Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every $L$-Lipschitz graph over a subset $k$-dimensional horizontal subgroup $\mathbb{V}$ $\mathbb{H}^n$ can be extended to an $L'$-Lipschitz entire $\mathbb{V}$, where $L'$ depends only on $L$, $k$, $n$. We further prove $1$-dimensional $1$-Lipschitz graphs \mathbb{N}$, admit corona decompositions by with smaller constants. complements results were known previously first $\mathbb{H}^1$. The main difference this case arises from fact for k<n$, complementary vertical subgroups are not commutative.