Multiscale analysis of 1-rectifiable measures: necessary conditions

Badger, Matthew; Schul, Raanan

doi:10.1007/s00208-014-1104-9

Cited by 32 publications

(47 citation statements)

References 23 publications

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“…x ∈ Γ (see Theorem 4.3). This result is some generalization and extension of the main result of the predecessor [5] of the current paper. In Section 5, we establish su cient conditions, which guarantee that a Radon measure is 1-recti able.…”

Section: Organizationsupporting

confidence: 73%

“…This is a variant of the density-normalized Jones function J (µ, x) used in [5], which was associated to the beta numbers β (µ, Q). Peter Jones conjectured circa 2000 that these types of weighted Jones functions could be used to characterize recti abilty of a measure (personal communication).…”

Section: Main Results and Organization Of The Papermentioning

confidence: 99%

“…The proof that we present is an adaptation of the proof of Proposition 3.1 in [5], the forerunner to this paper by the same name. For clarity, we develop the part of the proof that needs to be altered.…”

Section: Proposition 44 Let N ≥ If ν Is a Nite Borel Measure On Rmentioning

confidence: 97%

“…Examples of 1-recti able doubling measures µ on R n with spt µ = R n and µ ⊥ H , which satisfy the hypothesis of Theorem 1.5, were constructed by Garnett, Killip, and Schul [23]. Interestingly, such examples give measure zero to every bi-Lipschitz image of R m in R n (in particular, they give measure zero to every Lipschitz graph), but nevertheless give full measure to a countable family of Lipschitz images of R. Following [5], it is also known that such measures have density D (µ, x) = ∞ µ-a.e., with µ(B(x, r))/r → ∞ at a rapid rate as r → .…”

Section: Theorem 15 (Azzam and Mourgoglou [3]) Assume That µ Is A Dmentioning

confidence: 99%

“…see [13] and [14]). Much work has been done in connecting beta numbers and recti ability (highlights include [50], [32], [33], [24], [34], [15], [5], [2], [6], [46]), but we single out two recent papers, [58] and [4], and state one theorem, which is a combination of their main results. …”

Section: P Jones Beta Numbers and Recti Abilitymentioning

confidence: 99%

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Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Badger

Schul

2017

Analysis and Geometry in Metric Spaces

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A measure is 1-recti able if there is a countable union of nite length curves whose complement has zero measure. We characterize 1-recti able Radon measures µ in n-dimensional Euclidean space for all n ≥ in terms of positivity of the lower density and niteness of a geometric square function, which loosely speaking, records in an L 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 Hausdor measure H . We also characterize purely 1-unrecti able Radon measures, i.e. locally nite measures that give measure zero to every nite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L variant of P. Jones' traveling salesman construction, which is of independent interest.

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Section: Organizationsupporting

confidence: 73%

Section: Main Results and Organization Of The Papermentioning

confidence: 99%

Section: Proposition 44 Let N ≥ If ν Is a Nite Borel Measure On Rmentioning

confidence: 97%

Section: Theorem 15 (Azzam and Mourgoglou [3]) Assume That µ Is A Dmentioning

confidence: 99%

Section: P Jones Beta Numbers and Recti Abilitymentioning

confidence: 99%

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Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Badger

Schul

2017

Analysis and Geometry in Metric Spaces

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

2015

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Abstract. 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.

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An analyst’s traveling salesman theorem for sets of dimension larger than one

Azzam

Schul

2017

Math. Ann.

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In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of β-numbers. These βnumbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones' β-numbers. This variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set.In this paper we give a version of P. Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean space. Our main result is a lower bound for the d-dimensional Hausdorff measure of a set in terms of an analogous sum of β-type numbers. We also show an upper bound of this type. The combination of these results gives a Jones theorem for higher dimensional sets. While there is no assumption of Ahlfors regularity, or of a measure on the underlying set, there is an assumption of a lower bound on the Hausdorff content. We adapt David and Semmes' version of Jones' β-numbers by redefining them using a Choquet integral, allowing them to be defined for arbitrary sets (and not just sets of locally finite measure). A key tool in the proof is G. David and T. Toro's parametrization of Reifenberg flat sets (with holes). CONTENTS 2010 Mathematics Subject Classification. 28A75, 28A78, 28A12.

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Multiscale analysis of 1-rectifiable measures: necessary conditions

Cited by 32 publications

References 23 publications

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

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

An analyst’s traveling salesman theorem for sets of dimension larger than one

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