Lipschitz free spaces isomorphic to their infinite sums and geometric applications

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 from [E. Bruè, S. Di Marino and F. Stra, Linear Lipschitz and $C^1$ extension operators through random projection, arXiv:1801.07533] 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 $p=1$.

References in corpus (2)

• Free spaces over some proper metric spaces
• Gâteaux differentiability on infinite-dimensional Carnot groups

Cited by in corpus (10)

• Approximation properties in Lipschitz-free spaces over groups
• Delta-points and their implications for the geometry of Banach spaces
• Structure of the Lipschitz free $p$-spaces ${\mathcal{F}}_p({\mathbb{Z}}^d)$ and ${\mathcal{F}}_p({\mathbb{R}}^d)$ for $0<p\le 1$
• Canonical embedding of Lipschitz-free $p$-spaces
• Hyperbolic Metric Spaces and Stochastic Embeddings
• Projections in Lipschitz-free spaces induced by group actions
• Pełczyński's property (V${}^{\ast }$) in Lipschitz-free spaces
• Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces
• Tilings of the Hyperbolic Space and Lipschitz Functions
• Lipschitz spaces over non-porous sets