......Page 326
18.8 Type and cotype of Lp spaces......Page 328
18.9 The little Grothendieck theorem revisited......Page 330
18.10 More on cotype......Page 332
Exercises......Page 335
References......Page 337
Index of inequalities......Page 343
Index......Page 344