论文标题
叶状电晕分解
Foliated corona decompositions
论文作者
论文摘要
我们证明，$ l_4 $ norm的垂直周长的垂直周长$ 3 $尺寸的海森伯格集团$ \ mathbb {h} $最多是该子集的（海森伯格）多个的通用常数倍数。我们表明，这种等速度类型不平等是最佳的，因为在某些集合中，它无法用$ l_4 $ norm替换为任何$ q <4 $的$ l_q $ norm。这与$ 5 $维的设置相反，上面的结果与$ L_4 $ NORM替换为$ L_2 $ NORM。 上述等级不平等的证明引入了一种新的结构方法，用于理解$ \ mathbb {h} $中表面的几何形状。在先前的工作（2017年）中，我们展示了如何将AHLFORS定期表面的层次分解成近似固有的Lipschitz图。在这里，我们证明任何这样的图形都承认了叶状的电晕分解，该分解是一个嵌套的隔板家族，分成几个与统治表面接近的碎片。 除了这些结果的固有几何和分析意义外，Cheeger-Kleiner-Naor（2009）和Lafforgue-Naor（2012）提出的问题还具有几个值得注意的含义，包括$ l_1 $在Radius $ n \ ge 2 $ 3 $ -dimendementions the Insendementions the Insensimensions the Insensimensionge yesenge yisen nisenge yisensen nisensen nisensen nisenimensen the Is的事实中$ \ sqrt [4] {\ log n} $;这与高维的海森伯格组相反，在该组中，我们以前的工作表明，radius $ n \ ge 2 $的单词表的失真为$ \ sqrt {\ log n} $。
We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$.
