Nagata dimension and Lipschitz extensions into quasi-Banach spaces
Abstract
Given two metric spaces $\mathcal N \subseteq \mathcal M$ and $0 \lt p\leq 1$, 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 $\mathrm{Lip}\,f’ \leq \mathfrak {t}_p (\mathcal N, \mathcal M)\cdot \mathrm{Lip}\, f$. We prove that if $\mathcal N$ has finite Nagata dimension at most $d$ with constant $\gamma $, then $\mathfrak {t}_p (\mathcal N, \mathcal M) \lesssim _p \gamma \cdot (d+1)^{1/p -1} \cdot \log (d+2)$ for all $0 \lt p\leq 1$. 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.
