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Hardcover ISBN:  9780821815373 
Product Code:  SURV/38 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470412654 
Product Code:  SURV/38.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821815373 
eBook ISBN:  9781470412654 
Product Code:  SURV/38.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 38; 1993; 356 ppMSC: Primary 28; Secondary 42; 30; 49
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 scaleinvariant 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 selfintersections 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.
ReadershipHarmonic analysts, complex analysts, mathematicians working in geometric measure theory, and mathematicians studying bilipshitz and quasiconformal mappings.

Table of Contents

Part I. Background information and the statements of the main results

1. Reviews of various topics

2. A summary of the main results

3. Dyadic cubes and corona decompositions

Part II. New geometrical conditions related to uniform rectifiability

1. Onedimensional sets

2. The bilateral weak geometric lemma and its variants

3. The WHIP and related conditions

4. Other conditions in the codimension 1 case

Part III. Applications

1. Uniform rectifiability and singular integral operators

2. Uniform rectifiability and square function estimates for the Cauchy kernel

3. Square function estimates and uniform rectifiability in higher dimensions

4. Approximating Lipschitz functions by affine functions

5. The weak constant density condition

Part IV. Direct arguments for some stability results

1. Stability of various versions of the geometric lemma

2. Stability properties of the corona decomposition


Reviews

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


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

Part I. Background information and the statements of the main results

1. Reviews of various topics

2. A summary of the main results

3. Dyadic cubes and corona decompositions

Part II. New geometrical conditions related to uniform rectifiability

1. Onedimensional sets

2. The bilateral weak geometric lemma and its variants

3. The WHIP and related conditions

4. Other conditions in the codimension 1 case

Part III. Applications

1. Uniform rectifiability and singular integral operators

2. Uniform rectifiability and square function estimates for the Cauchy kernel

3. Square function estimates and uniform rectifiability in higher dimensions

4. Approximating Lipschitz functions by affine functions

5. The weak constant density condition

Part IV. Direct arguments for some stability results

1. Stability of various versions of the geometric lemma

2. Stability properties of the corona decomposition

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