分解空间的一种大尺度方法
A Large Scale Approach to Decomposition Spaces
摘要：
分解空间是一类由集合的良好覆盖和单位分割构造的函数空间。集合的覆盖结构决定了分解空间的性质。Besov空间，剪切波空间和调制空间是众所周知的分解空间。在本文中，我们专注于几何方面的分解空间，并利用这些自然被捕获的大规模属性的度量空间，覆盖空间，与覆盖的集合。我们证明了由拟等距覆盖空间构造的分解空间具有许多共同的几何特征。几何嵌入的概念被引入到形式化的方式，一个分解空间可以嵌入到另一个分解空间，同时尊重覆盖的几何特征。分解空间的大尺度方法的一些结果是（i）不同集合的覆盖的比较，（ii）基于覆盖的几何特征和对称性的分解空间的嵌入的研究，以及（iii）使用大尺度几何的概念，如渐近维数或双曲性，来研究分解空间的性质。
英文摘要：
Decomposition spaces are a class of function spaces constructed out of well-behaved coverings and partitions of unity of a set. The structure of the covering of the set determines the properties of the decomposition space. Besov spaces, shearlet spaces and modulation spaces are well-known decomposition spaces. In this paper we focus on the geometric aspects of decomposition spaces and utilize that these are naturally captured by the large scale properties of a metric space, the covered space, associated to a covering of a set. We demonstrate that decomposition spaces constructed out of quasi-isometric covered spaces have many geometric features in common. The notion of geometric embedding is introduced to formalize the way one decomposition space can be embedded into another decomposition space while respecting the geometric features of the coverings. Some consequences of the large scale approach to decomposition spaces are (i) comparison of coverings of different sets, (ii) study of embeddings of decomposition spaces based on the geometric features and the symmetries of the coverings and (iii) the use of notions from large scale geometry, such as asymptotic dimension or hyperbolicity, to study the properties of decomposition spaces.
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