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 \(1\leqslant k\leqslant n\), 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 \(1\leqslant k<n\), the complementary vertical subgroups of k-dimensional horizontal subgroups in \({\mathbb {H}}^n\) are not commutative.

本注释涉及海森堡群中的 Franchi、Serapioni 和 Serra Cassano 意义上的低维内在 Lipschitz 图\({\mathbb {H}}^n\) , \(n\in {\mathbb {N }}\)。对于\（1 \ leqslantķ\ leqslantÑ\） ，我们表明，每固有大号-Lipschitz图形上的一个子集ķ维水平子组\（{\ mathbb {V}} \）的\（{\ mathbb { H}}^n\)可以扩展到整个子群\({\mathbb {V}}\)上的内在\(L'\) -Lipschitz 图，其中\(L'\)仅取决于L，k , 和n. 我们进一步证明\({\mathbb {H}}^n\) , \(n\in {\mathbb {N}}\) 中的一维内在 1-Lipschitz 图，通过内在 Lipschitz 图承认电晕分解较小的 Lipschitz 常数。这补充了先前仅在第一个海森堡群\({\mathbb {H}}^1\)中已知的结果。这种情况的主要区别在于，对于\(1\leqslant k<n\)，\({\mathbb {H}}^n\)中k维水平子群的互补垂直子群不是可交换的.