**University Lecture Series**

Volume: 35;
2005;
155 pp;
Softcover

MSC: Primary 35; 31; 34; 28;

Print ISBN: 978-0-8218-2728-4

Product Code: ULECT/35

List Price: $41.00

Individual Member Price: $32.80

**Electronic ISBN: 978-1-4704-2180-9
Product Code: ULECT/35.E**

List Price: $41.00

Individual Member Price: $32.80

#### Supplemental Materials

# Harmonic Measure: Geometric and Analytic Points of View

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*Luca Capogna; Carlos E. Kenig; Loredana Lanzani*

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 at the University of Arkansas. 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.

#### Table of Contents

# Table of Contents

## Harmonic Measure: Geometric and Analytic Points of View

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Introduction ix10 free
- Chapter 1. Motivation and statement of the main results 112 free
- 1. Characterization (1)[sub(α)]: Approximation with planes 213
- 2. Characterization (2)[sub(α)]: Introducing BMO and VMO 314
- 3. Multiplicative vs. additive formulation: Introducing the doubling condition 314
- 4. Characterization (1)[sub(α)] and flatness 415
- 5. Doubling and asymptotically optimally doubling measures 718
- 6. Regularity of a domain and doubling character of its harmonic measure 819
- 7. Regularity of a domain and smoothness of its Poisson kernel 1021

- Chapter 2. The relation between potential theory and geometry for planar domains 1324
- Chapter 3. Preliminary results in potential theory 3950
- Chapter 4. Reifenberg flat and chord arc domains 5566
- Chapter 5. Further results on Reifenberg flat and chord arc domains 7384
- Chapter 6. From the geometry of a domain to its potential theory 8192
- Chapter 7. From potential theory to the geometry of a domain 113124
- Chapter 8. Higher codimension and further regularity results 147158
- Bibliography 153164
- Back Cover Back Cover1170

#### Readership

Graduate students and research mathematicians interested in analysis.

#### Reviews

This book is a good introduction to an exciting new research area on the interface of harmonic analysis and geometric measure theory. The book is very well written, with clear explanations and useful pictures. It negotiates a fine compromise between brevity and detail as it presents a subject that is necessarily somewhat technical.

-- Bulletin of the American Mathematical Society