Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
复制标题
通过定量拓扑寻找一般空间上的曲线，并应用于索博列夫和庞加莱不等式
DOI：
--
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
S. Semmes
中科院分区：
文献类型：
--
作者：
S. Semmes
在许多度量空间中，人们可以用一条有限长的曲线来连接任意一对点，但在欧几里得空间中，人们可以用许多可求长的曲线来连接一对点，这些曲线在一个区域上分布均匀。本文给出了度量空间中相似曲线族的几何判据。我们将找到这些曲线，首先解决一个“对偶”问题，从我们的度量空间到一个具有良好拓扑性质的球面建立Lipschitz映射。这些曲线族可以用来控制函数的梯度值（在一般度量空间中适当解释），并导出Sobolev和Poincaré不等式。
In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that are well distributed across a region. In the present paper we give geometric criteria on a metric space under which we can find similar families of curves. We shall find these curves by first solving a “dual” problem of building Lipschitz maps from our metric space into a sphere with good topological properties. These families of curves can be used to control the values of a function in terms of its gradient (suitably interpreted on a general metric space), and to derive Sobolev and Poincaré inequalities.