Generalized rectifiability of measures and the identification problem

Badger, Matthew

doi:10.1007/s40627-019-0027-3

Cited by 17 publications

(15 citation statements)

References 82 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One goal of geometric measure theory is to understand the structure of a measure in R N through its interaction with families of lower dimensional sets. For an extended introduction, see the survey [Bad19] by the first author. In this section, we use the Hölder Traveling Salesman theorem to establish criteria for fractional rectifiability of pointwise doubling measures in terms of L p Jones beta numbers.…”

Section: Fractional Rectifiability Of Measuresmentioning

confidence: 99%

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Badger

Naples

Vellis

2019

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a (1/s)-Hölder continuous map f : [0, 1] → l 2 , with s > 1. Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R N or l 2 in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higherdimensional curves such as Hölder curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves f k ([0, 1]). We then find conditions in terms of tube approximations that ensure the approximating curves converge to a Hölder curve. As an application to the geometry of measures, we identify conditions that guarantee fractional rectifiability of pointwise doubling measures in R N .

show abstract

Section: Fractional Rectifiability Of Measuresmentioning

confidence: 99%

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Badger

Naples

Vellis

2019

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Tolsa [Tol19] obtains an alternative proof of the result in [ENV16] using the techniques from [Tol15,AT15]. For a survey on generalized rectifiability of measures, including classical results and recent advances, see [Bad19].…”

mentioning

confidence: 99%

Sufficient conditions for C^1,α parametrization and rectifiability

Ghinassi¹

2020

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

We say a measure is C 1,α d-rectifiable if there is a countable union of C 1,α d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in R n to be C 1,α d-rectifiable, with α ∈ (0, 1]. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the C 1,α constants depend on such a function. Along the way we also give sufficient conditions for C 1,α parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and Toro's Reifenberg parametrizations of sets with holes in the Hölder and Lipschitz categories.

show abstract

“…The identification problem (see [9]) is to find pointwise defined properties P (µ, x) and Q(µ, x) such that µ N = µ {x ∈ X : P (µ, x) holds} and µ ⊥ N = µ {x ∈ X : Q(µ, x)} for every (locally) finite measure µ on X. An ideal solution should involve the geometry of the space X and the sets in N .…”

Section: 2mentioning

confidence: 99%

“…Rectifiability is a concept in geometric measure theory that supplies a finer notion of regularity of measure than dimension [9,56]. Given any metric space X, family A of Borel subsets of X, and Borel measure µ on X, we say that µ charges A if µ(A) > 0 for some A ∈ A.…”

Section: Introductionmentioning

confidence: 99%

Identifying 1-rectifiable measures in Carnot groups

Badger¹,

Li²,

Zimmerman³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R 2 (P. Jones, 1990), in R n (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' β-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R n that charges a rectifiable curve in an arbitrary complete, quasiconvex, doubling metric space.

show abstract

Generalized rectifiability of measures and the identification problem

Cited by 17 publications

References 82 publications

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Sufficient conditions for C^1,α parametrization and rectifiability

Identifying 1-rectifiable measures in Carnot groups

Contact Info

Product

Resources

About