论文标题
Lipschitz映射到度量空间的定量分解
Quantitative decompositions of Lipschitz mappings into metric spaces
论文作者
论文摘要
我们研究了Lipschitz映射从欧几里得空间到公制空间的定量特性。我们证明，始终有可能将这种映射的域分解为绘制的片段，并在其上映射“类似于投影映射”，以及“垃圾集”，从适当的意义上讲是任意小的。此外，我们的控制是定量的，即独立于特定的映射及其映射到的度量空间。这改善了纸上的“硬撒”定理，并回答了该论文中留下的问题。该证明使用定量分化的思想，以及如何通过其他坐标来补充Lipschitz映射以形成Bi-Lipschitz映射的详细研究。
We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which the mapping "behaves like a projection mapping" along with a "garbage set" that is arbitrarily small in an appropriate sense. Moreover, our control is quantitative, i.e., independent of both the particular mapping and the metric space it maps into. This improves a theorem of Azzam-Schul from the paper "Hard Sard", and answers a question left open in that paper. The proof uses ideas of quantitative differentiation, as well as a detailed study of how to supplement Lipschitz mappings by additional coordinates to form bi-Lipschitz mappings.
