HARMONIC MEASURE

John B. Garnett, UCLA, and Donald E. Marshall, University of Washington

I. JORDAN DOMAINS

1. The Halfplane and the Disc

2. Fatou's Theorem and Maximal Functions

3. Carathéodory's Theorem

4. Distortion and the Hyperbolic Metric

5. The Hayman-Wu Theorem

Notes

Exercises and Further Results

II. FINITELY CONNECTED DOMAINS

1. The Schwarz Alternating Method

2. Green's Functions and Poisson Kernels

3. Conjugate Functions

4. Boundary Smoothness

Notes

Exercises and Further Results

III. POTENTIAL THEORY

1. Capacity and Green's Functions

2. The Logarithmic Potential

3. The Energy Integral

4. The Equilibrium Distribution

5. Wiener's Solution to the Dirichlet Problem

6. Regular Points

7. Wiener Series

8. Capacity Zero and Harmonic Measure Zero

9. Estimates for Harmonic Measure

Notes

Exercises and Further Results

IV. EXTREMAL DISTANCE

1. Definitions and Examples

2. Uniqueness of Extremal Metrics

3. Four Rules for Extremal Length

4. Extremal Metrics for Extremal Distance

5. Extremal Distance and Harmonic Measure

6. The $\int{{d\theta(x)} \over {x}}$ Estimate

Notes

Exercises and Further Results

V. APPLICATIONS AND REVERSE INEQUALITIES

1. Asymptotic Values of Entire Functions

2. Lower Bounds

3. Reduced Extremal Distance

4. Teichmüller's Modulsatz

5. Boundary Conformality and Angular Derivatives

6. Conditions More Geometric

Notes

Exercises and Further Results

VI. SIMPLY CONNECTED DOMAINS, PART ONE

1. The F. and M. Riesz Theorem

2. Privalov's Theorem and Plessner's Theorem

3. Accessible Points

4. Cone Points and McMillan's Theorem

5. Compression and Expansion

6. Pommerenke's Theorem

Notes

Exercises and Further Results

VII. BLOCH FUNCTIONS AND QUASICIRCLES

1. Bloch Functions

2. Bloch Functions and Univalent Functions

3. Quasicircles

4. Chord-arc Curves and the $A^{\infty}$ Condition

5. BMO Domains

Notes

Exercises and Further Results

VIII. SIMPLY CONNECTED DOMAINS, PART TWO

1. The Law of the Iterated Logarithm for Bloch Functions

2. Harmonic Measure and Hausdorff Measure

3. The Number of Bad Discs

4. Brennan's Conjecture and Integral Means Spectra

5. $\beta$ Numbers and Polygonal Trees

6. The Dandelion Construction and (c) $\Longrightarrow$ (a)

7. Baernstein's Example

Notes

Exercises and Further Results

IX. INFINITELY CONNECTED DOMAINS

1. Some Cantor Sets

2. $\dim \om < 1$   for some $\Omega$

3. $\dim \om \leq 1$   for all $\Omega$

Notes

Exercises and Further Results

X. RECTIFIABILITY AND QUADRATIC EXPRESSIONS

1. The Lusin Area Function

2. Square Sums and Rectifiability

3. A Decomposition Theorem

4. Schwarzian Derivatives

5. Geometric Estimates of Schwarzian Derivatives

6. Schwarzian Derivatives and Rectifiable Quasicircles

7. The Bishop-Jones $H^{1/2 - \eta}$ Theorem

8. Schwarzian Derivatives and BMO Domains

9. Angular Derivatives

10. A Local F. and M. Riesz Theorem

11. Ahlfors Regular Sets and the Hayman-Wu Theorem

Notes

Exercises and Further Results

APPENDICES

A. Hardy Spaces

B. Mixed Boundary Value Problems

C. The Dirichlet Principle

D. Hausdorff Measure

E. Transfinite Diameter and Evans Functions

F. Martingales, Brownian Motion, and Kakutani's Theorem

G. Carleman's Method

H. Extremal Distance in Finitely Connected Domains

I. McMillan's Twist Point Theorem

J. Bloch Martingales and the Law of the Iterated Logarithm

K. A Dichotomy Theorem

L. Two Estimates on Integral Means

M. Calderón's Theorem and Chord-Arc Domains