Intrinsic Lipschitz Graphs and Vertical β-Numbers in the Heisenberg Group
Intrinsic Lipschitz Graphs and Vertical β-Numbers in the Heisenberg Group
复制标题
海森堡群中的本征 Lipschitz 图和垂直 β 数
DOI：
--
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发表时间：
2019
影响因子：
1.7
通讯作者：
Tuomas Orponen
中科院分区：
文献类型：
--
作者：
Vasileios Chousionis;Katrin Fässler;Tuomas Orponen
abstract:The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $Bbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 1990s. The theory in $Bbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in ${Bbb R}^3$.Our main object of study are the {it intrinsic Lipschitz graphs} in $Bbb{H}$, introduced by B. Franchi, R. Serapioni, and F. Serra Cassano in 2006. We claim that these $3$-dimensional sets in $Bbb{H}$, if any, deserve to be called quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a {it weak geometric lemma} with respect to {it vertical $eta$-numbers}. Conversely, extending a result of David and Semmes from ${Bbb R}^n$, we prove that a $3$-Ahlfors-David regular subset in $Bbb{H}$, which satisfies the weak geometric lemma and has {it big vertical projections}, necessarily has {it big pieces of intrinsic Lipschitz graphs}.
abstract:The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $Bbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 1990s. The theory in $Bbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in ${Bbb R}^3$.Our main object of study are the {it intrinsic Lipschitz graphs} in $Bbb{H}$, introduced by B. Franchi, R. Serapioni, and F. Serra Cassano in 2006. We claim that these $3$-dimensional sets in $Bbb{H}$, if any, deserve to be called quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a {it weak geometric lemma} with respect to {it vertical $eta$-numbers}. Conversely, extending a result of David and Semmes from ${Bbb R}^n$, we prove that a $3$-Ahlfors-David regular subset in $Bbb{H}$, which satisfies the weak geometric lemma and has {it big vertical projections}, necessarily has {it big pieces of intrinsic Lipschitz graphs}.