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
A new family of singular integral operators whose $L^2$-boundedness implies rectifiability
TL;DR: Chousionis, Mateu, Prat and Tolsa as discussed by the authors proved that the rectifiability property of the Cauchy kernel is valid for operators associated to a much wider class of kernels.
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Reifenberg Parameterizations for Sets with Holes
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TL;DR: In this paper, the authors extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of a set$E$ by a $d$-dimensional plane or smooth manifold.
Square functions and uniform rectifiability
TL;DR: Chousionis et al. as mentioned in this paper showed that the Dirichlet problem for elliptic systems in the upper-half space can be solved by the convergence of the Fourier series and integrals.
Effective Reifenberg theorems in Hilbert and Banach spaces
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Geometry of measures in real dimensions via Hölder parameterizations
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