Open AccessBook
Analysis of and on uniformly rectifiable sets
Guy David,Stephen Semmes +1 more
- 01 Jan 1993
515
TL;DR: The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways as mentioned in this paper, 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, as an achievable baseline for information about the structure of a set.
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Abstract: 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 behaviour of the set at most locations and scales. In addition to being a general reference on uniform rectifiability, the book also poses many open problems, some of which are quite basic.
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Citations
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Integral Menger Curvature and Rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space
TL;DR: In this article, it was shown that a Borel set in Euclidean space with finite integral Menger curvature is rectifiable, meaning that it can be covered by countably many Lipschitz continuous functions up to a null set in the sense of Hausdorff measure.
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A quantitative metric differentiation theorem
Jonas Azzam,Raanan Schul +1 more
- 29 Jan 2014
TL;DR: In this article, for a Lipschitz function from a Euclidean space into a metric space, the authors give a Carleson-type estimate for how often the pullback of the metric under $f$ is approximately a seminorm.
Lower and upper bounds for the waists of different spaces
TL;DR: In this paper, a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique was given, and the waist-type results in terms of the Hausdorff measure were established.
14
The isotonic cauchy transform
TL;DR: In this paper, the isotonic Cauchy transform and the Sokhotski-Plemelj formulae are established for the class of continuously differentiable solutions of the system.
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Rectifiability of harmonic measure in domains with porous boundaries
TL;DR: In this article, it was shown that if the harmonic measure is continuous with respect to the Hausdorff measure, then it is concentrated on an $n$-rectifiable set.
13