The Traveling Salesman Theorem 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 Calderon–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.