Lipschitz extensions into $p$-Banach spaces, and canonical embeddings of Lipschitz-free $p$-spaces for $0<p<1$

Abstract

We show that inclusions of $p$-metric spaces always produce genuine linear embeddings at the level of Lipschitz-free $p$-spaces. More precisely, for every $0<p<1$ and every inclusion $N\subset M$ of $p$-metric spaces, the canonical map from $F_p(N)$ into $F_p(M)$ is always an isomorphic embedding, as it plainly happens for $p=1$. Our proof relies on a versatile extension procedure for $p$-Banach-valued Lipschitz maps, allowing us to control the geometry of canonical molecules and uncover a rigidity principle governing the structure of Lipschitz free $p$-spaces. As an application, we prove that, given $0<p<q\le 1$, the natural envelope map from the Lipschitz-free $p$-space $F_p(M)$ to its $q$-Banach envelope $F_q(M)$ is one-to-one. These results give positive answers to two foundational questions that were originally raised by Kalton in [Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335], and provide tools for furthering the understanding of subspace structures, hereditary properties, and geometric invariants in Lipschitz-free $p$-spaces.