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arXiv · 1804.05646

The Traveling Salesman Theorem in Carnot Groups

Vasileios Chousionis·Sean Li·Scott Zimmerman
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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.

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Vasileios Chousionis, Sean Li, Scott Zimmerman. 2018-04-16. The Traveling Salesman Theorem in Carnot Groups. https://arxiv.org/abs/1804.05646

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