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

• Autorzy: Matthew Badger , Raanan Schul
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

• Wydawca: De Gruyter Open
• Czasopismo: Analysis and Geometry in Metric Spaces
• Rocznik: 2017
• Tom: 5
• Numer: 1
• Strony: 1-39

Daty

wydano

2017-03-01

otrzymano

2016-07-25

poprawiono

2017-01-22

online

2017-03-16

Twórcy

autor

Matthew Badger

• Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009,

autor

Raanan Schul

• Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651,

Identyfikatory

DOI

10.1515/agms-2017-0001