Harmonic Measure ：Geometric and Analytic Points of View ( University Lecture Series )

Publication subTitle ：Geometric and Analytic Points of View

Publication series ：University Lecture Series

Author： Luca Capogna;Carlos E. Kenig;Loredana Lanzani

Publisher： American Mathematical Society‎

Publication year： 2005

E-ISBN: 9781470421809

P-ISBN(Paperback): 9780821827284

Subject： O174.3 harmonic functions and potential theory

Keyword： Analysis

Language： ENG

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Harmonic Measure

Description

Recent developments in geometric measure theory and harmonic analysis have led to new and deep results concerning the regularity of the support of measures which behave “asymptotically” (for balls of small radius) as the Euclidean volume. A striking feature of these results is that they actually characterize flatness of the support in terms of the asymptotic behavior of the measure. Such characterizations have led to important new progress in the study of harmonic measure for non-smooth domains. This volume provides an up-to-date overview and an introduction to the research literature in this area. The presentation follows a series of five lectures given by Carlos Kenig at the 2000 Arkansas Spring Lecture Series. The original lectures have been expanded and updated to reflect the rapid progress in this field. A chapter on the planar case has been added to provide a historical perspective. Additional background has been included to make the material accessible to advanced graduate students and researchers in harmonic analysis and geometric measure theory.

Chapter

Cover

Title

Contents

Introduction

Chapter 1. Motivation and statement of the main results

1. Characterization (1)[sub(α)]: Approximation with planes

2. Characterization (2)[sub(α)]: Introducing BMO and VMO

3. Multiplicative vs. additive formulation: Introducing the doubling condition

4. Characterization (1)[sub(α)] and flatness

5. Doubling and asymptotically optimally doubling measures

6. Regularity of a domain and doubling character of its harmonic measure

7. Regularity of a domain and smoothness of its Poisson kernel

Chapter 2. The relation between potential theory and geometry for planar domains

1. Smooth domains

2. Non smooth domains

3. Preliminaries to the proofs of Theorems 2.7 and 2.8

4. Proof of Theorem 2.7

5. Proof of Theorem 2.8

6. Notes

Chapter 3. Preliminary results in potential theory

1. Potential theory in NTA domains

2. A brief review of the real variable theory of weights

3. The spaces BMO and VMO

4. Potential theory in C[sup(1)] domains

5. Notes

Chapter 4. Reifenberg flat and chord arc domains

1. Geometry of Reifenberg flat domains

2. Small constant chord arc domains

3. Notes

Chapter 5. Further results on Reifenberg flat and chord arc domains

1. Improved boundary regularity for δReifenberg flat domains

2. Approximation and Rellich identity

3. Notes

Chapter 6. From the geometry of a domain to its potential theory

1. Potential theory for Reifenberg domains with vanishing constant

2. Potential theory for vanishing chord arc domains

3. Notes

Chapter 7. From potential theory to the geometry of a domain

1. Asymptotically optimally doubling implies Reifenberg vanishing

2. Back to chord arc domains

3. log k ε VMO implies vanishing chord arc; Step I

4. log k ε VMO implies vanishing chord arc; Step II

5. Notes

Chapter 8. Higher codimension and further regularity results

1. Notes

Bibliography

Back Cover

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