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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