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, on one hand 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 $p=1$ achieved in [P. Hájek and E. Pernecká, On Schauder bases in Lipschitz-free spaces, J. Math. Anal. Appl. 416 (2014), no. 2, 629--646] and answering in the positive a question that was raised in [F. Albiac, J. L. Ansorena, M. Cúth, and M. Doucha, Embeddability of lp and bases in Lipschitz free $p$-spaces for $0<p\le 1$, J. Funct. Anal. 278 (2020), no. 4, 108354, 33]. Explicit formulas for the bases of both ${\mathcal{F}}_p({\mathbb{R}}^d)$ and its isomorphic space ${\mathcal{F}}_p([0,1{]}^d)$ are given. On the other hand we 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 $0<p<1$.

References in corpus (2)

• Lipschitz free spaces isomorphic to their infinite sums and geometric applications
• Approximation properties in Lipschitz-free spaces over groups

Cited by in corpus (1)

• Tilings of the Hyperbolic Space and Lipschitz Functions