Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Abstract

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.

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

“…The theorem is valid if the length of a curve Γ = f ([0, 1]) is interpreted either as the 1-dimensional Hausdorff measure of the set Γ or as the total variation of the parameterization f . A curious feature of the known proofs of the sufficient half of the Analyst's TST (see [Jon90] or [BS17]) is that a rectifiable curve Γ containing the set E is constructed as the limit of piecewise linear curves Γ k containing a 2 −k -net for E without constructing a parameterization of Γ k or Γ. This aspect of the proof breaks the analogy with the classical TSP, in which one is asked to find a minimal tour of a graph.…”

The analyst's traveling salesman theorem in graph inverse limits

“…We will show that each point of Γ n is contained in at most N η = 1 + log(20C η m)/ log(m) elements of T n . The constant N η is bounded above in terms of η, by 2 + log(20Cη) log (2) . Suppose T and T are in T n and distinct and intersect.…”

“…Significant work in the direction of singular integrals was done by David and Semmes (see for instance [10,11] and the references within), as part of a field which now falls under the name "quantitative rectifiability", to separate it from the older field of "qualitative rectifiability". Several works tying the two fields exist, most recently by Tolsa [28], Azzam and Tolsa [1], and Badger and the second author [2]. (This last reference has a detailed exposition of the fields, to which we refer the interested reader.)…”

“…(This last reference has a detailed exposition of the fields, to which we refer the interested reader.) We remark that a variant of Theorem 1.1 was needed to do the work in [2].…”

Characterization of rectifiable measures in terms of 𝛼-numbers

“…Unfortunately, these necessary conditions are not sufficient, as shown by an example of Martikainen and Orponen [MO16]. However, see [BS17] for a characterization for measures with positive lower density using a different β-type quantity.…”