Identifying 1-rectifiable measures in Carnot groups

DOI10.1515/agms-2023-0102arXiv2109.06753MaRDI QIDQ6189131

Matthew Badger, Sean Li, Scott Zimmerman

Publication date: 12 January 2024

Published in: Analysis and Geometry in Metric Spaces (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2109.06753

zbMATH Keywords

doubling measuresCarnot groupsrectifiable curvesrectifiable measuresJones' \( \beta\) numbers

Mathematics Subject Classification ID

Harmonic analysis on homogeneous spaces (43A85) Length, area, volume, other geometric measure theory (28A75) Curves in Euclidean and related spaces (53A04) Sub-Riemannian geometry (53C17)

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• On the structure of continua with finite length and Gołąb's semicontinuity theorem
• Multiscale analysis of 1-rectifiable measures. II: Characterizations
• A counterexample for the geometric traveling salesman problem in the Heisenberg group
• An upper bound for the length of a traveling salesman path in the Heisenberg group
• Rectifiable sets and the traveling salesman problem
• Generalized rectifiability of measures and the identification problem
• Morceaux de graphes lipschitziens et intégrales singulières sur une surface
• Subsets of rectifiable curves in Hilbert space-the analyst's TSP
• Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
• Lectures on analysis on metric spaces
• The traveling salesman theorem in Carnot groups
• Boundedness of the density normalised Jones' square function does not imply 1-rectifiability
• An analyst's traveling salesman theorem for sets of dimension larger than one
• The density theorem and Hausdorff inequality for packing measure in general metric spaces
• A sharp necessary condition for rectifiable curves in metric spaces
• The restricted content and the \(d\)-dimensional Analyst's travelling salesman theorem for general sets
• Characterising rectifiable metric spaces using tangent spaces
• The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups
• On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion
• On rectifiable measures in Carnot groups: existence of density
• Box-counting by Hölder's traveling salesman
• Purely unrectifiable metric spaces and perturbations of Lipschitz functions
• Effective Reifenberg theorems in Hilbert and Banach spaces
• Multiscale analysis of 1-rectifiable measures: necessary conditions
• Hölder curves and parameterizations in the Analyst's traveling salesman theorem
• Geometry of measures in real dimensions via Hölder parameterizations
• The geometric traveling salesman problem in the Heisenberg group
• Rectifiable sets, densities and tangent measures
• A characterization of 1-rectifiable doubling measures with connected supports
• On rectifiable measures in Carnot groups: representation
• Dimension of a measure
• The traveling salesman problem in the Heisenberg group: Upper bounding curvature
• Two sufficient conditions for rectifiable measures
• Existence of doubling measures via generalised nested cubes
• What is a cube?
• Characterizations of rectifiable metric measure spaces
• A smooth subadditive homogeneous norm on a homogeneous group
• A doubling measure on $\mathbb {R}^d$ can charge a rectifiable curve
• Upper conical density results for general measures on ℝⁿ
• Stratified Lie Groups and Potential Theory for their Sub-Laplacians
• k-Dimensional Regularity Classifications for s-Fractals
• Quantifying curvelike structures of measures by usingL2 Jones quantities
• Menger curvature and Lipschitz parametrizations in metric spaces
• Characterization of Subsets of Rectifiable Curves in R ⁿ
• A T(b) theorem with remarks on analytic capacity and the Cauchy integral
• Radon measures and Lipschitz graphs
• Rectifiability
• A New Theory of Dimension
• The φ Rectifiable Subsets of the Plane
• Subsets of rectifiable curves in Banach spaces. I: Sharp exponents in traveling salesman theorems
• Subsets of rectifiable curves in Banach spaces. II: Universal estimates for almost flat arcs
• Stratified β$\beta$‐numbers and traveling salesman in Carnot groups