Nagata Dimension and Lipschitz Extensions Into Quasi-Banach Spaces

Abstract

Given two metric spaces $\mathcal N \subseteq \mathcal M$ in inclusion and $0<p\leq 1$, we wish to determine the smallest constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ such that any Lipschitz map $f: \mathcal N \to Z$ into any $p$-Banach space $Z$ can be extended to a Lipschitz map $f' : \mathcal M \to Z$ satisfying $\operatorname{Lip} f' \leq \mathfrak{t}_p (\mathcal N, \mathcal M)\cdot \operatorname{Lip} f$. In this article, we prove that if $\mathcal N$ has finite Nagata dimension at most $d$ with constant $γ$, then $\mathfrak{t}_p (\mathcal N, \mathcal M) \lesssim_p γ\cdot (d+1)^{1/p -1} \cdot \log (d+2)$ for all $0<p\leq 1$. We show that examples of spaces with finite Nagata dimension include doubling spaces, as well as minor-excluded metric graphs. We also establish that the constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ generally increases as $p$ approaches zero.

Theorems & Definitions (45)

• Definition 1
• Definition 2
• Theorem I: cf. \ref{['thm:t_p_counterexample']}
• Theorem 3: Johnson1986
• Definition 4
• Theorem 5: Lee2004
• Theorem 6: Albiac2021sums
• Theorem II: for doubling spaces; cf. \ref{['cor:ae_p_doubling']}
• Definition 8
• Theorem II: for spaces with finite Nagata dimension; cf. \ref{['thm:ae_p_nagata']}