论文标题
海森堡集团中的耐寒空间和准文化地图
Hardy spaces and quasiconformal maps in the Heisenberg group
论文作者
论文摘要
我们定义Hardy Spaces $ H^p $，$ 0 <P <\ infty $，用于Korányi单位球上的Quasiconformal映射$ b $在第一个Heisenberg Group $ \ Mathbb {H}^1 $中。我们的定义是根据Korányi和Reimann以及Balogh和Tyson引入的Heisenberg Polar坐标所陈述的。首先，我们证明存在$ p_0（k）> 0 $，因此每个$ k $ -quasiconformal映射$ f：b \ to f（b）\ subset \ mathbb {h}^1 $均属于$ h^p $，均为$ h^p $，全部$ 0 <p <p_0（k）$。其次，我们为$ H^p $的$ H^p $成员提供了两个等效条件，其中一个是$ f $的径向限制，一种使用$ f $的非义务最大函数。作为应用程序，我们通过$ b $及其径向限制的准信息映射的积分不等式来表征$ b $的Carleson措施。因此，我们的论文将Astala和Koskela，Jerison和Weitsman，Nolder和Zinsmeister扩展到$ \ Mathbb {R}^n $到$ \ Mathbb {H}^1 $。 $ \ Mathbb {r}^n $和$ \ mathbb {h}^1 $中的证明之间的关键差异是由Korányi单位球形的非异分性质引起的，其两个特征点。
We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Korányi unit ball $B$ in the first Heisenberg group $\mathbb{H}^1$. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of $p_0(K)>0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential maximal function of $f$. As an application, we characterize Carleson measures on $B$ via integral inequalities for quasiconformal mappings on $B$ and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from $\mathbb{R}^n$ to $\mathbb{H}^1$. A crucial difference between the proofs in $\mathbb{R}^n$ and $\mathbb{H}^1$ is caused by the nonisotropic nature of the Korányi unit sphere with its two characteristic points.
