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

Abstract

We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a $(1/s)$-Hölder continuous map $f:[0,1]\to l^2$, 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 well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as Hölder curves. To overcome this obstacle, we reimagine Jones' non-parametric 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 ${\mathbb{R}}^N$.

74 pages, 4 figures (v3: corrected Lemma 4.12 and simplified Appendix A, other small improvements, final version)