发表机构
Public University of Navarre; Universidad de La Rioja; Charles University（纳瓦拉公立大学; 拉里奥哈大学; 查理大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Figiel定理在非局部凸空间（$0<p<1$）中完全失效，构造了无线性左逆的等距嵌入，并利用Lipschitz-free $p$-空间及定量估计，得出等距$p$-提升性质仅在一维空间成立。
AI 中文摘要
Figiel的一个经典定理断言：实巴拿赫空间之间的每个等距嵌入$\varphi\colon X\to Y$，满足$\varphi(0)=0$时，在$\varphi(X)$的闭线性张成上存在一个线性压缩左逆。我们证明，一旦失去局部凸性，这一现象完全崩溃。更精确地，对于每个$0<p<1$和每个非零实巴拿赫空间$X$，我们构造一个$p$-巴拿赫空间$Y$和一个等距嵌入$\varphi\colon X\to Y$，使得$\varphi(0)=0$且$\varphi(X)$的闭线性张成是整个空间$Y$，但在$\varphi(X)$的线性张成上不存在任何线性映射（无论有界与否）作为$\varphi$的左逆。该构造利用Lipschitz-free空间的$p$-巴拿赫对应物完成。事实上，$Y$可以选择为与$X$上的Lipschitz-free $p$-空间同构。主要成分是一个定量估计，控制Lipschitz-free $p$-空间中到基本分子的距离，这是$0<p<1$范围内特有的现象。作为进一步推论，我们证明一个实巴拿赫空间具有等距$p$-提升性质当且仅当其维数至多为1。
英文摘要
A classical theorem of Figiel asserts that every isometric embedding $φ\colon X\to Y$ between real Banach spaces, with $φ(0)=0$, admits a linear contractive left inverse on the closed linear span of $φ(X)$. We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every $0<p<1$ and every nonzero real Banach space $X$, we construct a $p$-Banach space $Y$ and an isometric embedding $φ\colon X\to Y$ such that $φ(0)=0$ and the closed linear span of $φ(X)$ is the entire space $Y$, but there is no linear map, bounded or not, on the linear span of $φ(X)$ which is a left inverse of $φ$. The construction is carried out using the $p$-Banach counterpart of Lipschitz-free spaces. In fact, $Y$ may be chosen isomorphic to the Lipschitz-free $p$-space over $X$. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free $p$-spaces, a phenomenon specific to the range $0<p<1$. As a further consequence, we show that a real Banach space has the isometric $p$-lifting property if and only if its dimension is at most one.
