Mathematical Surveys and Monographs 1993; 356 pp; hardcover Volume: 38 ISBN10: 0821815377 ISBN13: 9780821815373 List Price: US$129 Member Price: US$103.20 Order Code: SURV/38
 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. Readership Harmonic analysts, complex analysts, mathematicians working in geometric measure theory, and mathematicians studying bilipshitz and quasiconformal mappings. Reviews "A mixture of geometric measure theory and harmonic analysis ... a remarkable development of these researches."  Zentralblatt MATH Table of Contents Part I. Background information and the statements of the main results  Reviews of various topics
 A summary of the main results
 Dyadic cubes and corona decompositions
Part II. New geometrical conditions related to uniform rectifiability  Onedimensional sets
 The bilateral weak geometric lemma and its variants
 The WHIP and related conditions
 Other conditions in the codimension 1 case
Part III. Applications  Uniform rectifiability and singular integral operators
 Uniform rectifiability and square function estimates for the Cauchy kernel
 Square function estimates and uniform rectifiability in higher dimensions
 Approximating Lipschitz functions by affine functions
 The weak constant density condition
Part IV. Direct arguments for some stability results  Stability of various versions of the geometric lemma
 Stability properties of the corona decomposition
 References
 Table of selected notation
 Table of acronyms
 Table of theorems
 Index
