Chapter 1．Lp Spaces and Banach Spaces 1
1 Lp spaces 2
1.1 The H？lder and Minkowski inequalities 3
1.2 Completeness of Lp 5
1.3 Further remarks 7
2 The case p＝∞ 7
3 Banach spaces 9
3.1 Examples 9
3.2 Linear functionals and the dual of a Banach space 11
4 The dual space of Lp when 1≤p＜∞ 13
5 More about linear functionals 16
5.1 Separation of convex sets 16
5.2 The Hahn-Banach Theorem 20
5.3 Some consequences 21
5.4 The problem of measure 23
6 Complex Lp and Banach spaces 27
7 Appendix:The dual of C(X) 28
7.1 The case of positive linear functionals 29
7.2 The main result 32
7.3 An extension 33
8 Exercises 34
9 Problems 43
Chapter 2．Lp Spaces in Harmonic Analysis 47
1 Early Motivations 48
2 The Riesz interpolation theorem 52
2.1 Some examples 57
3 The Lp theory of the Hilbert transform 61
3.1 The L2 formalism 61
3.2 The Lp theorem 64
3.3 Proof of Theorem 3.2 66
4 The maximal function and weak-type estimates 70
4.1 The Lp inequality 71
5 The Hardy space Hl r 73
5.1 Atomic decomposition of Hl r 74
5.2 An alternative definition of Hl r 81
5.3 Application to the Hilbert transform 82
6 The space Hl r and maximal functions 84
6.1 The space BMO 86
7 Exercises 90
8 Problems 94
Chapter 3．Distributions:Generalized Functions 98
1 Elementary properties 99
1.1 Definitions 100
1.2 Operations on distributions 102
1.3 Supports of distributions 104
1.4 Tempered distributions 105
1.5 Fourier transform 107
1.6 Distributions with point supports 110
2 Important examples of distributions 111
2.1 The Hilbert transform and pv(1/x) 111
2.2 Homogeneous distributions 115
2.3 Fundamental solutions 125
2.4 Fundamental solution to general partial differential equations with constant coefficients 129
2.5 Parametrices and regularity for elliptic equations 131
3 Calderón-Zygmund distributions and Lp estimates 134
3.1 Defining properties 134
3.2 The Lp theory 138
4 Exercises 145
5 Problems 153
Chapter 4．Applications of the Baire Category Theorem 157
1 The Baire category theorem 158
1.1 Continuity of the limit of a sequence of continuous functions 160
1.2 Continuous functions that are nowhere differentiable 163
2 The uniform boundedness principle 166
2.1 Divergence of Fourier series 167
3 The open mapping theorem 170
3.1 Decay of Fourier coefficients of L1-functions 173
4 The closed graph theorem 174
4.1 Grothendieck's theorem on closed subspaces of Lp 174
5 Besicovitch sets 176
6 Exercises 181
7 Problems 185
Chapter 5．Rudiments of Probability Theory 188
1 Bernoulli trials 189
1.1 Coin flips 189
1.2 The case N＝∞ 191
1.3 Behavior of SN as N→∞,first results 194
1.4 Central limit theorem 195
1.5 Statement and proof of the theorem 197
1.6 Random series 199
1.7 Random Fourier series 202
1.8 Bernoulli trials 204
2 Sums of independent random variables 205
2.1 Law of large numbers and ergodic theorem 205
2.2 The role of martingales 208
2.3 The zero-one law 215
2.4 The central limit theorem 215
2.5 Random variables with values in Rd 220
2.6 Random walks 222
3 Exercises 227
4 Problems 235
Chapter 6．An Introduction to Brownian Motion 238
1 The Framework 239
2 Technical Preliminaries 241
3 Construction of Brownian motion 246
4 Some further properties of Brownian motion 251
5 Stopping times and the strong Markov property 253
5.1 Stopping times and the Blumenthal zero-one law 254
5.2 The strong Markov property 258
5.3 Other forms of the strong Markov Property 260
6 Solution of the Dirichlet problem 264
7 Exercises 268
8 Problems 273
Chapter 7．A Glimpse into Several Complex Variables 276
1 Elementary properties 276
2 Hartogs' phenomenon:an example 280
3 Hartogs'theorem:the inhomogeneous Cauchy-Riemann equations 283
4 A boundary version:the tangential Cauchy-Riemann equa-tions 288
5 The Levi form 293
6 A maximum principle 296
7 Approximation and extension theorems 299
8 Appendix:The upper half-space 307
8.1 Hardy space 308
8.2 Cauchy integral 311
8.3 Non-solvability 313
9 Exercises 314
10 Problems 319
Chapter 8．Oscillatory Integrals in Fourier Analysis 321
1 An illustration 322
2 Oscillatory integrals 325
3 Fourier transform of surface-carried measures 332
4 Return to the averaging operator 337
5 Restriction theorems 343
5.1 Radial functions 343
5.2 The problem 345
5.3 The theorem 345
6 Application to some dispersion equations 348
6.1 The Schr？dinger equation 348
6.2 Another dispersion equation 352
6.3 The non-homogeneous Schr？dinger equation 355
6.4 A critical non-linear dispersion equation 359
7 A look back at the Radon transform 363
7.1 A variant of the Radon transform 363
7.2 Rotational curvature 365
7.3 Oscillatory integrals 367
7.4 Dyadic decomposition 370
7.5 Almost-orthogonal sums 373
7.6 Proof of Theorem 7.1 374
8 Counting lattice points 376
8.1 Averages of arithmetic functions 377
8.2 Poisson summation formula 379
8.3 Hyperbolic measure 384
8.4 Fourier transforms 389
8.5 A summation formula 392
9 Exercises 398
10 Problems 405
Notes and References 409
Bibliography 413
Symbol Glossary 417
Index 419