Analysis

Cover  ......Page 1
Contents  ......Page 6
Preface to the First Edition  ......Page 14
Preface to the Second Edition  ......Page 18
1.1 Introduction  ......Page 20
1.2 Basic notions of measure theory  ......Page 23
1.3 Monotone class theorem  ......Page 28
1.4 Uniqueness of measures  ......Page 30
1.5 Definition of measurable functions and integrals  ......Page 31
1.6 Monotone convergence  ......Page 36
1.7 Fatou\'s lemma  ......Page 37
1.8 Dominated convergence  ......Page 38
1.9 Missing term in Fatou\'s lemma  ......Page 40
1.10 Product measure  ......Page 42
1.11 Commutativity and associativity of product measures  ......Page 43
1.12 Fubini\'s theorem  ......Page 44
1.13 Layer cake representation  ......Page 45
1.14 Bathtub principle  ......Page 47
1.15 Constructing a measure from an outer measure  ......Page 48
1.16 Uniform convergence except on small sets  ......Page 50
1.17 Simple functions and really simple functions  ......Page 51
1.18 Approximation by really simple functions  ......Page 53
1.19 Approximation by C8 functions  ......Page 55
Exercises  ......Page 56
2.1 Definition of Lp-spaces  ......Page 60
2.2 Jensen\'s inequality  ......Page 63
2.3 Holder\'s inequality  ......Page 64
2.4 Minkowski\'s inequality  ......Page 66
2.5 Hanner\'s inequality  ......Page 68
2.6 Differentiability of norms  ......Page 70
2.7 Completeness of Lp-spaces  ......Page 71
2.8 Projection on convex sets  ......Page 72
2.9 Continuous linear functionals and weak convergence  ......Page 73
2.10 Linear functionals separate  ......Page 75
2.11 Lower semicontinuity of norms  ......Page 76
2.12 Uniform boundedness principle  ......Page 77
2.13 Strongly convergent convex combinations  ......Page 79
2.14 The dual of Lp  ......Page 80
2.16 Approximation by C8-functions  ......Page 83
2.17 Separability of Lp(Rn)  ......Page 86
2.18 Bounded sequences have weak limits  ......Page 87
2.19 Approximation by C8-functions  ......Page 88
2.20 Convolutions of functions in dual Lp(Rn)-spaces are continuous  ......Page 89
2.21 Hilbert-spaces  ......Page 90
Exercises  ......Page 94
3.1 Introduction  ......Page 98
3.3 Rearrangements of sets and functions  ......Page 99
3.4 The simplest rearrangement inequality  ......Page 101
3.5 Nonexpansivity of rearrangement  ......Page 102
3.6 Riesz\'s rearrangement inequality in one-dimension  ......Page 103
3.7 Riesz\'s rearrangement inequality  ......Page 106
3.9 Strict rearrangement inequality  ......Page 112
Exercises  ......Page 114
4.1 Introduction  ......Page 116
4.2 Young\'s inequality  ......Page 117
4.3 Hardy-Littlewood-Sobolev inequality  ......Page 125
4.4 Conformal transformations and stereographic projection  ......Page 129
4.5 Conformal invariance of the Hardy-Littlewood-Sobolev inequality  ......Page 133
4.6 Competing symmetries  ......Page 136
4.7 Proof of Theorem 4.3.. Sharp version of the Hardy-Littlewood-Sobolev inequality  ......Page 138
4.8 Action of the conformal group on optimizers  ......Page 139
Exercises  ......Page 140
5.1 Definition of the L1 Fourier transform  ......Page 142
5.2 Fourier transform of a Gaussian  ......Page 144
5.3 PlancherePs theorem  ......Page 145
5.4 Definition of the L2 Fourier transform  ......Page 146
5.6 The Fourier transform in Lp(Rn)  ......Page 147
5.7 The sharp Hausdorff-Young inequality  ......Page 148
5.9 Fourier transform of |x|a_n  ......Page 149
5.10 Extension of 5.9 to Lp(Rn)  ......Page 150
Exercises  ......Page 152
6.1 Introduction  ......Page 154
6.3 Definition of distributions and their convergence  ......Page 155
6.4 Locally summable functions, Lp-loc()  ......Page 156
6.5 Functions are uniquely determined by distributions  ......Page 157
6.6 Derivatives of distributions  ......Page 158
6.7 Definition of W1p(Q) and W1p-loc(Q)  ......Page 159
6.8 Interchanging convolutions with distributions  ......Page 161
6.9 Fundamental theorem of calculus for distributions  ......Page 162
6.10 Equivalence of classical and distributional derivatives  ......Page 163
6.12 Multiplication and convolution of distributions by C8-functions  ......Page 165
6.13 Approximation of distributions by C8-functions  ......Page 166
6.14 Linear dependence of distributions  ......Page 167
6.15 C8() is \'dense\' in W1p-loc()  ......Page 168
6.16 Chain rule  ......Page 169
6.17 Derivative of the absolute value  ......Page 171
6.18 Min and Max of W1p-functions are in W1p  ......Page 172
6.19 Gradients vanish on the inverse of small sets  ......Page 173
6.20 Distributional Laplacian of Green\'s functions  ......Page 175
6.21 Solution of Poisson\'s equation  ......Page 176
6.22 Positive distributions are measures  ......Page 178
6.23 Yukawa potential  ......Page 182
6.24 The dual of W1p{Rn)  ......Page 185
Exercises  ......Page 186
7.2 Definition of H1  ......Page 190
7.3 Completeness of H1  ......Page 191
7.4 Multiplication by functions in C8(fi)  ......Page 192
7.6 Density of C8 in H1  ......Page 193
7.7 Partial integration for functions in H1(Rn)  ......Page 194
7.8 Convexity inequality for gradients  ......Page 196
7.9 Fourier characterization of H1(Rn)  ......Page 198
Heat kernel  ......Page 199
7.11 Definition of H1/2(Rn)  ......Page 200
7.12 Integral formulas for  and    ......Page 203
7.13 Convexity inequality for the relativistic kinetic energy  ......Page 204
7.15 Action of   and  on distributions  ......Page 205
7.16 Multiplication of H1/2 functions by C8-functions  ......Page 206
7.17 Symmetric decreasing rearrangement decreases kinetic energy  ......Page 207
7.18 Weak limits  ......Page 209
7.19 Magnetic fields: The H1A-spaces  ......Page 210
7.20 Definition of H1A(Rn)  ......Page 211
7.21 Diamagnetic inequality  ......Page 212
7.22 Cc8(Rn) is dense in H1A(Rn)  ......Page 213
Exercises  ......Page 214
8.1 Introduction  ......Page 218
8.2 Definition of D1(Rn) and D1/2(Rn)  ......Page 220
8.3 Sobolev\'s inequality for gradients  ......Page 221
8.4 Sobolev\'s inequality for |p|  ......Page 223
8.5 Sobolev inequalities in 1 and 2 dimensions  ......Page 224
8.6 Weak convergence implies strong convergence on small sets  ......Page 227
8.7 Weak convergence implies a.e. convergence  ......Page 231
8.8 Sobolev inequalities for Wmp()  ......Page 232
8.9 Rellich-Kondrashov theorem  ......Page 233
8.10 Nonzero weak convergence after translations  ......Page 234
8.11 Poincare\'s inequalities for Wmp()  ......Page 237
8.12 Poincare-Sobolev inequality for Wmp{)  ......Page 238
8.13 Nash\'s inequality  ......Page 239
8.14 The logarithmic Sobolev inequality  ......Page 242
8.15 A glance at contraction semigroups  ......Page 244
8.16 Equivalence of Nash\'s inequality and smoothing estimates  ......Page 246
8.17 Application to the heat equation  ......Page 248
8.18 Derivation of the heat kernel via logarithmic Sobolev  inequalities  ......Page 251
Exercises  ......Page 254
9.1 Introduction  ......Page 256
9.2 Definition of harmonic, subharmonic, and superharmonic functions  ......Page 257
9.3 Properties of harmonic, subharmonic, and superharmonic functions  ......Page 258
9.4 The strong maximum principle  ......Page 263
9.5 Harnack\'s inequality  ......Page 264
9.6 Subharmonic functions are potentials  ......Page 265
9.7 Spherical charge distributions are \'equivalent\' to point charges  ......Page 268
9.8 Positivity properties of the Coulomb energy  ......Page 269
9.9 Mean value inequality for    ......Page 270
9.10 Lower bounds on Schrodinger \'wave\' functions  ......Page 273
9.11 Unique solution of Yukawa\'s equation  ......Page 274
Exercises  ......Page 275
10.1 Introduction  ......Page 276
10.2 Continuity and first differentiability of solutions of Poisson\'s equation  ......Page 279
10.3 Higher differentiability of solutions of Poisson\'s equation  ......Page 281
11.1 Introduction  ......Page 286
11.2 Schrodinger\'s equation  ......Page 288
11.3 Domination of the potential energy by the kinetic energy  ......Page 289
11.4 Weak continuity of the potential energy  ......Page 293
11.5 Existence of a minimizer for E0  ......Page 294
11.6 Higher eigenvalues and eigenfunctions  ......Page 297
11.7 Regularity of solutions  ......Page 298
11.8 Uniqueness of minimizers  ......Page 299
11.9 Uniqueness of positive solutions  ......Page 300
11.10 The hydrogen atom  ......Page 301
11.11 The Thomas-Fermi problem  ......Page 303
11.12 Existence of an unconstrained Thomas-Fermi minimizer  ......Page 304
11.13 Thomas-Fermi equation  ......Page 305
11.14 The Thomas-Fermi minimizer  ......Page 306
11.15 The capacitor problem  ......Page 308
11.16 Solution of the capacitor problem  ......Page 312
11.17 Balls have smallest capacity  ......Page 315
Exercises  ......Page 316
CHAPTER 12. More about Eigenvalues  ......Page 318
12.1 Min-max principles  ......Page 319
12.2 Generalized min-max  ......Page 321
12.3 Bound for eigenvalue sums in a domain  ......Page 323
12.4 Bound for Schrodinger eigenvalue sums  ......Page 325
12.5 Kinetic energy with antisymmetry  ......Page 330
12.6 The semiclassical approximation  ......Page 333
12.7 Definition of coherent states  ......Page 335
12.8 Resolution of the identity  ......Page 336
12.10 Bounds for the relativistic kinetic energy  ......Page 338
12.11 Large N eigenvalue sums in a domain  ......Page 339
12.12 Large N asymptotics of Schrodinger eigenvalue sums  ......Page 342
Exercises  ......Page 346
List of Symbols  ......Page 350
References  ......Page 354
Index  ......Page 360