Lipschitz extensions into $p$-Banach spaces, and canonical embeddings of Lipschitz-free $p$-spaces for $0<p<1$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915898850279424 |
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| author | Albiac, Fernando Ansorena, José L. |
| author_facet | Albiac, Fernando Ansorena, José L. |
| contents | 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 $ \mathit{N}\subset \mathit{M}$ of $p$-metric spaces, the canonical map from $ \mathit{F}_p(\mathit{N})$ into $ \mathit{F}_p( \mathit{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 $ \mathit{F}_p( \mathit{M})$ to its $q$-Banach envelope $ \mathit{F}_q( \mathit{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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28193 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lipschitz extensions into $p$-Banach spaces, and canonical embeddings of Lipschitz-free $p$-spaces for $0<p<1$ Albiac, Fernando Ansorena, José L. Functional Analysis 46B20 (Primary) 46B03, 46B07, 46A35, 46A16 (Secondary) 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 $ \mathit{N}\subset \mathit{M}$ of $p$-metric spaces, the canonical map from $ \mathit{F}_p(\mathit{N})$ into $ \mathit{F}_p( \mathit{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 $ \mathit{F}_p( \mathit{M})$ to its $q$-Banach envelope $ \mathit{F}_q( \mathit{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. |
| title | Lipschitz extensions into $p$-Banach spaces, and canonical embeddings of Lipschitz-free $p$-spaces for $0<p<1$ |
| topic | Functional Analysis 46B20 (Primary) 46B03, 46B07, 46A35, 46A16 (Secondary) |
| url | https://arxiv.org/abs/2603.28193 |