Structure of the Lipschitz free p-spaces ${\mathcal{F}}_p({\mathbb{Z}}^d)$ and ${\mathcal{F}}_p({\mathbb{R}}^d)$ for $0<p\le 1$

Abstract

Our aim in this article is to contribute to the theory of Lipschitz free p-spaces for $0<p\le 1$ over the Euclidean spaces ${\mathbb{R}}^d$ and ${\mathbb{Z}}^d$. To that end, we show that ${\mathcal{F}}_p({\mathbb{R}}^d)$ admits a Schauder basis for every $p\in (0,1]$, thus generalizing the corresponding result for the case $p=1$ by Hájek and Pernecká (J Math Anal Appl 416(2):629–646, 2014, Theorem 3.1) and answering in the positive a question that was raised by Albiac et al. in (J Funct Anal 278(4):108354, 2020). Explicit formulas for the bases of ${\mathcal{F}}_p({\mathbb{R}}^d)$ and its isomorphic space ${\mathcal{F}}_p([0,1]^d)$ are given. We also show that the well-known fact that ${\mathcal{F}}({\mathbb{Z}})$ is isomorphic to $\ell _{1}$ does not extend to the case when $p<1$, that is, ${\mathcal{F}}_{p}({\mathbb{Z}})$ is not isomorphic to $\ell _{p}$ when $0<p<1$.