Representation Theorems in Hardy Spaces Book

Preface xi

1 Fourier series 1

1.1 The Laplacian 1

1.2 Some function spaces and sequence spaces 5

1.3 Fourier coefficients 8

1.4 Convolution on T 13

1.5 Young's inequality 16

2 Abel-Poisson means 21

2.1 Abel-Poisson means of Fourier series 21

2.2 Approximate identities on T 25

2.3 Uniform convergence and pointwise convergence 32

2.4 Weak* convergence of measures 39

2.5 Convergence in norm 43

2.6 Weak* convergence of bounded functions 47

2.7 Parseval's identity 49

3 Harmonic functions in the unit disc 55

3.1 Series representation of harmonic functions 55

3.2 Hardy spaces on D 59

3.3 Poisson representation of h&infinity;(D) functions 60

3.4 Poisson representation of hp(D) functions (1 < p < &infinity;) 65

3.5 Poisson representation of h1(D) functions 66

3.6 Radial limits of hp(D) functions (1 <q; p <q; &infinity;) 70

3.7 Series representation of the harmonic conjugate 77

4 Logarithmic convexity 81

4.1 Subharmonic functions 81

4.2 The maximum principle 84

4.3 A characterization of subharmonic functions 88

4.4 Various means of subharmonic functions 90

4.5 Radial subharmonic functions 95

4.6 Hardy's convexity theorem 97

4.7 A complete characterization of hp(D) spaces 99

5 Analytic functions in the unit disc 103

5.1 Representation of Hp(D) functions (1 < p <q; &infinity;) 103

5.2 The Hilbert transform on T 106

5.3 Radial limits of the conjugate function 110

5.4 The Hilbert transform of C1(T) functions 113

5.5 Analytic measures on T 116

5.6 Representations of H1(D) functions 120

5.7 The uniqueness theorem and its applications 123

6 Norm inequalities for the conjugate function 131

6.1 Kolmogorov'stheorems 131

6.2 Harmonic conjugate of h2(D) functions 135

6.3 M. Riesz's theorem 136

6.4 The Hilbert transform of bounded functions 142

6.5 The Hilbert transform of Dini continuous functions 144

6.6 Zygmund's L log L theorem 149

6.7 M. Riesz's L log L theorem 153

7 Blaschke products and their applications 155

7.1 Automorphisms of the open unit disc 155

7.2 Blaschke products for the open unit disc 158

7.3 Jensen's formula 162

7.4 Riesz's decomposition theorem 166

7.5 Representation of Hp(D) functions (0 < p < 1) 168

7.6 The canonical factorization in Hp(D) (0 < p <q; &infinity;) 172

7.7 The Nevanlinna class 175

7.8 The Hardy and Fejér-Riesz inequalities 181

8 Interpolating linear operators 187

8.1 Operators on Lebesgue spaces 187

8.2 Hadamard's three-line theorem 189

8.3 The Riesz-Thorin interpolation theorem 191

8.4 The Hausdorff-Young theorem 197

8.5 An interpolation theorem for Hardy spaces 200

8.6 The Hardy-Littlewood inequality 205

9 The Fourier transform 207

9.1 Lebesgue spaces on the real line 207

9.2 The Fourier transform on L1(R) 209

9.3 The multiplication formula on L1 (R) 218

9.4 Convolution on R 219

9.5 Young's inequality 221

10 Poisson integrals 225

10.1 An application of the multiplication formula on L1(R) 225

10.2 The conjugate Poisson kernel 227

10.3 Approximate identities on R 229

10.4 Uniform convergence and pointwise convergence 232

10.5 Weak* convergence of measures 238

10.6 Convergence in norm 241

10.7 Weak* convergence of bounded functions 243

11 Harmonic functions in the upper half plane 247

11.1 Hardy spaces on C+ 247

11.2 Poisson representation for semidiscs 248

11.3 Poisson representation of h(&Cbar;+) functions 250

11.4 Poisson representation of hp(C+) functions (1 <q; p <q; &infinity;) 252

11.5 A correspondence between &Cbar;+ and &Dbar; 253

11.6 Poisson representation of positive harmonic functions 255

11.7 Vertical limits of hp (C+) functions (1 <q; p <q; &infinity;) 258

12 The Plancherel transform 263

12.1 The inversion formula 263

12.2 The Fourier-Plancherel transform 266

12.3 The multiplication formula on Lp(R) (1 <q; p <q; 2) 271

12.4 The Fourier transform on Lp(R) (1 <q; p <q; 2) 273

12.5 An application of the multiplication formula on Lp(R) (1 <q; p <q; 2) 274

12.6 A complete characterization of hp(C+) spaces 276

13 Analytic functions in the upper half plane 279

13.1 Representation of Hp(C+) functions (1 < p <q; &infinity;) 279

13.2 Analytic measures on R 284

13.3 Representation of H1 (C+) functions 286

13.4 Spectral analysis of Hp(R) (1 <q; p <q; 2) 287

13.5 A contraction from Hp(C+) into Hp(D) 289

13.6 Blaschke products for the upper half plane 293

13.7 The canonical factorization in Hp(C+) (0 < p <q; &infinity;) 294

13.8 A correspondence between Hp(C+) and Hp(D) 298

14 The Hilbert transform on R 301

14.1 Various definitions of the Hilbert transform 301

14.2 The Hilbert transform of C1c(R) functions 303

14.3 Almost everywhere existence of the Hilbert transform 305

14.4 Kolmogorov's theorem 308

14.5 M. Riesz's theorem 311

14.6 The Hilbert transform of Lipα(t) functions 321

14.7 Maximal functions 329

14.8 The maximal Hilbert transform 336

A Topics from real analysis 339

A.1 A very concise treatment of measure theory 339

A.2 Riesz representation theorems 344

A.3 Weak* convergence of measures 345

A.4 C(T) is dense in Lp(T) (0 < p < &infinity;) 346

A.5 The distribution function 347

A.6 Minkowski's inequality 348

A.7 Jensen's inequality 349

B A panoramic view of the representation theorems 351

B.1 hp(D) 352

B.1.1 h1(D) 352

B.1.2 hp(D) (1 < p < &infinity;) 354

B.1.3 h&infinity;(D) 355

B.2 Hp(D) 356

B.2.1 Hp(D) (1 <q; p < &infinity;) 356

B.2.2 H&infinity;(D) 358

B.3 hp(C+) 359

B.3.1 h1(C+) 359

B.3.2 hp(C+) (1 < p <q; 2) 361

B.3.3 hp(C+) (2 < p < &infinity;) 362

B.3.4 h&infinity;(C+) 363

B.3.5 h+(C+) 363

B.4 Hp(C+) 364

B.4.1 Hp(C+) (1 <q; p <q; 2) 364

B.4.2 Hp(C+) (2 < p < &infinity;) 365

B.4.3 H&infinity;(C+) 366

Bibliography 367

Index 369