Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups
Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups
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DOI：
10.1007/s00208-002-0334-4
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发表时间：
2002-09-01
影响因子：
1.4
通讯作者：
Tyson, JT
中科院分区：
文献类型：
--
作者：
Balogh, ZM;Holopainen, I;Tyson, JT
在齐次维数为Q的任意Carnot（幂零层李群）群G中，Q-Laplace方程的绿色函数u存在且唯一.我们证明了存在一个常数gamma = gamma（G），使得N = e（-gammau）是G中的齐次范数.然后，可以计算球形环域的极值长度（相对于N测量），并用于估计相对于Carnot-Caratheodory度量测量的环域的极值长度。应用包括拟共形映射的正则性和双Lipschitz映射的几何特征。
In any Carnot (nilpotent stratified Lie) group G of homogeneous dimension Q, Green's function u for the Q-Laplace equation exists and is unique. We prove that there exists a constant gamma = gamma(G) so that N = e(-gammau) is a homogeneous norm in G. Then the extremal lengths of spherical ring domains (measured with respect to N) can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the Carnot-Caratheodory metric. Applications include regularity properties of quasiconformal mappings and a geometric characterization of bi-Lipschitz mappings.