论文标题
Lipschitz操作员的注入性
Injectivity of Lipschitz operators
论文作者
论文摘要
Any Lipschitz map $f\colon M \to N$ between metric spaces can be "linearised" in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$.本说明的目的是探索$ f $的注射率与$ \ wideHat {f} $的注入性之间的连接。虽然很明显，如果$ \ widehat {f} $是注入的，那么$ f $也是如此，相反的情况就不太清楚。的确，我们确定了某些含义不存在的情况，但我们还证明，对于某些类别的度量空间$ m $，任何注射器Lipschitz Map $ f \ colon m \ colon m \ to n $（对于任何$ n $）都可以承认注射性线性化。在我们的途中，我们研究Lipschitz地图如何在自由空间中携带元素的支持，并且还为$ f $提供了更强的条件，这些条件确保了$ \ widehat {f} $是注入的。
Any Lipschitz map $f\colon M \to N$ between metric spaces can be "linearised" in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $\widehat{f}$. While it is obvious that if $\widehat{f}$ is injective then so is $f$, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces $M$, any injective Lipschitz map $f\colon M \to N$ (for any $N$) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on $f$ which ensure that $\widehat{f}$ is injective.
