This book provides an enlightening survey of remarkable new results that have only recently been discovered in the past two decades about harmonic measure in the complex plane. Many of these results, due to Bishop, Carleson, Jones, Makarov, Wolff and others, appear here in paperback for the first time. The book is accessible to students who have completed standard graduate courses in real and complex analysis. The first four chapters provide the needed background material on univalent functions, potential theory, and extremal length, and each chapter has many exercises to further inform and teach the readers.
Author(s): John B. Garnett, Donald E. Marshall
Series: New Mathematical Monographs
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 589
Cover......Page 1
Half-title......Page 3
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Preface......Page 15
1. The Half-Plane and the Disc......Page 19
2. Fatou’s Theorem and Maximal Functions......Page 24
3. Carathéodory’s Theorem......Page 31
4. Distortion and the Hyperbolic Metric......Page 34
5. The Hayman–Wu Theorem......Page 41
Notes......Page 43
Exercises and Further Results......Page 44
1. The Schwarz Alternating Method......Page 55
2. Green’s Functions and Poisson Kernels......Page 59
3. Conjugate Functions......Page 68
4. Boundary Smoothness......Page 77
Exercises and Further Results......Page 84
III Potential Theory......Page 91
1. Capacity and Green’s Functions......Page 92
2. The Logarithmic Potential......Page 95
3. The Energy Integral......Page 97
4. The Equilibrium Distribution......Page 100
5. Wiener’s Solution to the Dirichlet Problem......Page 107
6. Regular Points......Page 111
7. Wiener Series......Page 115
8. Polar Sets and Sets of Harmonic Measure Zero......Page 120
9. Estimates for Harmonic Measure......Page 122
Exercises and Further Results......Page 130
1. Definitions and Examples......Page 147
2. Uniqueness of Extremal Metrics......Page 151
3. Four Rules for Extremal Length......Page 152
4. Extremal Metrics for Extremal Distance......Page 157
5. Extremal Distance and Harmonic Measure......Page 161
6. The… Estimate......Page 164
Notes......Page 167
Exercises and Further Results......Page 168
1. Asymptotic Values of Entire Functions......Page 175
2. Lower Bounds......Page 177
3. Reduced Extremal Distance......Page 180
4. Teichmüller’s Modulsatz......Page 185
5. Boundary Conformality and Angular Derivatives......Page 191
6. Conditions More Geometric......Page 203
Notes......Page 211
Exercises and Further Results......Page 212
1. The F. and M. Riesz Theorem......Page 218
2. Privalov’s Theorem and Plessner’s Theorem......Page 221
3. Accessible Points......Page 223
4. Cone Points and McMillan’s Theorem......Page 225
5. Compression and Expansion......Page 230
6. Pommerenke’s Extension......Page 234
Exercises and Further Results......Page 239
1. Bloch Functions......Page 247
2. Bloch Functions and Univalent Functions......Page 250
3. Quasicircles......Page 259
4. Chord-Arc Curves and the A Condition......Page 264
5. BMO Domains......Page 271
Notes......Page 275
Exercises and Further Results......Page 276
1. The Law of the Iterated Logarithm for Bloch Functions......Page 287
2. Harmonic Measure and Hausdorff Measure......Page 290
3. The Number of Bad Discs......Page 299
5. Beta Numbers and Polygonal Trees......Page 307
6. The Dandelion Construction and (c)…......Page 314
7. Baernstein’s Example on the Hayman–Wu Theorem......Page 320
Notes......Page 323
Exercises and Further Results......Page 325
1. Cantor Sets......Page 333
2. For Certain Tones, dim Omega < 1......Page 342
3.For All Tones,dim Omega ≤ 1......Page 349
Notes......Page 359
Exercises and Further Results......Page 360
X Rectifiability and Quadratic Expressions......Page 365
1. The Lusin Area Function......Page 366
2. Squares Sums and Rectifiability......Page 379
3. A Decomposition Theorem......Page 390
4. Schwarzian Derivatives......Page 398
5. Geometric Estimates of Schwarzian Derivatives......Page 402
6. Schwarzian Derivatives and Rectifiable Quasicircles......Page 411
7. The Bishop–Jones… Theorem......Page 415
8. Schwarzians and BMO Domains......Page 426
9. Angular Derivatives......Page 429
10. A Local F. and M. Riesz Theorem......Page 433
11. Ahlfors Regular Sets and the Hayman–Wu Theorem......Page 438
Notes......Page 443
Exercises and Further Results......Page 444
A. Hardy Spaces......Page 453
B. Mixed Boundary Value Problems......Page 459
C. The Dirichlet Principle......Page 465
D. Hausdorff Measure......Page 474
E. Transfinite Diameter and Evans Functions......Page 484
F. Martingales, Brownian Motion, and Kakutani’s Theorem......Page 488
G. Carleman’s Method......Page 498
H. Extremal Distance in Finitely Connected Jordan Domains......Page 502
Notes......Page 514
I. McMillan’s Twist Point Theorem......Page 515
J. Bloch Martingales and the Law of the Iterated Logarithm......Page 521
K. A Dichotomy Theorem......Page 530
L. Two Estimates on Integral Means......Page 536
M. Calderón’s Theorem and Chord-Arc Domains......Page 538
Bibliography......Page 549
Author Index......Page 573
Symbol Index......Page 577
Subject Index......Page 579
