H\"older curves and parameterizations in the Analyst's Traveling Salesman theorem

TL;DR

This paper extends the Analyst's Traveling Salesman theorem to characterize when sets in Euclidean and Hilbert spaces can be parameterized by H"older continuous maps, generalizing rectifiability concepts to higher-dimensional curves.

Contribution

It introduces new sufficient conditions for sets to be contained in H"older curves, generalizing the theorem to infinite-dimensional spaces and higher-dimensional curves.

Findings

Established conditions for sets to be contained in H"older continuous images.

Generalized the Traveling Salesman theorem to infinite-dimensional Hilbert spaces.

Provided criteria for fractional rectifiability of measures in Euclidean spaces.

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 $(1/s)$-H\"older 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 well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as H\"older 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 H\"older curve. As an application, we provide sufficient conditions that guarantee fractional rectifiability of pointwise doubling measures in $\mathbb{R}^N$.