Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Matthew Badger, and Raanan Schul. "Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations." Analysis and Geometry in Metric Spaces 5.1 (2017): 1-39. <http://eudml.org/doc/288049>.

@article{MatthewBadger2017,
abstract = {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.},
author = {Matthew Badger, Raanan Schul},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {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; 1-rectifiable measures; purely 1-unrectifiable measures; analyst's traveling salesman theorem},
language = {eng},
number = {1},
pages = {1-39},
title = {Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations},
url = {http://eudml.org/doc/288049},
volume = {5},
year = {2017},
}

TY - JOUR
AU - Matthew Badger
AU - Raanan Schul
TI - Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations
JO - Analysis and Geometry in Metric Spaces
PY - 2017
VL - 5
IS - 1
SP - 1
EP - 39
AB - 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.
LA - eng
KW - 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; 1-rectifiable measures; purely 1-unrectifiable measures; analyst's traveling salesman theorem
UR - http://eudml.org/doc/288049
ER -