Multiscale Analysis of 1-rectifiable Measures II: Characterizations

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function,...

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Autores principales:	Badger Matthew, Schul Raanan
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2017
Materias:	1 rectifiable measures purely 1 unrectifiable measures rectifiable curves jones beta numbers jones square functions analyst’s traveling salesman theorem doubling measures hausdorff densities hausdorff measures Analysis QA299.6-433
Acceso en línea:	https://doaj.org/article/b8b65633af424a67aff0410686fe5c7b
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spelling	oai:doaj.org-article:b8b65633af424a67aff0410686fe5c7b2021-12-05T14:10:38ZMultiscale Analysis of 1-rectifiable Measures II: Characterizations2299-327410.1515/agms-2017-0001https://doaj.org/article/b8b65633af424a67aff0410686fe5c7b2017-03-01T00:00:00Zhttps://doi.org/10.1515/agms-2017-0001https://doaj.org/toc/2299-3274A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.Badger MatthewSchul RaananDe Gruyterarticle1 rectifiable measurespurely 1 unrectifiable measuresrectifiable curvesjones beta numbersjones square functionsanalyst’s traveling salesman theoremdoubling measureshausdorff densitieshausdorff measuresAnalysisQA299.6-433ENAnalysis and Geometry in Metric Spaces, Vol 5, Iss 1, Pp 1-39 (2017)
institution	DOAJ
collection	DOAJ
language	EN
topic	1 rectifiable measures purely 1 unrectifiable measures rectifiable curves jones beta numbers jones square functions analyst’s traveling salesman theorem doubling measures hausdorff densities hausdorff measures Analysis QA299.6-433
spellingShingle	1 rectifiable measures purely 1 unrectifiable measures rectifiable curves jones beta numbers jones square functions analyst’s traveling salesman theorem doubling measures hausdorff densities hausdorff measures Analysis QA299.6-433 Badger Matthew Schul Raanan Multiscale Analysis of 1-rectifiable Measures II: Characterizations
description	A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.
format	article
author	Badger Matthew Schul Raanan
author_facet	Badger Matthew Schul Raanan
author_sort	Badger Matthew
title	Multiscale Analysis of 1-rectifiable Measures II: Characterizations
title_short	Multiscale Analysis of 1-rectifiable Measures II: Characterizations
title_full	Multiscale Analysis of 1-rectifiable Measures II: Characterizations
title_fullStr	Multiscale Analysis of 1-rectifiable Measures II: Characterizations
title_full_unstemmed	Multiscale Analysis of 1-rectifiable Measures II: Characterizations
title_sort	multiscale analysis of 1-rectifiable measures ii: characterizations
publisher	De Gruyter
publishDate	2017
url	https://doaj.org/article/b8b65633af424a67aff0410686fe5c7b
work_keys_str_mv	AT badgermatthew multiscaleanalysisof1rectifiablemeasuresiicharacterizations AT schulraanan multiscaleanalysisof1rectifiablemeasuresiicharacterizations
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Multiscale Analysis of 1-rectifiable Measures II: Characterizations

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