Nagata Dimension and Lipschitz Extensions Into Quasi-Banach Spaces

Given two metric spaces $\mathcal{N}\subseteq \mathcal{M}$ in inclusion and $0<p\le 1$, 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 $f^{\mathrm{\prime }}:\mathcal{M}\to Z$ satisfying $\mathrm{Lip}{f}^{\mathrm{\prime }}\le {\mathfrak{t}}_p(\mathcal{N},\mathcal{M})\cdot \mathrm{Lip}f$. In this article, we prove that if $\mathcal{N}$ has finite Nagata dimension at most $d$ with constant $\gamma$, then ${\mathfrak{t}}_p(\mathcal{N},\mathcal{M}){\lesssim }_p\gamma \cdot (d+1{)}^{1\mathrm{/}p-1}\cdot \log (d+2)$ for all $0<p\le 1$. 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.