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Small constant uniform rectifiability

Cole Jeznach

Abstract

We provide several equivalent characterizations of locally flat, $d$-Ahlfors regular, uniformly rectifiable sets $E$ in $\mathbb{R}^n$ with density close to $1$ for any dimension $d \in \mathbb{N}$ with $1 \le d \le n-1$. In particular, we show that when $E$ is Reifenberg flat with small constant and has Ahlfors regularity constant close to $1$, then the Tolsa alpha coefficients associated to $E$ satisfy a small constant Carleson measure estimate. This estimate is new, even when $d = n-1$, and gives a new characterization of chord-arc domains with small constant.

Small constant uniform rectifiability

Abstract

We provide several equivalent characterizations of locally flat, -Ahlfors regular, uniformly rectifiable sets in with density close to for any dimension with . In particular, we show that when is Reifenberg flat with small constant and has Ahlfors regularity constant close to , then the Tolsa alpha coefficients associated to satisfy a small constant Carleson measure estimate. This estimate is new, even when , and gives a new characterization of chord-arc domains with small constant.
Paper Structure (12 sections, 16 theorems, 148 equations)

This paper contains 12 sections, 16 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Main result and outline of the paper
  3. An application to chord-arc domains
  4. Preliminary definitions
  5. The Christ-David dyadic lattice
  6. Small-constant Corona Decompositions
  7. Conventions for constants
  8. delta-UR measures admit small-Corona decompositions
  9. delta-Corona decompositions imply alpha are small
  10. Reifenberg flatness implies oscillation of tangents is small
  11. oscillation of tangents small implies good Lipschitz approximations to E
  12. A comment on local results and chord-arc domains with small constant

Key Result

Theorem 1.9

Fix $n, d \in \mathbb{N}$ with $0 < d <n$ and $C_E >0$. Then there are constants $\delta_0 >0$ and $\theta_0 \in (0,1)$ depending only on $n$, $d$, and $C_E >0$, so that the following holds. Whenever $0 < \delta < \delta_0$, $E \subset \mathbb{R}^n$ is $d$-Ahlfors regular with constant $C_E$, and on then all of the others also hold with constant $\delta^{\theta_0}$ in place of $\delta$.

Theorems & Definitions (26)

  • Theorem 1.9
  • Theorem 1.15
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: see Lemma 5.4, DLMAinfty
  • Theorem 3.3
  • proof
  • Theorem 4.2: Theorem 1.2 in TOLSAALPHA
  • ...and 16 more