# Singular integrals and rectifiable sets in R[n] : au-delà des graphes lipschitziens

@article{David1991SingularIA, title={Singular integrals and rectifiable sets in R[n] : au-del{\`a} des graphes lipschitziens}, author={Guy David and S. Semmes}, journal={Ast{\'e}risque}, year={1991}, pages={7-145} }

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## 191 Citations

Sets with topology, the Analyst's TST, and applications

- Mathematics
- 2019

This paper was motivated by three questions. First: in a recent paper, Azzam and Schul asked what sort of sets could play the role of curves in the context of the higher dimensional analyst's…

Intrinsic graphs and big pieces of parabolic Lipschitz images

- Mathematics
- 2019

Let $\mathbb{H}$ be the first Heisenberg group, and let $S \subset \mathbb{H}$ be an $\mathbb{H}$-regular surface. Can almost all of $S$ be covered by countably many Lipschitz images of subsets of…

A d-DIMENSIONAL ANALYST’S TRAVELLING SALESMAN THEOREM FOR GENERAL SETS IN R

- Mathematics
- 2021

In his 1990 paper, Jones proved the following: given E ⊆ R2, there exists a curve Γ such that E ⊆ Γ and H (Γ) ∼ diamE + ∑

An Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$

- Mathematics
- 2020

In his 1990 paper, Jones proved the following: given $E \subseteq \mathbb{R}^2$, there exists a curve $\Gamma$ such that $E \subseteq \Gamma$ and \[ \mathscr{H}^1(\Gamma) \sim \text{diam}\, E +…

Wild examples of rectifiable sets

- Mathematics
- 2019

We study the geometry of sets based on the behavior of the Jones function, $J_{E}(x) = \int_{0}^{1} \beta_{E;2}^{1}(x,r)^{2} \frac{dr}{r}$. We construct two examples of countably $1$-rectifiable sets…

A sufficient condition for having big pieces of bilipschitz images of subsets of euclidean space in Heisenberg groups

- Mathematics
- 2012

In this article we extend a euclidean result of David and Semmes to the Heisenberg group by giving a sufficient condition for a $k$-Ahlfors-regular subset to have big pieces of bilipschitz images of…

Regularizing and self-avoidance effects of integral Menger curvature

- Mathematics
- 2010

We investigate geometric curvature energies on closed curves involving integral versions of Menger curvature. In particular, we prove geometric variants of Morrey-Sobolev and Morrey-space imbedding…

A parabolic version of Corona decompositions

- Mathematics
- 2009

. Let E be a subset in ( n + 1)-dimensional Euclid-ian space with parabolic homogeneity, codimension 1, and with an appropriate surface measure σ associated to it. We deﬁne a parabolic version of…

Singular Integrals and Elliptic Boundary Problems on Regular Semmes–Kenig–Toro Domains

- Mathematics
- 2009

We develop the theory of layer potentials and related singular integral operators as a tool to study a variety of elliptic boundary problems on a family of domains introduced by Semmes [105, 106] and…

Reifenberg Parameterizations for Sets with Holes

- Mathematics
- 2009

We 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…