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[资源] Functional Analysis: Introduction 【Elias M. Stein & Rami Shakarchi 】

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Princeton Lectures in Analysis
IV
FUNCTIONAL ANALYSIS
Introduction to Further Topics in
Analysis Princeton Lectures in Analysis
IV
FUNCTIONAL ANALYSIS
Introduction to Further Topics in
Analysis

Contents
Foreword
Preface
Chapter 1. LP Spaces and Banach Spaces
1 LP spaces
1.1 The Holder and Minkowski inequalities
1.2 Completeness of LP
1.3 Further remarks
2 The case p = oo
3 Banach spaces
3.1 Examples
3.2 Linear functionals and the dual of a Banach
4 The dual space of LP when 1 < p < oo
5 More about linear functionals
5.1 Separation of convex sets
5.2 The Hahn-Banach Theorem
5.3 Some consequences
5.4 The problem of measure
6 Complex LP and Banach spaces
7 Appendix: The dual of C(X)
7.1 The case of positive linear functionals
7.2 The main result
7.3 An extension
8 Exercises
9 Problems
Chapter 2. LP Spaces in Harmonic Analysis
1 Early Motivations
2 The Riesz interpolation theorem
2.1 Some examples
3 The LP theory of the Hilbert transform
3.1 The L2 formalism
3.2 The LP theorem
3.3 Proof of Theorem 3.2
4 The maximal function and weak-type estimates
4.1 The Lp inequality

5 The Hardy space HJ 73
5.1 Atomic decomposition of H* 74
5.2 An alternative definition of H* 81
5.3 Application to the Hilbert transform 82
6 The space H* 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(￡) 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 Calderon-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 Z^-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 = oo 191
1.3 Behavior of Sn as N —> oo, 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
equations 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 Schrodinger equation 348
6.2 Another dispersion equation 352
6.3 The non-homogeneous Schrodinger 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 ogo
10 Problems 4q5
Notes and References 4qq
Bibliography 413
Symbol Glossary 417
Index 419

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