AI 中文总结
该研究构造了$c_0$到可分对偶巴拿赫空间的粗李普希茨嵌入，证明最优失真为2，且利用Kalton环形分解实现了任意小的失真逼近。
AI 中文摘要
设$G=\mathbb{Z}^{<\omega}\subset c_0$，且$G_R=G\cap R B_{c_0}$（$R\in\mathbb{N}$），其中度量$d$继承自$c_0$。在每个$G_R$上，我们构造一个交换收缩族，将其收缩到$G_R$某特殊序的有限初始段，李普希茨常数至多为2。这得到李普希茨自由空间$\mathcal{F}(G_R)$的有界完备基，其基常数至多为2，且与$\ell_1$的单位向量基$2R$等价。因此该空间与一个对偶空间是2同构的。常数2是最优的：半径为2的网格$G_2$无法以严格小于2的失真嵌入可分对偶巴拿赫空间。利用Kalton的环形分解，我们随后将$\mathcal{F}(G)$嵌入一个固定的可分对偶空间，对任意$\varepsilon>0$，失真至多为$2(1+\varepsilon)$。由于$G$是$c_0$中的整数网，这给出$c_0$到可分对偶空间的粗李普希茨嵌入。以所有可分对偶目标的下确界衡量，最优粗李普希茨失真等于2。
英文摘要
We prove that $c_0$ admits a coarse Lipschitz embedding into a separable dual Banach space and that the optimal coarse Lipschitz distortion is equal to two. Let $$ G=\mathbb Z^{<ω}\subset c_0, \qquad G_R=G\cap R B_{c_0}, \quad R\in\mathbb N, $$ with the metric inherited from $c_0$. On each $G_R$ we construct a commuting family of retractions onto finite initial segments of a special ordering of $G_R$, with Lipschitz constant at most two. Associated to these retractions there is a boundedly complete Schauder basis of $\mathcal F(G_R)$ whose basis constant is at most two and which is $2R$-equivalent to the unit vector basis of $\ell_1$. Consequently, each $\mathcal F(G_R)$ is $2$-isomorphic to a separable dual Banach space, uniformly in $R$. Kalton's annular decomposition then gives an embedding of $\mathcal F(G)$ into a separable dual space with distortion at most $2(1+\varepsilon)$ for every $\varepsilon>0$. A decomposition result of Aliaga and Medina further shows that \[ \mathcal F(G) \cong \Big( \bigoplus_{n\geq0}\mathcal F(G_{2^n}) \Big)_{\ell_1}, \] and hence $\mathcal F(G)$ itself is isomorphic to a separable dual Banach space. The constant two is sharp: if $G_2$ embeds into $X^*$ with distortion strictly smaller than two, then $X$ contains an isomorphic copy of $\ell_1$. It follows that the infimum of the coarse Lipschitz distortions of embeddings of $c_0$ into separable dual Banach spaces is exactly two.
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