Multiscale analysis of 1-rectifiable measures: necessary conditions
نویسندگان
چکیده
منابع مشابه
Multiscale Analysis of 1-rectifiable Measures: Necessary Conditions
We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R, n ≥ 2. To each locally finite Borel measure μ, we associate a function J̃2(μ, x) which uses a weighted sum to record how closely the mass of μ is concentrated near a line in the triples of dyadic cubes containing x. We show that J̃2(μ, ·) < ∞ μ-a.e. is a necessary condition ...
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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 L gauge the extent to which μ admits approximate tangent l...
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Contents Chapter 1. Introduction 5 Chapter 2. Notation and preliminaries 9 1. General notation and measures 9 2. Weak * convergence of measures 10 3. Covering theorems and differentiation of measures 13 4. Hausdorff measures 13 5. Lipschitz functions 15 6. The Stone–Weierstrass Theorem 16 Chapter 3. Marstrand's Theorem and tangent measures 17 1.
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ژورنال
عنوان ژورنال: Mathematische Annalen
سال: 2014
ISSN: 0025-5831,1432-1807
DOI: 10.1007/s00208-014-1104-9