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

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