Abstract:We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct $\ell _1$-sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over balls and spheres of the same finite dimensions are isomorphic, that the Lipschitz-free space over $\mathbb {Z}^d$ is isomorphic to its $\ell _1$-sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to $\ell _1$. Moreover, following new ideas of Bruè et al. from [J. Funct. Anal. 280 (2021), pp. 108868, 21] we provide an elementary self-contained proof that Lipschitz-free spaces over doubling metric spaces are complemented in Lipschitz-free spaces over their superspaces and they have BAP. Everything, including the results about doubling metric spaces, is explored in the more comprehensive setting of $p$-Banach spaces, which allows us to appreciate the similarities and differences of the theory between the cases $p<1$ and $p=1$.

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中文翻译：

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Lipschitz 自由空间同构于它们的无穷和和几何应用

摘要：我们发现了在度量空间上的 Lipschitz-free 空间与其无限直接 $\ell_1$-sum 同构的一般条件，并展示了几种应用。作为此类应用的示例，我们有相同有限维度的球和球体上的 Lipschitz-free 空间是同构的，$\mathbb {Z}^d$ 上的 Lipschitz-free 空间与其 $\ell _1$- 同构总和，或者倍增度量空间的任何雪花上的 Lipschitz-free 空间与 $\ell_1$ 同构。此外，遵循 Bruè 等人的新想法。来自 [J. 功能。肛门。280 (2021), pp. 108868, 21] 我们提供了一个基本的自包含证明，证明加倍度量空间上的 Lipschitz-free 空间在其超空间上的 Lipschitz-free 空间中得到补充，并且它们具有 BAP。一切，包括关于度量空间加倍的结果，

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