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

TL;DR

This paper investigates the structure of Lipschitz free p-spaces over Euclidean spaces for 0<p≤1, establishing the existence of Schauder bases and clarifying their isomorphic properties, thus extending and refining prior results in the field.

Contribution

It proves that Lipschitz free p-spaces over Euclidean spaces admit Schauder bases for all 0<p≤1 and provides explicit formulas, while also showing these spaces are not isomorphic to ℓ_p when p<1.

Findings

$_p( r^d)$ admits a Schauder basis for all $0<p extless 1$

Explicit formulas for bases of $_p( r^d)$ and $_p([0,1]^d)$ are provided

$_p(z)$ is not isomorphic to $ell_p$ for $0<p<1$

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. H\'ajek and E. Perneck\'a, 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. C\'uth, 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 $0<p<1$.