The Traveling Salesman Theorem in Carnot Groups

TL;DR

This paper extends the Traveling Salesman Theorem to Carnot groups, establishing geometric conditions for rectifiable curves and analyzing singular integral operators with new curvature inequalities.

Contribution

It proves one direction of the Traveling Salesman Theorem in Carnot groups and introduces Alexandrov-type curvature inequalities for Hebisch-Sikora metrics.

Findings

Sets in Carnot groups contained in rectifiable curves satisfy Peter Jones' geometric lemma.

Existence of specific Calderón-Zygmund kernels with bounded singular integrals on 1-regular curves.

New curvature inequalities for Hebisch-Sikora metrics in Carnot groups.

Abstract

Let $\mathbb{G}$ be any Carnot group. We prove that, if a subset of $\mathbb{G}$ is contained in a rectifiable curve, then it satisfies Peter Jones' geometric lemma with some natural modifications. We thus prove one direction of the Traveling Salesman Theorem in $\mathbb{G}$. Our proof depends on new Alexandrov-type curvature inequalities for the Hebisch-Sikora metrics. We also apply the geometric lemma to prove that, in every Carnot group, there exist $-1$-homogeneous Calder\'on-Zygmund kernels such that, if a set $E \subset \mathbb{G}$ is contained in a 1-regular curve, then the corresponding singular integral operators are bounded in $L^2(E)$. In contrast to the Euclidean setting, these kernels are nonnegative and symmetric.