The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups

We show that the $\beta$--numbers of intrinsic Lipschitz graphs of Heisenberg groups $\mathbb{H}_n$ are locally Carleson integrable when $n \geq 2$. Our technique relies on a recent Dorronsoro inequality \cite{FO} as well as a novel slicing argument. A key ingredient in our proof is a Euclidean inequality bounding the $\beta$--number of a function on a cube of $\mathbb{R}^n$ using the $\beta$--number of the restriction of the function to codimension--1 slices of the cube.