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

Fernando Albiac, José L. Ansorena

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 $ \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.

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

Abstract

We show that inclusions of -metric spaces always produce genuine linear embeddings at the level of Lipschitz-free -spaces. More precisely, for every and every inclusion of -metric spaces, the canonical map from into is always an isomorphic embedding, as it plainly happens for . Our proof relies on a versatile extension procedure for -Banach-valued Lipschitz maps, allowing us to control the geometry of canonical molecules and uncover a rigidity principle governing the structure of Lipschitz free -spaces. As an application, we prove that, given , the natural envelope map from the Lipschitz-free -space to its -Banach envelope 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 -spaces.

Paper Structure

This paper contains 7 sections, 13 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Lipschitz extensions of mappings into -Banach spaces
  3. Canonical embeddings of Lipschitz-free -spaces
  4. Faithfulness of the scale of Lipschitz-free spaces over quasi-metric spaces
  5. -spaces
  6. Finite-dimensional decompositions
  7. Lipschitz-free spaces over vs. Lipschitz-free spaces over ,

Key Result

Lemma 2.2

Let $\mathcal{N}$ be a nonempty closed set of a metric space $(\mathcal{M},\rho)$. Suppose that $V:=\mathcal{M}\setminus\mathcal{N}$, $\gamma\in[1,\infty)$, $\beta\in(0,\infty)$ and $\kappa\in\mathbb{N}$, $\kappa\ge 2$, satisfy conditions it:W1, it:W2 and it:W3 in the definition of a Whitney cover. In fact, we can choose $\mu= 2 e \log(2) \gamma \log(2\kappa)$ and any $\nu>2+\beta$.

Theorems & Definitions (25)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 3.1: AACD2020*Proposition 4.1 and Theorem 4.10
  • Lemma 3.2: CuthRaunig2024*Theorem 4.16
  • Theorem 3.3: LangSch2005*Proposition 3.2
  • ...and 15 more