**Mathematical Surveys and Monographs**

Volume: 38;
1993;
356 pp;
Hardcover

MSC: Primary 28;
Secondary 42; 30; 49

Print ISBN: 978-0-8218-1537-3

Product Code: SURV/38

List Price: $129.00

Individual Member Price: $103.20

**Electronic ISBN: 978-1-4704-1265-4
Product Code: SURV/38.E**

List Price: $129.00

Individual Member Price: $103.20

# Analysis of and on Uniformly Rectifiable Sets

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*Guy David; Stephen Semmes*

The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant substitute for the classical notion of rectifiability; as the answer (sometimes only conjecturally) to certain geometric questions in complex and harmonic analysis; as a condition which ensures the parametrizability of a given set, with estimates, but with some holes and self-intersections allowed; and as an achievable baseline for information about the structure of a set. This book is about understanding uniform rectifiability of a given set in terms of the approximate behavior of the set at most locations and scales. In addition to being the only general reference available on uniform rectifiability, this book also poses many open problems, some of which are quite basic.

#### Readership

Harmonic analysts, complex analysts, mathematicians working in geometric measure theory, and mathematicians studying bilipshitz and quasiconformal mappings.

#### Reviews & Endorsements

A mixture of geometric measure theory and harmonic analysis … a remarkable development of these researches.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Analysis of and on Uniformly Rectifiable Sets

- Table of Contents v6 free
- Preface ix10 free
- Notation and Conventions xi12 free
- Part I: Background Information and the Statements of the Main Results 114 free
- Chapter 1. Reviews of Various Topics 316
- 1.1 Review from geometric measure theory 316
- 1.2 Review of some topics concerning singular integral operators and rectifiability 720
- 1.3 Review of some aspects of Littlewood-Paley theory in connection with rectifiability 1629
- 1.4 Various characterizations of uniform rectifiability 2134
- 1.5 The weak geometric lemma and its relatives 2639

- Chapter 2. A Summary of the Main Results 3144
- Chapter 3. Dyadic Cubes and Corona Decompositions 5366

- Part II: New Geometrical Conditions Related to Uniform Rectifiability 6780
- Chapter 1. One-Dimensional Sets 6982
- Chapter 2. The Bilateral Weak Geometric Lemma and its Variants 97110
- 2.1 Introduction; the corona method 97110
- 2.2 Big projections in codimension 1 104117
- 2.3 Big projections in the higher codimension case 110123
- 2.4 The local convexity condition LCV 120133
- 2.5 The weaker local convexity condition WLCV 126139
- 2.6 Weak starlikeness 129142
- 2.7 Some questions about variants of the LCV and the LS 131144

- Chapter 3. The WHIP and Related Conditions 135148
- 3.1 The WHIP, the WTP, and uniform rectifiability 135148
- 3.2 The WHIP and weaker versions of the BWGL 138151
- 3.3 The weak exterior convexity condition and the GWEC 141154
- 3.4 The weak-no-mugs, weak-no-boxes, and weak-no-reels conditions 147160
- 3.5 The proof of Theorem 3.9 (part 1) 154167
- 3.6 Part 2 of the proof: The stopping-time argument 165178

- Chapter 4. Other Conditions in the Codimension 1 Case 183196

- Part III: Applications 205218
- Chapter 1. Uniform Rectifiability and Singular Integral Operators 207220
- Chapter 2. Uniform Rectifiability and Square Function Estimates for the Cauchy Kernel 217230
- 2.1 Some general comments about square function estimates 217230
- 2.2 Uniform rectifiability implies the USFE when d = 1 219232
- 2.3 From square function estimates to uniform rectifiability: Preliminary reductions and the plan of the proof 226239
- 2.4 The proof of Lemma 2.36 229242
- 2.5 A topological lemma 232245
- 2.6 The main step in the proof of Proposition 2.38 234247
- 2.7 The end of the proof of Proposition 2.38 244257

- Chapter 3. Square Function Estimates and Uniform Rectifiability in Higher Dimensions 249262
- 3.1 A brief review of Clifford analysis 249262
- 3.2 Clifford analysis and square function estimates 251264
- 3.3 From square functions to uniform rectifiability: Preliminary reductions 252265
- 3.4 Cauchy flatness implies rectifiability 253266
- 3.5 The analogue of Proposition 2.59 256269
- 3.6 Cauchy flatness implies weak flatness 261274
- 3.7 Weak flatness implies exterior convexity 265278
- 3.8 Some remarks about the higher-codimension case 267280

- Chapter 4. Approximating Lipschitz Functions by Affine Functions 269282
- Chapter 5. The Weak Constant Density Condition 297310

- Part IV: Direct Arguments for Some Stability Results 311324
- References 345358
- Table of Selected Notation 349362
- Table of Acronyms 351364
- Table of Theorems 353366
- Index 355368 free