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
Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress
TL;DR: Azzam, Mourgoglou and Tolsa as discussed by the authors showed that scale invariant absolute continuity of harmonic measure with respect to surface measure is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂ Ω) for some p < ∞.
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The Riesz transform and quantitative rectifiability for general Radon measures
TL;DR: In this paper, it was shown that the Borel measure of the Riesz transform is continuous with respect to a uniformly rectifiable subset of the Euclidean space with a constant ε > 0.
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The Traveling Salesman Theorem in Carnot groups
TL;DR: In this article, the Traveling Salesman Theorem in any Carnot group has been shown to satisfy the geometric lemma with some natural modifications, based on new Alexandrov-type curvature inequalities for the Hebisch and Sikora metrics.
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On the Notion of the Bochner–Martinelli Integral for Domains with Rectifiable Boundary
TL;DR: The notion of the Bochner-Martinelli kernel for domains with rectifiable boundary in C 2, by expressing the kernel in terms of the exterior normal due to Federer, was discussed in this article.
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The Riesz transform of codimension smaller than one and the Wolff energy
TL;DR: In this article, the non-negative locally finite Borel measures with bounded Riesz transform were characterized in terms of the Wolff energy, and a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator was given.
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