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The weak lower density condition and uniform rectifiability
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Azzam, Jonas and Hyde, Matthew (2022) The weak lower density condition and uniform rectifiability. Annales Fennici Mathematici, 47 (2). pp. 791-819. doi:10.54330/afm.119478 ISSN 2737-0690.
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Official URL: http://dx.doi.org/10.54330/afm.119478
Abstract
We show that an Ahlfors d-regular set E in Rn is uniformly rectifiable if the
set of pairs (x, r) ∈ E × (0, ∞) for which there exists y ∈ B(x, r) and 0 < t < r satisfying
H d
∞(E ∩ B(y, t)) < (2t)d − ε(2r)d is a Carleson set for every ε > 0. To prove this, we generalize
a result of Schul by proving, if X is a C-doubling metric space, ε, ρ ∈ (0, 1), A > 1, and Xn is a
sequence of maximal 2−n-separated sets in X, and B = {B(x, 2−n) : x ∈ Xn, n ∈ N}, then
∑ {
rs
B : B ∈ B, H s
ρrB (X ∩ AB)
(2rAB )s > 1 + ε
}
.C,A,ε,ρ,s H s(X).
This is a quantitative version of the classical result that for a metric space X of finite s-dimensional
Hausdorff measure, the upper s-dimensional densities are at most 1 H s-almost everywhere.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Minimal surfaces, Geometric measure theory, Fourier analysis, Measure theory | ||||||||
Journal or Publication Title: | Annales Fennici Mathematici | ||||||||
Publisher: | Finnish Mathematical Society | ||||||||
ISSN: | 2737-0690 | ||||||||
Official Date: | 2022 | ||||||||
Dates: |
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Volume: | 47 | ||||||||
Number: | 2 | ||||||||
Page Range: | pp. 791-819 | ||||||||
DOI: | 10.54330/afm.119478 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Re-use Statement: | The final version of the article is published in Ann. Fenn. Math., 47 (2). pp. 791-819. doi:10.54330/afm.119478. | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 27 June 2022 | ||||||||
Date of first compliant Open Access: | 27 June 2022 |
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