The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups

Abstract

Abstract We show that the β-numbers of intrinsic Lipschitz graphs of Heisenberg groups ℍ n {\mathbb{H}_{n}} are locally Carleson integrable when n ≥ 2 {n\geq 2} . Our main bound uses a novel slicing argument to decompose intrinsic Lipschitz graphs into graphs of Lipschitz functions. A key ingredient in our proof is a Euclidean inequality that bounds the β-numbers of the original graph in terms of the β-numbers of many families of slices. This allows us to use recent work of Fässler and Orponen (2020) which asserts that Lipschitz functions satisfy a Dorronsoro inequality.