论文标题
海森堡组中低维固有Lipschitz图的延伸和电晕分解
Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups
论文作者
论文摘要
本说明涉及在Heisenberg Group $ \ Mathbb {H}^n $，$ n \ in \ Mathbb {n} $中，在Franchi，Serapioni和Serra Cassano的意义上涉及低维的Lipschitz图。对于$ 1 \ leq k \ leq n $，我们表明，在$ k $ - 维度的水平水平亚组$ \ mathbb {v} $ $ \ mathbb { $ l'$仅取决于$ l $，$ k $和$ n $。我们进一步证明，$ 1 $ - 维固有的$ 1 $ -Lipschitz图形在$ \ mathbb {h}^n $，$ n \ in \ mathbb {n} $中，允许固有的lipschitz图形与较小的LipsChitz Comantants的固有lipschitz图形。这补充了以前仅在Heisenberg Group $ \ Mathbb {H}^1 $中知道的结果。与这种情况的主要区别在于以下事实，即$ 1 \ leq k <n $，$ k $ dimensional水平亚组的互补垂直亚组中的$ \ m \ mathbb {h}^n $不是交换的。
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.
