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We show that the β-numbers of intrinsic Lipschitz graphs of Heisenberg groupsℍ  n{\mathbb{H}_{n}}    are locally Carleson integrable whenn  ≥  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.

The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups