We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton’s interlaced graphs ${({[\mathbb{N}]}^k,d_{𝕂})}_k$ into dual spaces. Notably, we define and study a modification of Kalton’s property ${𝒬}$ that we call property ${{𝒬}}_p$ (with $p\in (1,+\infty ]$). We show that if ${({[\mathbb{N}]}^k,d_{𝕂})}_k$ equi-coarse Lipschitzly embeds into $X^{*}$, then the Szlenk index of $X$ is greater than $\omega$, and that this is optimal, i.e. there exists a separable dual space $Y^{*}$ that contains ${({[\mathbb{N}]}^k,d_{𝕂})}_k$ equi-Lipschitzly and so that $Y$ has Szlenk index ${\omega }^2$. We prove that $c_0$ does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than $\frac{3}{2}$. We also show that neither $c_0$ nor $L_1$ coarsely embeds into a separable dual by a weak-to-weak${*}^{}$ sequentially continuous map.

我们研究了 Banach 空间的非线性可嵌入性和 Kalton 交错图族的等可嵌入性${({[\mathbb{N}]}^k,d_{𝕂})}_k$进入双重空间。值得注意的是，我们定义并研究了 Kalton 属性的修改${𝒬}$我们称之为财产${{𝒬}}_p$（和$p\in (\mathrm{1个},+\infty ]$). 我们证明如果${({[\mathbb{N}]}^k,d_{𝕂})}_k$等粗 Lipschitzly 嵌入到$X^{*}$, 然后是 Szlenk 指数$X$大于$\omega$，并且这是最优的，即存在一个可分离的对偶空间${是}^{*}$包含${({[\mathbb{N}]}^k,d_{𝕂})}_k$equi-Lipschitzly 所以$是$有 Szlenk 指数${\omega }^{\mathrm{2个}}$. 我们证明$C_0$不粗糙 Lipschitzly 嵌入到一个可分离的对偶空间中，映射的失真严格小于$\frac{\mathrm{3个}}{\mathrm{2个}}$. 我们还表明，无论是$C_0$也不${\mathrm{大号}}_{\mathrm{1个}}$通过弱到弱粗略地嵌入到一个可分离的对偶中${*}^{}$顺序连续映射。