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

TL;DR

This paper studies low-dimensional intrinsic Lipschitz graphs in Heisenberg groups, proving extension and corona decomposition results that generalize previous work from the first Heisenberg group to higher dimensions.

Contribution

It establishes extension theorems for intrinsic Lipschitz graphs over horizontal subgroups and proves corona decompositions for 1-dimensional graphs in higher Heisenberg groups.

Findings

Extension of intrinsic Lipschitz graphs over entire subgroups

Corona decompositions for 1-dimensional intrinsic Lipschitz graphs

Results extend known cases from $ ext{H}^1$ to higher-dimensional 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 $1\leq k\leq n$, 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 $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ 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 $1\leq k<n$, the complementary vertical subgroups of $k$-dimensional horizontal subgroups in $\mathbb{H}^n$ are not commutative.