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

Abstract

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

30 pages; v2: minor revision, results unchanged