# Embeddability of $\ell_{p}$ and bases in Lipschitz free $p$-spaces for $0<p\leq 1$

## TL;DR

This paper investigates the geometric structure of Lipschitz free $p$-spaces for $0<p extless 1$, demonstrating embeddings of $ ext{ell}_p$, exploring basis properties, and providing new examples of $p$-Banach spaces with bases but no unconditional bases.

## Contribution

It develops new techniques to show $ ext{ell}_p$ embeds in Lipschitz free $p$-spaces for $0<p<1$, and constructs the first examples of $p$-Banach spaces with bases lacking unconditional bases.

## Key findings

- $ ext{ell}_p$ embeds isomorphically in $ ext{Lip}_0$ free $p$-spaces for $0<p<1$
- Embeddings may not be complemented without restrictions
- Constructed $p$-Banach spaces with bases but no unconditional bases

## Abstract

Our goal in this paper is to continue the study initiated by the authors in [Lipschitz free $p$-spaces for $0<p<1$; arXiv:1811.01265 [math.FA]] of the geometry of the Lipschitz free $p$-spaces over quasimetric spaces for $0<p\le1$, denoted $\mathcal F_{p}(\mathcal M)$. Here we develop new techniques to show that, by analogy with the case $p=1$, the space $\ell_{p}$ embeds isomorphically in $\mathcal F_{p}(\mathcal M)$ for $0<p<1$. Going further we see that despite the fact that, unlike the case $p=1$, this embedding need not be complemented in general, complementability of $\ell_{p}$ in a Lipschitz free $p$-space can still be attained by imposing certain natural restrictions to $\mathcal M$. As a by-product of our discussion on basis in $\mathcal F_{p}([0,1])$, we obtain the first-known examples of $p$-Banach spaces for $p<1$ that possess a basis but fail to have an unconditional basis.