Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem
Abstract
We investigate the geometry of sets in Euclidean and infinitedimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a $(1/s)$Hölder continuous map $f:[0,1]\rightarrow l^2$, 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 $\mathbb{R}^N$ 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 wellknown 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' nonparametric 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, we provide sufficient conditions that guarantee fractional rectifiability of pointwise doubling measures in $\mathbb{R}^N$.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 DOI:
 10.48550/arXiv.1806.01197
 arXiv:
 arXiv:1806.01197
 Bibcode:
 2018arXiv180601197B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Metric Geometry;
 28A75 (Primary);
 26A16;
 28A80;
 30L05;
 65D10 (Secondary)
 EPrint:
 74 pages, 4 figures (v3: corrected Lemma 4.12 and simplified Appendix A, other small improvements, final version)