Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Identifying 1-rectifiable measures in Carnot groups   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 $\mathbb{R}^2$ (P. Jones, 1990), in $\mathbb{R}^n$ (K.
Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of
local geometric least squares data called Jones' $\beta$-numbers. In a
secondary result, we implement the Garnett-Killip-Schul construction of a
doubling measure in $\mathbb{R}^n$ that charges a rectifiable curve in an
arbitrary complete, quasiconvex, doubling metric space.

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The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph