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.
read more
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.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
Hermitian decomposition of continuous functions on a fractal surface
TL;DR: In this paper, the Theodoresco transform is used to show that each Holder continuous function defined on the boundary Γ of a fractal domain can be expressed as f = Ψ+ − Ψ−, where Ψ± are Holder continuous functions on Γ and Hermitian monogenically extendable to Ω and to ℝ2n ∖ (Ω ∪ Γ) respectively.
10
A free boundary problem for the parabolic Poisson kernel
TL;DR: In this article, it was shown that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc (i.e. satisfies a vanishing Carleson measure condition).
10
Plenty of big projections imply big pieces of Lipschitz graphs
TL;DR: In this paper, it was shown that a closed regular set with Lipschitz graphs has big pieces of regular sets, i.e., a regular set that has a large number of Lipschi-graphs.
A Martinelli-Bochner formula on fractal domains
TL;DR: A new perspective on a Cauchy integral formula for Clifford algebras valued functions on domains with quite smooth boundaries was discussed in this article, where a new perspective was proposed for Clifford integral formulas for valued functions.
10
Nonexistence Results for Semilinear Equations in Carnot Groups
Fausto Ferrari,Andrea Pinamonti +1 more
TL;DR: In this article, the authors provided some nonexistence results for semilinear equations in the class of Carnot groups of type?, which contains all groups of step 2; like the Heisenberg group, and also Carnot group of arbitrarly large step.
10