Characterization of n-rectifiability in terms of Jones’ square function: Part II

We show that a Radon measure μ in R d which is absolutely continuous with respect to the n -dimensional Hausdorff measure H n is n -rectifiable if the so called Jones’ square function is finite μ -almost everywhere. The converse of this result is proven in a companion paper by the second author, and...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 25; no. 5; pp. 1371 - 1412
Main Authors Azzam, Jonas, Tolsa, Xavier
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.10.2015
Subjects
Analysis
Mathematics
Mathematics and Statistics
Absolute Constant
Radon Measure
Pairwise Disjoint
Analytic Capacity
Main Lemma
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Summary:We show that a Radon measure μ in R d which is absolutely continuous with respect to the n -dimensional Hausdorff measure H n is n -rectifiable if the so called Jones’ square function is finite μ -almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n -rectifiable measures which are absolutely continuous with respect to H n . Further, in this paper we also investigate the relationship between the Jones’ square function and the so called Menger curvature of a measure with linear growth, and we show an application to the study of analytic capacity.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-015-0334-7