泛函分析 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Chapter 1．Lp Spaces and Banach Spaces1

1 Lp spaces2

1.1 The H？lder and Minkowski inequalities3

1.2 Completeness of Lp5

1.3 Further remarks7

2 The case p＝∞7

3 Banach spaces9

3.1 Examples9

3.2 Linear functionals and the dual of a Banach space11

4 The dual space of Lp when 1≤p＜∞13

5 More about linear functionals16

5.1 Separation of convex sets16

5.2 The Hahn-Banach Theorem20

5.3 Some consequences21

5.4 The problem of measure23

6 Complex Lp and Banach spaces27

7 Appendix:The dual of C(X)28

7.1 The case of positive linear functionals29

7.2 The main result32

7.3 An extension33

8 Exercises34

9 Problems43

Chapter 2．Lp Spaces in Harmonic Analysis47

1 Early Motivations48

2 The Riesz interpolation theorem52

2.1 Some examples57

3 The Lp theory of the Hilbert transform61

3.1 The L2 formalism61

3.2 The Lp theorem64

3.3 Proof of Theorem 3.266

4 The maximal function and weak-type estimates70

4.1 The Lp inequality71

5 The Hardy space Hl r73

5.1 Atomic decomposition of Hl r74

5.2 An alternative definition of Hl r81

5.3 Application to the Hilbert transform82

6 The space Hl r and maximal functions84

6.1 The space BMO86

7 Exercises90

8 Problems94

Chapter 3．Distributions:Generalized Functions98

1 Elementary properties99

1.1 Definitions100

1.2 Operations on distributions102

1.3 Supports of distributions104

1.4 Tempered distributions105

1.5 Fourier transform107

1.6 Distributions with point supports110

2 Important examples of distributions111

2.1 The Hilbert transform and pv(1/x)111

2.2 Homogeneous distributions115

2.3 Fundamental solutions125

2.4 Fundamental solution to general partial differential equations with constant coefficients129

2.5 Parametrices and regularity for elliptic equations131

3 Calderón-Zygmund distributions and Lp estimates134

3.1 Defining properties134

3.2 The Lp theory138

4 Exercises145

5 Problems153

Chapter 4．Applications of the Baire Category Theorem157

1 The Baire category theorem158

1.1 Continuity of the limit of a sequence of continuous functions160

1.2 Continuous functions that are nowhere differentiable163

2 The uniform boundedness principle166

2.1 Divergence of Fourier series167

3 The open mapping theorem170

3.1 Decay of Fourier coefficients of L1-functions173

4 The closed graph theorem174

4.1 Grothendieck's theorem on closed subspaces of Lp174

5 Besicovitch sets176

6 Exercises181

7 Problems185

Chapter 5．Rudiments of Probability Theory188

1 Bernoulli trials189

1.1 Coin flips189

1.2 The case N＝∞191

1.3 Behavior of SN as N→∞,first results194

1.4 Central limit theorem195

1.5 Statement and proof of the theorem197

1.6 Random series199

1.7 Random Fourier series202

1.8 Bernoulli trials204

2 Sums of independent random variables205

2.1 Law of large numbers and ergodic theorem205

2.2 The role of martingales208

2.3 The zero-one law215

2.4 The central limit theorem215

2.5 Random variables with values in Rd220

2.6 Random walks222

3 Exercises227

4 Problems235

Chapter 6．An Introduction to Brownian Motion238

1 The Framework239

2 Technical Preliminaries241

3 Construction of Brownian motion246

4 Some further properties of Brownian motion251

5 Stopping times and the strong Markov property253

5.1 Stopping times and the Blumenthal zero-one law254

5.2 The strong Markov property258

5.3 Other forms of the strong Markov Property260

6 Solution of the Dirichlet problem264

7 Exercises268

8 Problems273

Chapter 7．A Glimpse into Several Complex Variables276

1 Elementary properties276

2 Hartogs' phenomenon:an example280

3 Hartogs'theorem:the inhomogeneous Cauchy-Riemann equations283

4 A boundary version:the tangential Cauchy-Riemann equa-tions288

5 The Levi form293

6 A maximum principle296

7 Approximation and extension theorems299

8 Appendix:The upper half-space307

8.1 Hardy space308

8.2 Cauchy integral311

8.3 Non-solvability313

9 Exercises314

10 Problems319

Chapter 8．Oscillatory Integrals in Fourier Analysis321

1 An illustration322

2 Oscillatory integrals325

3 Fourier transform of surface-carried measures332

4 Return to the averaging operator337

5 Restriction theorems343

5.1 Radial functions343

5.2 The problem345

5.3 The theorem345

6 Application to some dispersion equations348

6.1 The Schr？dinger equation348

6.2 Another dispersion equation352

6.3 The non-homogeneous Schr？dinger equation355

6.4 A critical non-linear dispersion equation359

7 A look back at the Radon transform363

7.1 A variant of the Radon transform363

7.2 Rotational curvature365

7.3 Oscillatory integrals367

7.4 Dyadic decomposition370

7.5 Almost-orthogonal sums373

7.6 Proof of Theorem 7.1374

8 Counting lattice points376

8.1 Averages of arithmetic functions377

8.2 Poisson summation formula379

8.3 Hyperbolic measure384

8.4 Fourier transforms389

8.5 A summation formula392

9 Exercises398

10 Problems405

Notes and References409

Bibliography413

Symbol Glossary417

Index419